Elliptic curves over the rational numbers

AUTHORS:

  • William Stein (2005): first version
  • William Stein (2006-02-26): fixed Lseries_extended which didn’t work because of changes elsewhere in Sage.
  • David Harvey (2006-09): Added padic_E2, padic_sigma, padic_height, padic_regulator methods.
  • David Harvey (2007-02): reworked padic-height related code
  • Christian Wuthrich (2007): added padic sha computation
  • David Roe (2007-09): moved sha, l-series and p-adic functionality to separate files.
  • John Cremona (2008-01)
  • Tobias Nagel and Michael Mardaus (2008-07): added integral_points
  • John Cremona (2008-07): further work on integral_points
  • Christian Wuthrich (2010-01): moved Galois reps and modular parametrization in a separate file
  • Simon Spicer (2013-03): Added code for modular degrees and congruence numbers of higher level
  • Simon Spicer (2014-08): Added new analytic rank computation functionality
class sage.schemes.elliptic_curves.ell_rational_field.EllipticCurve_rational_field(ainvs, **kwds)

Bases: sage.schemes.elliptic_curves.ell_number_field.EllipticCurve_number_field

Elliptic curve over the Rational Field.

INPUT:

  • ainvs – a list or tuple \([a_1, a_2, a_3, a_4, a_6]\) of Weierstrass coefficients.

Note

This class should not be called directly; use sage.constructor.EllipticCurve to construct elliptic curves.

EXAMPLES:

Construction from Weierstrass coefficients (\(a\)-invariants), long form:

sage: E = EllipticCurve([1,2,3,4,5]); E
Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field

Construction from Weierstrass coefficients (\(a\)-invariants), short form (sets \(a_1 = a_2 = a_3 = 0\)):

sage: EllipticCurve([4,5]).ainvs()
(0, 0, 0, 4, 5)

Constructor from a Cremona label:

sage: EllipticCurve('389a1')
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field

Constructor from an LMFDB label:

sage: EllipticCurve('462.f3')
Elliptic Curve defined by y^2 + x*y = x^3 - 363*x + 1305 over Rational Field
CPS_height_bound()

Return the Cremona-Prickett-Siksek height bound. This is a floating point number B such that if P is a rational point on the curve, then \(h(P) \le \hat{h}(P) + B\), where \(h(P)\) is the naive logarithmic height of \(P\) and \(\hat{h}(P)\) is the canonical height.

See also

silverman_height_bound() for a bound that also works for points over number fields.

EXAMPLES:

sage: E = EllipticCurve("11a")
sage: E.CPS_height_bound()
2.8774743273580445
sage: E = EllipticCurve("5077a")
sage: E.CPS_height_bound()
0.0
sage: E = EllipticCurve([1,2,3,4,1])
sage: E.CPS_height_bound()
Traceback (most recent call last):
...
RuntimeError: curve must be minimal.
sage: F = E.quadratic_twist(-19)
sage: F
Elliptic Curve defined by y^2 + x*y + y = x^3 - x^2 + 1376*x - 130 over Rational Field
sage: F.CPS_height_bound()
0.6555158376972852

IMPLEMENTATION:

Call the corresponding mwrank C++ library function. Note that the formula in the [CPS] paper is given for number fields. It is only the implementation in Sage that restricts to the rational field.

Lambda(s, prec)

Return the value of the Lambda-series of the elliptic curve E at s, where s can be any complex number.

IMPLEMENTATION: Fairly slow computation using the definitions and implemented in Python.

Uses prec terms of the power series.

EXAMPLES:

sage: E = EllipticCurve('389a')
sage: E.Lambda(1.4+0.5*I, 50)
-0.354172680517... + 0.874518681720...*I
Np(p)

The number of points on \(E\) modulo \(p\).

INPUT:

  • p (int) – a prime, not necessarily of good reduction.

OUTPUT:

(int) The number ofpoints on the reduction of \(E\) modulo \(p\) (including the singular point when \(p\) is a prime of bad reduction).

EXAMPLES:

sage: E = EllipticCurve([0, -1, 1, -10, -20])
sage: E.Np(2)
5
sage: E.Np(3)
5
sage: E.conductor()
11
sage: E.Np(11)
11

This even works when the prime is large:

sage: E = EllipticCurve('37a')
sage: E.Np(next_prime(10^30))
1000000000000001426441464441649
S_integral_points(S, mw_base='auto', both_signs=False, verbose=False, proof=None)

Compute all S-integral points (up to sign) on this elliptic curve.

INPUT:

  • S - list of primes
  • mw_base - list of EllipticCurvePoint generating the Mordell-Weil group of E (default: ‘auto’ - calls gens())
  • both_signs - True/False (default False): if True the output contains both P and -P, otherwise only one of each pair.
  • verbose - True/False (default False): if True, some details of the computation are output.
  • proof - True/False (default True): if True ALL S-integral points will be returned. If False, the MW basis will be computed with the proof=False flag, and also the time-consuming final call to S_integral_x_coords_with_abs_bounded_by(abs_bound) is omitted. Use this only if the computation takes too long, but be warned that then it cannot be guaranteed that all S-integral points will be found.

OUTPUT:

A sorted list of all the S-integral points on E (up to sign unless both_signs is True)

Note

The complexity increases exponentially in the rank of curve E and in the length of S. The computation time (but not the output!) depends on the Mordell-Weil basis. If mw_base is given but is not a basis for the Mordell-Weil group (modulo torsion), S-integral points which are not in the subgroup generated by the given points will almost certainly not be listed.

EXAMPLES:

A curve of rank 3 with no torsion points:

sage: E = EllipticCurve([0,0,1,-7,6])
sage: P1 = E.point((2,0))
sage: P2 = E.point((-1,3))
sage: P3 = E.point((4,6))
sage: a = E.S_integral_points(S=[2,3], mw_base=[P1,P2,P3], verbose=True);a
max_S: 3 len_S: 3 len_tors: 1
lambda 0.485997517468...
k1,k2,k3,k4 7.65200453902598e234 1.31952866480763 3.54035317966420e9 2.42767548272846e17
p= 2 : trying with p_prec =  30
mw_base_p_log_val =  [2, 2, 1]
min_psi =  2 + 2^3 + 2^6 + 2^7 + 2^8 + 2^9 + 2^11 + 2^12 + 2^13 + 2^16 + 2^17 + 2^19 + 2^20 + 2^21 + 2^23 + 2^24 + 2^28 + O(2^30)
p= 3 : trying with p_prec =  30
mw_base_p_log_val =  [1, 2, 1]
min_psi =  3 + 3^2 + 2*3^3 + 3^6 + 2*3^7 + 2*3^8 + 3^9 + 2*3^11 + 2*3^12 + 2*3^13 + 3^15 + 2*3^16 + 3^18 + 2*3^19 + 2*3^22 + 2*3^23 + 2*3^24 + 2*3^27 + 3^28 + 3^29 + O(3^30)
mw_base [(1 : -1 : 1), (2 : 0 : 1), (0 : -3 : 1)]
mw_base_log [0.667789378224099, 0.552642660712417, 0.818477222895703]
mp [5, 7]
mw_base_p_log [[2^2 + 2^3 + 2^6 + 2^7 + 2^8 + 2^9 + 2^14 + 2^15 + 2^18 + 2^19 + 2^24 + 2^29 + O(2^30), 2^2 + 2^3 + 2^5 + 2^6 + 2^9 + 2^11 + 2^12 + 2^14 + 2^15 + 2^16 + 2^18 + 2^20 + 2^22 + 2^23 + 2^26 + 2^27 + 2^29 + O(2^30), 2 + 2^3 + 2^6 + 2^7 + 2^8 + 2^9 + 2^11 + 2^12 + 2^13 + 2^16 + 2^17 + 2^19 + 2^20 + 2^21 + 2^23 + 2^24 + 2^28 + O(2^30)], [2*3^2 + 2*3^5 + 2*3^6 + 2*3^7 + 3^8 + 3^9 + 2*3^10 + 3^12 + 2*3^14 + 3^15 + 3^17 + 2*3^19 + 2*3^23 + 3^25 + 3^28 + O(3^30), 2*3 + 2*3^2 + 2*3^3 + 2*3^4 + 2*3^6 + 2*3^7 + 2*3^8 + 3^10 + 2*3^12 + 3^13 + 2*3^14 + 3^15 + 3^18 + 3^22 + 3^25 + 2*3^26 + 3^27 + 3^28 + O(3^30), 3 + 3^2 + 2*3^3 + 3^6 + 2*3^7 + 2*3^8 + 3^9 + 2*3^11 + 2*3^12 + 2*3^13 + 3^15 + 2*3^16 + 3^18 + 2*3^19 + 2*3^22 + 2*3^23 + 2*3^24 + 2*3^27 + 3^28 + 3^29 + O(3^30)]]
k5,k6,k7 0.321154513240... 1.55246328915... 0.161999172489...
initial bound 2.8057927340...e117
bound_list [58, 58, 58]
bound_list [8, 9, 9]
bound_list [9, 7, 7]
starting search of points using coefficient bound  9
x-coords of S-integral points via linear combination of mw_base and torsion:
[-3, -26/9, -8159/2916, -2759/1024, -151/64, -1343/576, -2, -7/4, -1, -47/256, 0, 1/4, 4/9, 9/16, 58/81, 7/9, 6169/6561, 1, 17/16, 2, 33/16, 172/81, 9/4, 25/9, 3, 31/9, 4, 25/4, 1793/256, 8, 625/64, 11, 14, 21, 37, 52, 6142/81, 93, 4537/36, 342, 406, 816, 207331217/4096]
starting search of extra S-integer points with absolute value bounded by 3.89321964979420
x-coords of points with bounded absolute value
[-3, -2, -1, 0, 1, 2]
Total number of S-integral points: 43
[(-3 : 0 : 1), (-26/9 : 28/27 : 1), (-8159/2916 : 233461/157464 : 1), (-2759/1024 : 60819/32768 : 1), (-151/64 : 1333/512 : 1), (-1343/576 : 36575/13824 : 1), (-2 : 3 : 1), (-7/4 : 25/8 : 1), (-1 : 3 : 1), (-47/256 : 9191/4096 : 1), (0 : 2 : 1), (1/4 : 13/8 : 1), (4/9 : 35/27 : 1), (9/16 : 69/64 : 1), (58/81 : 559/729 : 1), (7/9 : 17/27 : 1), (6169/6561 : 109871/531441 : 1), (1 : 0 : 1), (17/16 : -25/64 : 1), (2 : 0 : 1), (33/16 : 17/64 : 1), (172/81 : 350/729 : 1), (9/4 : 7/8 : 1), (25/9 : 64/27 : 1), (3 : 3 : 1), (31/9 : 116/27 : 1), (4 : 6 : 1), (25/4 : 111/8 : 1), (1793/256 : 68991/4096 : 1), (8 : 21 : 1), (625/64 : 14839/512 : 1), (11 : 35 : 1), (14 : 51 : 1), (21 : 95 : 1), (37 : 224 : 1), (52 : 374 : 1), (6142/81 : 480700/729 : 1), (93 : 896 : 1), (4537/36 : 305425/216 : 1), (342 : 6324 : 1), (406 : 8180 : 1), (816 : 23309 : 1), (207331217/4096 : 2985362173625/262144 : 1)]

It is not necessary to specify mw_base; if it is not provided, then the Mordell-Weil basis must be computed, which may take much longer.

sage: a = E.S_integral_points([2,3])
sage: len(a)
43

An example with negative discriminant:

sage: EllipticCurve('900d1').S_integral_points([17], both_signs=True)
[(-11 : -27 : 1), (-11 : 27 : 1), (-4 : -34 : 1), (-4 : 34 : 1), (4 : -18 : 1), (4 : 18 : 1), (2636/289 : -98786/4913 : 1), (2636/289 : 98786/4913 : 1), (16 : -54 : 1), (16 : 54 : 1)]

Output checked with Magma (corrected in 3 cases):

sage: [len(e.S_integral_points([2], both_signs=False)) for e in cremona_curves([11..100])] # long time (17s on sage.math, 2011)
[2, 0, 2, 3, 3, 1, 3, 1, 3, 5, 3, 5, 4, 1, 1, 2, 2, 2, 3, 1, 2, 1, 0, 1, 3, 3, 1, 1, 5, 3, 4, 2, 1, 1, 5, 3, 2, 2, 1, 1, 1, 0, 1, 3, 0, 1, 0, 1, 1, 3, 7, 1, 3, 3, 3, 1, 1, 2, 3, 1, 2, 3, 1, 2, 1, 3, 3, 1, 1, 1, 0, 1, 3, 3, 1, 1, 7, 1, 0, 1, 1, 0, 1, 2, 0, 3, 1, 2, 1, 3, 1, 2, 2, 4, 5, 3, 2, 1, 1, 6, 1, 0, 1, 3, 1, 3, 3, 1, 1, 1, 1, 1, 3, 1, 5, 1, 2, 4, 1, 1, 1, 1, 1, 0, 1, 0, 2, 2, 0, 0, 1, 0, 1, 1, 6, 1, 0, 1, 1, 0, 4, 3, 1, 2, 1, 2, 3, 1, 1, 1, 1, 8, 3, 1, 2, 1, 2, 0, 8, 2, 0, 6, 2, 3, 1, 1, 1, 3, 1, 3, 2, 1, 3, 1, 2, 1, 6, 9, 3, 3, 1, 1, 2, 3, 1, 1, 5, 5, 1, 1, 0, 1, 1, 2, 3, 1, 1, 2, 3, 1, 3, 1, 1, 1, 1, 0, 0, 1, 3, 3, 1, 3, 1, 1, 2, 2, 0, 0, 6, 1, 0, 1, 1, 1, 1, 3, 1, 2, 6, 3, 1, 2, 2, 1, 1, 1, 1, 7, 5, 4, 3, 3, 1, 1, 1, 1, 1, 1, 8, 5, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 2, 3, 6, 1, 1, 7, 3, 3, 4, 5, 9, 6, 1, 0, 7, 1, 1, 3, 1, 1, 2, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 7, 8, 2, 3, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1]

An example from [PZGH]:

sage: E = EllipticCurve([0,0,0,-172,505])
sage: E.rank(), len(E.S_integral_points([3,5,7]))  # long time (5s on sage.math, 2011)
(4, 72)

This is curve “7690e1” which failed until trac ticket #4805 was fixed:

sage: EllipticCurve([1,1,1,-301,-1821]).S_integral_points([13,2])
[(-13 : 16 : 1),
(-9 : 20 : 1),
(-7 : 4 : 1),
(21 : 30 : 1),
(23 : 52 : 1),
(63 : 452 : 1),
(71 : 548 : 1),
(87 : 756 : 1),
(2711 : 139828 : 1),
(7323 : 623052 : 1),
(17687 : 2343476 : 1)]

REFERENCES:

[PZGH]Petho A., Zimmer H.G., Gebel J. and Herrmann E., Computing all S-integral points on elliptic curves Math. Proc. Camb. Phil. Soc. (1999), 127, 383-402
  • Some parts of this implementation are partially based on the function integral_points()

AUTHORS:

  • Tobias Nagel (2008-12)
  • Michael Mardaus (2008-12)
  • John Cremona (2008-12)
abelian_variety()

Return self as a modular abelian variety.

OUTPUT:

  • a modular abelian variety

EXAMPLES:

sage: E = EllipticCurve('11a')
sage: E.abelian_variety()
Abelian variety J0(11) of dimension 1

sage: E = EllipticCurve('33a')
sage: E.abelian_variety()
Abelian subvariety of dimension 1 of J0(33)
an(n)

The n-th Fourier coefficient of the modular form corresponding to this elliptic curve, where n is a positive integer.

EXAMPLES:

sage: E = EllipticCurve('37a1')
sage: [E.an(n) for n in range(20) if n>0]
[1, -2, -3, 2, -2, 6, -1, 0, 6, 4, -5, -6, -2, 2, 6, -4, 0, -12, 0]
analytic_rank(algorithm='pari', leading_coefficient=False)

Return an integer that is probably the analytic rank of this elliptic curve.

INPUT:

  • algorithm – (default: ‘pari’), String
    • 'pari' – use the PARI library function.
    • 'sympow' – use Watkins’s program sympow
    • 'rubinstein' – use Rubinstein’s L-function C++ program lcalc.
    • 'magma' – use MAGMA
    • 'zero_sum' – Use the rank bounding zero sum method implemented in self.analytic_rank_upper_bound()
    • 'all' – compute with PARI, sympow and lcalc, check that the answers agree, and return the common answer.
  • leading_coefficient – (default: False) Boolean; if set to True, return a tuple \((rank, lead)\) where \(lead\) is the value of the first non-zero derivative of the L-function of the elliptic curve. Only implemented for algorithm=’pari’.

Note

If the curve is loaded from the large Cremona database, then the modular degree is taken from the database.

Of the first three algorithms above, probably Rubinstein’s is the most efficient (in some limited testing done). The zero sum method is often much faster, but can return a value which is strictly larger than the analytic rank. For curves with conductor <=10^9 using default parameters, testing indicates that for 99.75% of curves the returned rank bound is the true rank.

Note

If you use set_verbose(1), extra information about the computation will be printed when algorithm=’zero_sum’.

Note

It is an open problem to prove that any particular elliptic curve has analytic rank \(\geq 4\).

EXAMPLES:

sage: E = EllipticCurve('389a')
sage: E.analytic_rank(algorithm='pari')
2
sage: E.analytic_rank(algorithm='rubinstein')
2
sage: E.analytic_rank(algorithm='sympow')
2
sage: E.analytic_rank(algorithm='magma')    # optional - magma
2
sage: E.analytic_rank(algorithm='zero_sum')
2
sage: E.analytic_rank(algorithm='all')
2

With the optional parameter leading_coefficient set to True, a tuple of both the analytic rank and the leading term of the L-series at \(s = 1\) is returned. This only works for algorithm==’pari’:

sage: EllipticCurve([0,-1,1,-10,-20]).analytic_rank(leading_coefficient=True)
(0, 0.25384186085591068...)
sage: EllipticCurve([0,0,1,-1,0]).analytic_rank(leading_coefficient=True)
(1, 0.30599977383405230...)
sage: EllipticCurve([0,1,1,-2,0]).analytic_rank(leading_coefficient=True)
(2, 1.518633000576853...)
sage: EllipticCurve([0,0,1,-7,6]).analytic_rank(leading_coefficient=True)
(3, 10.39109940071580...)
sage: EllipticCurve([0,0,1,-7,36]).analytic_rank(leading_coefficient=True)
(4, 196.170903794579...)
analytic_rank_upper_bound(max_Delta=None, adaptive=True, N=None, root_number='compute', bad_primes=None, ncpus=None)

Return an upper bound for the analytic rank of self, conditional on the Generalized Riemann Hypothesis, via computing the zero sum \(\sum_{\gamma} f(\Delta\gamma),\) where \(\gamma\) ranges over the imaginary parts of the zeros of \(L(E,s)\) along the critical strip, \(f(x) = (\sin(\pi x)/(\pi x))^2\), and \(\Delta\) is the tightness parameter whose maximum value is specified by max_Delta. This computation can be run on curves with very large conductor (so long as the conductor is known or quickly computable) when \(\Delta\) is not too large (see below). Uses Bober’s rank bounding method as described in [Bob13].

INPUT:

  • max_Delta – (default: None) If not None, a positive real value specifying the maximum Delta value used in the zero sum; larger values of Delta yield better bounds - but runtime is exponential in Delta. If left as None, Delta is set to \(\min\{\frac{1}{\pi}(\log(N+1000)/2-\log(2\pi)-\eta), 2.5\}\), where \(N\) is the conductor of the curve attached to self, and \(\eta\) is the Euler-Mascheroni constant \(= 0.5772...\); the crossover point is at conductor around \(8.3 \cdot 10^8\). For the former value, empirical results show that for about 99.7% of all curves the returned value is the actual analytic rank.
  • adaptive – (default: True) Boolean
    • True – the computation is first run with small and then successively larger \(\Delta\) values up to max_Delta. If at any point the computed bound is 0 (or 1 when root_number is -1 or True), the computation halts and that value is returned; otherwise the minimum of the computed bounds is returned.
    • False – the computation is run a single time with \(\Delta\) equal to max_Delta, and the resulting bound returned.
  • N – (default: None) If not None, a positive integer equal to the conductor of self. This is passable so that rank estimation can be done for curves whose (large) conductor has been precomputed.
  • root_number – (default: “compute”) String or integer
    • "compute" – the root number of self is computed and used to (possibly) lower ther analytic rank estimate by 1.
    • "ignore" – the above step is omitted
    • 1 – this value is assumed to be the root number of self. This is passable so that rank estimation can be done for curves whose root number has been precomputed.
    • -1 – this value is assumed to be the root number of self. This is passable so that rank estimation can be done for curves whose root number has been precomputed.
  • bad_primes – (default: None) If not None, a list of the primes of bad reduction for the curve attached to self. This is passable so that rank estimation can be done for curves of large conductor whose bad primes have been precomputed.
  • ncpus - (default: None) If not None, a positive integer defining the maximum number of CPUs to be used for the computation. If left as None, the maximum available number of CPUs will be used. Note: Due to parallelization overhead, multiple processors will only be used for Delta values \(\ge 1.75\).

Note

Output will be incorrect if the incorrect conductor or root number is specified.

Warning

Zero sum computation time is exponential in the tightness parameter \(\Delta\), roughly doubling for every increase of 0.1 thereof. Using \(\Delta=1\) (and adaptive=False) will yield a runtime of a few milliseconds; \(\Delta=2\) takes a few seconds, and \(\Delta=3\) may take upwards of an hour. Increase beyond this at your own risk!

OUTPUT:

A non-negative integer greater than or equal to the analytic rank of self.

Note

If you use set_verbose(1), extra information about the computation will be printed.

EXAMPLES:

For most elliptic curves with small conductor the central zero(s) of \(L_E(s)\) are fairly isolated, so small values of \(\Delta\) will yield tight rank estimates.

sage: E = EllipticCurve("11a")
sage: E.rank()
0
sage: E.analytic_rank_upper_bound(max_Delta=1,adaptive=False)
0
sage: E = EllipticCurve([-39,123])
sage: E.rank()
1
sage: E.analytic_rank_upper_bound(max_Delta=1,adaptive=True)
1

This is especially true for elliptic curves with large rank.

sage: for r in range(9):
....:     E = elliptic_curves.rank(r)[0]
....:     print((r, E.analytic_rank_upper_bound(max_Delta=1,
....:     adaptive=False,root_number="ignore")))
(0, 0)
(1, 1)
(2, 2)
(3, 3)
(4, 4)
(5, 5)
(6, 6)
(7, 7)
(8, 8)

However, some curves have \(L\)-functions with low-lying zeroes, and for these larger values of \(\Delta\) must be used to get tight estimates.

sage: E = EllipticCurve("974b1")
sage: r = E.rank(); r
0
sage: E.analytic_rank_upper_bound(max_Delta=1,root_number="ignore")
1
sage: E.analytic_rank_upper_bound(max_Delta=1.3,root_number="ignore")
0

Knowing the root number of \(E\) allows us to use smaller Delta values to get tight bounds, thus speeding up runtime considerably.

sage: E.analytic_rank_upper_bound(max_Delta=0.6,root_number="compute")
0

There are a small number of curves which have pathologically low-lying zeroes. For these curves, this method will produce a bound that is strictly larger than the analytic rank, unless very large values of Delta are used. The following curve (“256944c1” in the Cremona tables) is a rank 0 curve with a zero at 0.0256…; the smallest Delta value for which the zero sum is strictly less than 2 is ~2.815.

sage: E = EllipticCurve([0, -1, 0, -7460362000712, -7842981500851012704])
sage: N,r = E.conductor(),E.analytic_rank(); N, r
(256944, 0)
sage: E.analytic_rank_upper_bound(max_Delta=1,adaptive=False)
2
sage: E.analytic_rank_upper_bound(max_Delta=2,adaptive=False)
2

This method is can be called on curves with large conductor.

sage: E = EllipticCurve([-2934,19238])
sage: E.analytic_rank_upper_bound()
1

And it can bound rank on curves with very large conductor, so long as you know beforehand/can easily compute the conductor and primes of bad reduction less than \(e^{2\pi\Delta}\). The example below is of the rank 28 curve discovered by Elkies that is the elliptic curve of (currently) largest known rank.

sage: a4 = -20067762415575526585033208209338542750930230312178956502
sage: a6 = 34481611795030556467032985690390720374855944359319180361266008296291939448732243429
sage: E = EllipticCurve([1,-1,1,a4,a6])
sage: bad_primes = [2,3,5,7,11,13,17,19,48463]
sage: N = 3455601108357547341532253864901605231198511505793733138900595189472144724781456635380154149870961231592352897621963802238155192936274322687070
sage: E.analytic_rank_upper_bound(max_Delta=2.37,adaptive=False, # long time
....: N=N,root_number=1,bad_primes=bad_primes,ncpus=2)           # long time
32

REFERENCES:

[Bob13]J.W. Bober. Conditionally bounding analytic ranks of elliptic curves. ANTS 10. http://msp.org/obs/2013/1-1/obs-v1-n1-p07-s.pdf
anlist(n, python_ints=False)

The Fourier coefficients up to and including \(a_n\) of the modular form attached to this elliptic curve. The i-th element of the return list is a[i].

INPUT:

  • n - integer
  • python_ints - bool (default: False); if True return a list of Python ints instead of Sage integers.

OUTPUT: list of integers

EXAMPLES:

sage: E = EllipticCurve([0, -1, 1, -10, -20])
sage: E.anlist(3)
[0, 1, -2, -1]
sage: E = EllipticCurve([0,1])
sage: E.anlist(20)
[0, 1, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 8, 0]
antilogarithm(z, max_denominator=None)

Return the rational point (if any) associated to this complex number; the inverse of the elliptic logarithm function.

INPUT:

  • z – a complex number representing an element of \(\CC/L\) where \(L\) is the period lattice of the elliptic curve
  • max_denominator (int or None) – parameter controlling the attempted conversion of real numbers to rationals. If None, simplest_rational() will be used; otherwise, nearby_rational() will be used with this value of max_denominator.

OUTPUT:

  • point on the curve: the rational point which is the image of \(z\) under the Weierstrass parametrization, if it exists and can be determined from \(z\) and the given value of max_denominator (if any); otherwise a ValueError exception is raised.

EXAMPLES:

sage: E = EllipticCurve('389a')
sage: P = E(-1,1)
sage: z = P.elliptic_logarithm()
sage: E.antilogarithm(z)
(-1 : 1 : 1)
sage: Q = E(0,-1)
sage: z = Q.elliptic_logarithm()
sage: E.antilogarithm(z)
Traceback (most recent call last):
...
ValueError: approximated point not on the curve
sage: E.antilogarithm(z, max_denominator=10)
(0 : -1 : 1)

sage: E = EllipticCurve('11a1')
sage: w1,w2 = E.period_lattice().basis()
sage: [E.antilogarithm(a*w1/5,1) for a in range(5)]
[(0 : 1 : 0), (16 : -61 : 1), (5 : -6 : 1), (5 : 5 : 1), (16 : 60 : 1)]
ap(p)

The p-th Fourier coefficient of the modular form corresponding to this elliptic curve, where p is prime.

EXAMPLES:

sage: E = EllipticCurve('37a1')
sage: [E.ap(p) for p in prime_range(50)]
[-2, -3, -2, -1, -5, -2, 0, 0, 2, 6, -4, -1, -9, 2, -9]
aplist(n, python_ints=False)

The Fourier coefficients \(a_p\) of the modular form attached to this elliptic curve, for all primes \(p\leq n\).

INPUT:

  • n - integer
  • python_ints - bool (default: False); if True return a list of Python ints instead of Sage integers.

OUTPUT: list of integers

EXAMPLES:

sage: e = EllipticCurve('37a')
sage: e.aplist(1)
[]
sage: e.aplist(2)
[-2]
sage: e.aplist(10)
[-2, -3, -2, -1]
sage: v = e.aplist(13); v
[-2, -3, -2, -1, -5, -2]
sage: type(v[0])
<... 'sage.rings.integer.Integer'>
sage: type(e.aplist(13, python_ints=True)[0])
<... 'int'>
cm_discriminant()

Return the associated quadratic discriminant if this elliptic curve has Complex Multiplication over the algebraic closure.

A ValueError is raised if the curve does not have CM (see the function has_cm()).

EXAMPLES:

sage: E = EllipticCurve('32a1')
sage: E.cm_discriminant()
-4
sage: E = EllipticCurve('121b1')
sage: E.cm_discriminant()
-11
sage: E = EllipticCurve('37a1')
sage: E.cm_discriminant()
Traceback (most recent call last):
...
ValueError: Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field does not have CM
conductor(algorithm='pari')

Return the conductor of the elliptic curve.

INPUT:

  • algorithm - str, (default: “pari”)
    • "pari" - use the PARI C-library ellglobalred implementation of Tate’s algorithm
    • "mwrank" - use Cremona’s mwrank implementation of Tate’s algorithm; can be faster if the curve has integer coefficients (TODO: limited to small conductor until mwrank gets integer factorization)
    • "gp" - use the GP interpreter.
    • "generic" - use the general number field implementation
    • "all" - use all four implementations, verify that the results are the same (or raise an error), and output the common value.

EXAMPLES:

sage: E = EllipticCurve([1, -1, 1, -29372, -1932937])
sage: E.conductor(algorithm="pari")
3006
sage: E.conductor(algorithm="mwrank")
3006
sage: E.conductor(algorithm="gp")
3006
sage: E.conductor(algorithm="generic")
3006
sage: E.conductor(algorithm="all")
3006

Note

The conductor computed using each algorithm is cached separately. Thus calling E.conductor('pari'), then E.conductor('mwrank') and getting the same result checks that both systems compute the same answer.

congruence_number(M=1)

The case \(M==1\) corresponds to the classical definition of congruence number: Let \(X\) be the subspace of \(S_2(\Gamma_0(N))\) spanned by the newform associated with this elliptic curve, and \(Y\) be orthogonal compliment of \(X\) under the Petersson inner product. Let \(S_X\) and \(S_Y\) be the intersections of \(X\) and \(Y\) with \(S_2(\Gamma_0(N), \ZZ)\). The congruence number is defined to be \([S_X \oplus S_Y : S_2(\Gamma_0(N),\ZZ)]\). It measures congruences between \(f\) and elements of \(S_2(\Gamma_0(N),\ZZ)\) orthogonal to \(f\).

The congruence number for higher levels, when M>1, is defined as above, but instead considers \(X\) to be the subspace of \(S_2(\Gamma_0(MN))\) spanned by embeddings into \(S_2(\Gamma_0(MN))\) of the newform associated with this elliptic curve; this subspace has dimension \(\sigma_0(M)\), i.e. the number of divisors of \(M\). Let \(Y\) be the orthogonal complement in \(S_2(\Gamma_0(MN))\) of \(X\) under the Petersson inner product, and \(S_X\) and \(S_Y\) the intersections of \(X\) and \(Y\) with \(S_2(\Gamma_0(MN), \ZZ)\) respectively. Then the congruence number at level \(MN\) is \([S_X \oplus S_Y : S_2(\Gamma_0(MN),\ZZ)]\).

INPUT:

  • \(M\) – Non-negative integer; congruence number is computed at level \(MN\), where \(N\) is the conductor of self.

EXAMPLES:

sage: E = EllipticCurve('37a')
sage: E.congruence_number()
2
sage: E.congruence_number()
2
sage: E = EllipticCurve('54b')
sage: E.congruence_number()
6
sage: E.modular_degree()
2
sage: E = EllipticCurve('242a1')
sage: E.modular_degree()
16
sage: E.congruence_number()  # long time (4s on sage.math, 2011)
176

Higher level cases:

sage: E = EllipticCurve('11a')
sage: for M in range(1,11): print(E.congruence_number(M)) # long time (20s on 2009 MBP)
1
1
3
2
7
45
12
4
18
245

It is a theorem of Ribet that the congruence number (at level \(N\)) is equal to the modular degree in the case of square free conductor. It is a conjecture of Agashe, Ribet, and Stein that \(ord_p(c_f/m_f) \le ord_p(N)/2\).

cremona_label(space=False)

Return the Cremona label associated to (the minimal model) of this curve, if it is known. If not, raise a LookupError exception.

EXAMPLES:

sage: E = EllipticCurve('389a1')
sage: E.cremona_label()
'389a1'

The default database only contains conductors up to 10000, so any curve with conductor greater than that will cause an error to be raised. The optional package database_cremona_ellcurve contains many more curves.

sage: E = EllipticCurve([1, -1, 0, -79, 289])
sage: E.conductor()
234446
sage: E.cremona_label()  # optional - database_cremona_ellcurve
'234446a1'
sage: E = EllipticCurve((0, 0, 1, -79, 342))
sage: E.conductor()
19047851
sage: E.cremona_label()
Traceback (most recent call last):
...
LookupError: Cremona database does not contain entry for Elliptic Curve defined by y^2 + y = x^3 - 79*x + 342 over Rational Field
database_attributes()

Return a dictionary containing information about self in the elliptic curve database.

If there is no elliptic curve isomorphic to self in the database, a LookupError is raised.

EXAMPLES:

sage: E = EllipticCurve((0, 0, 1, -1, 0))
sage: data = E.database_attributes()
sage: data['conductor']
37
sage: data['cremona_label']
'37a1'
sage: data['rank']
1
sage: data['torsion_order']
1

sage: E = EllipticCurve((8, 13, 21, 34, 55))
sage: E.database_attributes()
Traceback (most recent call last):
...
LookupError: Cremona database does not contain entry for Elliptic Curve defined by y^2 + 8*x*y + 21*y = x^3 + 13*x^2 + 34*x + 55 over Rational Field
database_curve()

Return the curve in the elliptic curve database isomorphic to this curve, if possible. Otherwise raise a LookupError exception.

Since trac ticket #11474, this returns exactly the same curve as minimal_model(); the only difference is the additional work of checking whether the curve is in the database.

EXAMPLES:

sage: E = EllipticCurve([0,1,2,3,4])
sage: E.database_curve()
Elliptic Curve defined by y^2  = x^3 + x^2 + 3*x + 5 over Rational Field

Note

The model of the curve in the database can be different from the Weierstrass model for this curve, e.g., database models are always minimal.

elliptic_exponential(z, embedding=None)

Compute the elliptic exponential of a complex number with respect to the elliptic curve.

INPUT:

  • z (complex) – a complex number
  • embedding - ignored (for compatibility with the period_lattice function for elliptic_curve_number_field)

OUTPUT:

The image of \(z\) modulo \(L\) under the Weierstrass parametrization \(\CC/L \to E(\CC)\).

Note

The precision is that of the input z, or the default precision of 53 bits if z is exact.

EXAMPLES:

sage: E = EllipticCurve([1,1,1,-8,6])
sage: P = E([1,-2])
sage: z = P.elliptic_logarithm() # default precision is 100 here
sage: E.elliptic_exponential(z)
(1.0000000000000000000000000000 : -2.0000000000000000000000000000 : 1.0000000000000000000000000000)
sage: z = E([1,-2]).elliptic_logarithm(precision=201)
sage: E.elliptic_exponential(z)
(1.00000000000000000000000000000000000000000000000000000000000 : -2.00000000000000000000000000000000000000000000000000000000000 : 1.00000000000000000000000000000000000000000000000000000000000)
sage: E = EllipticCurve('389a')
sage: Q = E([3,5])
sage: E.elliptic_exponential(Q.elliptic_logarithm())
(3.0000000000000000000000000000 : 5.0000000000000000000000000000 : 1.0000000000000000000000000000)
sage: P = E([-1,1])
sage: P.elliptic_logarithm()
0.47934825019021931612953301006 + 0.98586885077582410221120384908*I
sage: E.elliptic_exponential(P.elliptic_logarithm())
(-1.0000000000000000000000000000 : 1.0000000000000000000000000000 : 1.0000000000000000000000000000)

Some torsion examples:

sage: w1,w2 = E.period_lattice().basis()
sage: E.two_division_polynomial().roots(CC,multiplicities=False)
[-2.0403022002854..., 0.13540924022175..., 0.90489296006371...]
sage: [E.elliptic_exponential((a*w1+b*w2)/2)[0] for a,b in [(0,1),(1,1),(1,0)]]
[-2.0403022002854..., 0.13540924022175..., 0.90489296006371...]

sage: E.division_polynomial(3).roots(CC,multiplicities=False)
[-2.88288879135...,
1.39292799513...,
0.078313731444316... - 0.492840991709...*I,
0.078313731444316... + 0.492840991709...*I]
sage: [E.elliptic_exponential((a*w1+b*w2)/3)[0] for a,b in [(0,1),(1,0),(1,1),(2,1)]]
[-2.8828887913533..., 1.39292799513138,
0.0783137314443... - 0.492840991709...*I,
0.0783137314443... + 0.492840991709...*I]

Observe that this is a group homomorphism (modulo rounding error):

sage: z = CC.random_element()
sage: 2 * E.elliptic_exponential(z)
(-1.52184235874404 - 0.0581413944316544*I : 0.948655866506124 - 0.0381469928565030*I : 1.00000000000000)
sage: E.elliptic_exponential(2 * z)
(-1.52184235874404 - 0.0581413944316562*I : 0.948655866506128 - 0.0381469928565034*I : 1.00000000000000)
eval_modular_form(points, order)

Evaluate the modular form of this elliptic curve at points in \(\CC\).

INPUT:

  • points – a list of points in the upper half-plane
  • order – a nonnegative integer

The order parameter is the number of terms used in the summation.

OUTPUT: A list of values for \(s\) in points

EXAMPLES:

sage: E = EllipticCurve('37a1')
sage: E.eval_modular_form([1.5+I,2.0+I,2.5+I],100) # abs tol 1e-20
[-0.0018743978548152085771342944989052703431,
 0.0018604485340371083710285594393397945456,
-0.0018743978548152085771342944989052703431]

sage: E.eval_modular_form(2.1+I, 100) # abs tol 1e-20
[0.00150864362757267079 + 0.00109100341113449845*I]
galois_representation()

The compatible family of the Galois representation attached to this elliptic curve.

Given an elliptic curve \(E\) over \(\QQ\) and a rational prime number \(p\), the \(p^n\)-torsion \(E[p^n]\) points of \(E\) is a representation of the absolute Galois group of \(\QQ\). As \(n\) varies we obtain the Tate module \(T_p E\) which is a a representation of \(G_K\) on a free \(\ZZ_p\)-module of rank \(2\). As \(p\) varies the representations are compatible.

EXAMPLES:

sage: rho = EllipticCurve('11a1').galois_representation()
sage: rho
Compatible family of Galois representations associated to the Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
sage: rho.is_irreducible(7)
True
sage: rho.is_irreducible(5)
False
sage: rho.is_surjective(11)
True
sage: rho.non_surjective()
[5]
sage: rho = EllipticCurve('37a1').galois_representation()
sage: rho.non_surjective()
[]
sage: rho = EllipticCurve('27a1').galois_representation()
sage: rho.is_irreducible(7)
True
sage: rho.non_surjective()   # cm-curve
[0]
gens(proof=None, **kwds)

Return generators for the Mordell-Weil group E(Q) modulo torsion.

INPUT:

  • proof – bool or None (default None), see proof.elliptic_curve or sage.structure.proof
  • verbose - (default: None), if specified changes the
    verbosity of mwrank computations
  • rank1_search - (default: 10), if the curve has analytic rank 1, try to find a generator by a direct search up to this logarithmic height. If this fails, the usual mwrank procedure is called.
  • algorithm – one of the following:
    • 'mwrank_shell' (default) – call mwrank shell command
    • 'mwrank_lib' – call mwrank C library
  • only_use_mwrank – bool (default True) if False, first attempts to use more naive, natively implemented methods
  • use_database – bool (default True) if True, attempts to find curve and gens in the (optional) database
  • descent_second_limit – (default: 12) used in 2-descent
  • sat_bound – (default: 1000) bound on primes used in saturation. If the computed bound on the index of the points found by two-descent in the Mordell-Weil group is greater than this, a warning message will be displayed.

OUTPUT:

  • generators - list of generators for the Mordell-Weil
    group modulo torsion

Note

If you call this with proof=False, then you can use the gens_certain() method to find out afterwards whether the generators were proved.

IMPLEMENTATION: Uses Cremona’s mwrank C library.

EXAMPLES:

sage: E = EllipticCurve('389a')
sage: E.gens()                 # random output
[(-1 : 1 : 1), (0 : 0 : 1)]

A non-integral example:

sage: E = EllipticCurve([-3/8,-2/3])
sage: E.gens() # random (up to sign)
[(10/9 : 29/54 : 1)]

A non-minimal example:

sage: E = EllipticCurve('389a1')
sage: E1 = E.change_weierstrass_model([1/20,0,0,0]); E1
Elliptic Curve defined by y^2 + 8000*y = x^3 + 400*x^2 - 320000*x over Rational Field
sage: E1.gens() # random (if database not used)
[(-400 : 8000 : 1), (0 : -8000 : 1)]
gens_certain()

Return True if the generators have been proven correct.

EXAMPLES:

sage: E = EllipticCurve('37a1')
sage: E.gens()                   # random (up to sign)
[(0 : -1 : 1)]
sage: E.gens_certain()
True
global_integral_model()

Return a model of self which is integral at all primes.

EXAMPLES:

sage: E = EllipticCurve([0, 0, 1/216, -7/1296, 1/7776])
sage: F = E.global_integral_model(); F
Elliptic Curve defined by y^2 + y = x^3 - 7*x + 6 over Rational Field
sage: F == EllipticCurve('5077a1')
True
has_cm()

Return whether or not this curve has a CM \(j\)-invariant.

OUTPUT:

True if the \(j\)-invariant of this curve is the \(j\)-invariant of an imaginary quadratic order, otherwise False.

Note

Even if \(E\) has CM in this sense (that its \(j\)-invariant is a CM \(j\)-invariant), since the associated negative discriminant \(D\) is not a square in \(\QQ\), the extra endomorphisms will not be defined over \(\QQ\). See also the method has_rational_cm() which tests whether \(E\) has extra endomorphisms defined over \(\QQ\) or a given extension of \(\QQ\).

EXAMPLES:

sage: E = EllipticCurve('37a1')
sage: E.has_cm()
False
sage: E = EllipticCurve('32a1')
sage: E.has_cm()
True
sage: E.j_invariant()
1728
has_good_reduction_outside_S(S=[])

Test if this elliptic curve has good reduction outside \(S\).

INPUT:

  • \(S\) – list of primes (default: empty list).

Note

Primality of elements of S is not checked, and the output is undefined if S is not a list or contains non-primes.

This only tests the given model, so should only be applied to minimal models.

EXAMPLES:

sage: EllipticCurve('11a1').has_good_reduction_outside_S([11])
True
sage: EllipticCurve('11a1').has_good_reduction_outside_S([2])
False
sage: EllipticCurve('2310a1').has_good_reduction_outside_S([2,3,5,7])
False
sage: EllipticCurve('2310a1').has_good_reduction_outside_S([2,3,5,7,11])
True
has_rational_cm(field=None)

Return whether or not this curve has CM defined over \(\QQ\) or the given field.

INPUT:

  • field – a field, which should be an extension of \(\QQ\). If field is None (the default), it is taken to be \(\QQ\).

OUTPUT:

True if the ring of endomorphisms of this curve over the given field is larger than \(\ZZ\); otherwise False. If field is None the output will always be False. See also cm_discriminant() and has_cm().

Note

If \(E\) has CM but the discriminant \(D\) is not a square in the given field \(K\), which will certainly be the case for \(K=\QQ\) since \(D<0\), then the extra endomorphisms will not be defined over \(K\), and this function will return False. See also has_cm(). To obtain the CM discriminant, use cm_discriminant().

EXAMPLES:

sage: E = EllipticCurve(j=0)
sage: E.has_cm()
True
sage: E.has_rational_cm()
False
sage: D = E.cm_discriminant(); D
-3

If we extend scalars to a field in which the discriminant is a square, the CM becomes rational:

sage: E.has_rational_cm(QuadraticField(-3))
True

sage: E = EllipticCurve(j=8000)
sage: E.has_cm()
True
sage: E.has_rational_cm()
False
sage: D = E.cm_discriminant(); D
-8

Again, we may extend scalars to a field in which the discriminant is a square, where the CM becomes rational:

sage: E.has_rational_cm(QuadraticField(-2))
True

The field need not be a number field provided that it is an extension of \(\QQ\):

sage: E.has_rational_cm(RR)
False
sage: E.has_rational_cm(CC)
True

An error is raised if a field is given which is not an extension of \(\QQ\), i.e., not of characteristic \(0\):

sage: E.has_rational_cm(GF(2))
Traceback (most recent call last):
...
ValueError: Error in has_rational_cm: Finite Field of size 2 is not an extension field of QQ
heegner_discriminants(bound)

Return the list of self’s Heegner discriminants between -1 and -bound.

INPUT:

  • bound (int) - upper bound for -discriminant

OUTPUT: The list of Heegner discriminants between -1 and -bound for the given elliptic curve.

EXAMPLES:

sage: E=EllipticCurve('11a')
sage: E.heegner_discriminants(30)                     # indirect doctest
[-7, -8, -19, -24]
heegner_discriminants_list(n)

Return the list of self’s first \(n\) Heegner discriminants smaller than -5.

INPUT:

  • n (int) - the number of discriminants to compute

OUTPUT: The list of the first n Heegner discriminants smaller than -5 for the given elliptic curve.

EXAMPLES:

sage: E=EllipticCurve('11a')
sage: E.heegner_discriminants_list(4)                     # indirect doctest
[-7, -8, -19, -24]
heegner_index(D, min_p=2, prec=5, descent_second_limit=12, verbose_mwrank=False, check_rank=True)

Return an interval that contains the index of the Heegner point \(y_K\) in the group of \(K\)-rational points modulo torsion on this elliptic curve, computed using the Gross-Zagier formula and/or a point search, or possibly half the index if the rank is greater than one.

If the curve has rank > 1, then the returned index is infinity.

Note

If min_p is bigger than 2 then the index can be off by any prime less than min_p. This function returns the index divided by \(2\) exactly when the rank of \(E(K)\) is greater than 1 and \(E(\QQ)_{/tor} \oplus E^D(\QQ)_{/tor}\) has index \(2\) in \(E(K)_{/tor}\), where the second factor undergoes a twist.

INPUT:

  • D (int) - Heegner discriminant
  • min_p (int) - (default: 2) only rule out primes = min_p dividing the index.
  • verbose_mwrank (bool) - (default: False); print lots of mwrank search status information when computing regulator
  • prec (int) - (default: 5), use prec*sqrt(N) + 20 terms of L-series in computations, where N is the conductor.
  • descent_second_limit - (default: 12)- used in 2-descent when computing regulator of the twist
  • check_rank - whether to check if the rank is at least 2 by computing the Mordell-Weil rank directly.

OUTPUT: an interval that contains the index, or half the index

EXAMPLES:

sage: E = EllipticCurve('11a')
sage: E.heegner_discriminants(50)
[-7, -8, -19, -24, -35, -39, -40, -43]
sage: E.heegner_index(-7)
1.00000?
sage: E = EllipticCurve('37b')
sage: E.heegner_discriminants(100)
[-3, -4, -7, -11, -40, -47, -67, -71, -83, -84, -95]
sage: E.heegner_index(-95)          # long time (1 second)
2.00000?

This tests doing direct computation of the Mordell-Weil group.

sage: EllipticCurve('675b').heegner_index(-11)
3.0000?

Currently discriminants -3 and -4 are not supported:

sage: E.heegner_index(-3)
Traceback (most recent call last):
...
ArithmeticError: Discriminant (=-3) must not be -3 or -4.

The curve 681b returns the true index, which is \(3\):

sage: E = EllipticCurve('681b')
sage: I = E.heegner_index(-8); I
3.0000?

In fact, whenever the returned index has a denominator of \(2\), the true index is got by multiplying the returned index by \(2\). Unfortunately, this is not an if and only if condition, i.e., sometimes the index must be multiplied by \(2\) even though the denominator is not \(2\).

This example demonstrates the descent_second_limit option, which can be used to fine tune the 2-descent used to compute the regulator of the twist:

sage: E = EllipticCurve([1,-1,0,-1228,-16267])
sage: E.heegner_index(-8)
Traceback (most recent call last):
...
RuntimeError: ...

However when we search higher, we find the points we need:

sage: E.heegner_index(-8, descent_second_limit=16, check_rank=False)
2.00000?

Two higher rank examples (of ranks 2 and 3):

sage: E = EllipticCurve('389a')
sage: E.heegner_index(-7)
+Infinity
sage: E = EllipticCurve('5077a')
sage: E.heegner_index(-7)
+Infinity
sage: E.heegner_index(-7, check_rank=False)
0.001?
sage: E.heegner_index(-7, check_rank=False).lower() == 0
True
heegner_index_bound(D=0, prec=5, max_height=None)

Assume self has rank 0.

Return a list \(v\) of primes such that if an odd prime \(p\) divides the index of the Heegner point in the group of rational points modulo torsion, then \(p\) is in \(v\).

If 0 is in the interval of the height of the Heegner point computed to the given prec, then this function returns \(v = 0\). This does not mean that the Heegner point is torsion, just that it is very likely torsion.

If we obtain no information from a search up to max_height, e.g., if the Siksek et al. bound is bigger than max_height, then we return \(v = -1\).

INPUT:

  • D (int) - (default: 0) Heegner discriminant; if 0, use the first discriminant -4 that satisfies the Heegner hypothesis
  • verbose (bool) - (default: True)
  • prec (int) - (default: 5), use \(prec \cdot \sqrt(N) + 20\) terms of \(L\)-series in computations, where \(N\) is the conductor.
  • max_height (float) - should be = 21; bound on logarithmic naive height used in point searches. Make smaller to make this function faster, at the expense of possibly obtaining a worse answer. A good range is between 13 and 21.

OUTPUT:

  • v - list or int (bad primes or 0 or -1)
  • D - the discriminant that was used (this is useful if \(D\) was automatically selected).
  • exact - either False, or the exact Heegner index (up to factors of 2)

EXAMPLES:

sage: E = EllipticCurve('11a1')
sage: E.heegner_index_bound()
([2], -7, 2)
heegner_point(D, c=1, f=None, check=True)

Returns the Heegner point on this curve associated to the quadratic imaginary field \(K=\QQ(\sqrt{D})\).

If the optional parameter \(c\) is given, returns the higher Heegner point associated to the order of conductor \(c\).

INPUT:

  • \(D\) – a Heegner discriminant
  • \(c\) – (default: 1) conductor, must be coprime to \(DN\)
  • \(f\) – binary quadratic form or 3-tuple \((A,B,C)\) of coefficients of \(AX^2 + BXY + CY^2\)
  • check – bool (default: True)

OUTPUT:

The Heegner point \(y_c\).

EXAMPLES:

sage: E = EllipticCurve('37a')
sage: E.heegner_discriminants_list(10)
[-7, -11, -40, -47, -67, -71, -83, -84, -95, -104]
sage: P = E.heegner_point(-7); P                          # indirect doctest
Heegner point of discriminant -7 on elliptic curve of conductor 37
sage: P.point_exact()
(0 : 0 : 1)
sage: P.curve()
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
sage: P = E.heegner_point(-40).point_exact(); P
(a : -a + 1 : 1)
sage: P = E.heegner_point(-47).point_exact(); P
(a : a^4 + a - 1 : 1)
sage: P[0].parent()
Number Field in a with defining polynomial x^5 - x^4 + x^3 + x^2 - 2*x + 1

Working out the details manually:

sage: P = E.heegner_point(-47).numerical_approx(prec=200)
sage: f = algdep(P[0], 5); f
x^5 - x^4 + x^3 + x^2 - 2*x + 1
sage: f.discriminant().factor()
47^2

The Heegner hypothesis is checked:

sage: E = EllipticCurve('389a'); P = E.heegner_point(-5,7);
Traceback (most recent call last):
...
ValueError: N (=389) and D (=-5) must satisfy the Heegner hypothesis

We can specify the quadratic form:

sage: P = EllipticCurve('389a').heegner_point(-7, 5, (778,925,275)); P
Heegner point of discriminant -7 and conductor 5 on elliptic curve of conductor 389
sage: P.quadratic_form()
778*x^2 + 925*x*y + 275*y^2
heegner_point_height(D, prec=2, check_rank=True)

Use the Gross-Zagier formula to compute the Neron-Tate canonical height over \(K\) of the Heegner point corresponding to \(D\), as an interval (it is computed to some precision using \(L\)-functions).

If the curve has rank at least 2, then the returned height is the exact Sage integer 0.

INPUT:

  • D (int) - fundamental discriminant (=/= -3, -4)
  • prec (int) - (default: 2), use \(prec \cdot \sqrt(N) + 20\)
    terms of \(L\)-series in computations, where \(N\) is the conductor.
  • check_rank - whether to check if the rank is at least 2 by computing the Mordell-Weil rank directly.

OUTPUT: Interval that contains the height of the Heegner point.

EXAMPLES:

sage: E = EllipticCurve('11a')
sage: E.heegner_point_height(-7)
0.22227?

Some higher rank examples:

sage: E = EllipticCurve('389a')
sage: E.heegner_point_height(-7)
0
sage: E = EllipticCurve('5077a')
sage: E.heegner_point_height(-7)
0
sage: E.heegner_point_height(-7,check_rank=False)
0.0000?
heegner_sha_an(D, prec=53)

Return the conjectural (analytic) order of Sha for E over the field \(K=\QQ(\sqrt{D})\).

INPUT:

  • \(D\) – negative integer; the Heegner discriminant
  • prec – integer (default: 53); bits of precision to compute analytic order of Sha

OUTPUT:

(floating point number) an approximation to the conjectural order of Sha.

Note

Often you’ll want to do proof.elliptic_curve(False) when using this function, since often the twisted elliptic curves that come up have enormous conductor, and Sha is nontrivial, which makes provably finding the Mordell-Weil group using 2-descent difficult.

EXAMPLES:

An example where E has conductor 11:

sage: E = EllipticCurve('11a')
sage: E.heegner_sha_an(-7)                                  # long time
1.00000000000000

The cache works:

sage: E.heegner_sha_an(-7) is E.heegner_sha_an(-7)          # long time
True

Lower precision:

sage: E.heegner_sha_an(-7,10)                               # long time
1.0

Checking that the cache works for any precision:

sage: E.heegner_sha_an(-7,10) is E.heegner_sha_an(-7,10)    # long time
True

Next we consider a rank 1 curve with nontrivial Sha over the quadratic imaginary field \(K\); however, there is no Sha for \(E\) over \(\QQ\) or for the quadratic twist of \(E\):

sage: E = EllipticCurve('37a')
sage: E.heegner_sha_an(-40)                                 # long time
4.00000000000000
sage: E.quadratic_twist(-40).sha().an()                     # long time
1
sage: E.sha().an()                                          # long time
1

A rank 2 curve:

sage: E = EllipticCurve('389a')                             # long time
sage: E.heegner_sha_an(-7)                                  # long time
1.00000000000000

If we remove the hypothesis that \(E(K)\) has rank 1 in Conjecture 2.3 in [Gross-Zagier, 1986, page 311], then that conjecture is false, as the following example shows:

sage: E = EllipticCurve('65a')                              # long time
sage: E.heegner_sha_an(-56)                                 # long time
1.00000000000000
sage: E.torsion_order()                                     # long time
2
sage: E.tamagawa_product()                                  # long time
1
sage: E.quadratic_twist(-56).rank()                         # long time
2
height(precision=None)

Return the real height of this elliptic curve. This is used in integral_points()

INPUT:

  • precision - desired real precision of the result (default real precision if None)

EXAMPLES:

sage: E = EllipticCurve('5077a1')
sage: E.height()
17.4513334798896
sage: E.height(100)
17.451333479889612702508579399
sage: E = EllipticCurve([0,0,0,0,1])
sage: E.height()
1.38629436111989
sage: E = EllipticCurve([0,0,0,1,0])
sage: E.height()
7.45471994936400
integral_model()

Return a model of self which is integral at all primes.

EXAMPLES:

sage: E = EllipticCurve([0, 0, 1/216, -7/1296, 1/7776])
sage: F = E.global_integral_model(); F
Elliptic Curve defined by y^2 + y = x^3 - 7*x + 6 over Rational Field
sage: F == EllipticCurve('5077a1')
True
integral_points(mw_base='auto', both_signs=False, verbose=False)

Compute all integral points (up to sign) on this elliptic curve.

INPUT:

  • mw_base - list of EllipticCurvePoint generating the Mordell-Weil group of E (default: ‘auto’ - calls self.gens())
  • both_signs - True/False (default False): if True the output contains both P and -P, otherwise only one of each pair.
  • verbose - True/False (default False): if True, some details of the computation are output

OUTPUT: A sorted list of all the integral points on E (up to sign unless both_signs is True)

Note

The complexity increases exponentially in the rank of curve E. The computation time (but not the output!) depends on the Mordell-Weil basis. If mw_base is given but is not a basis for the Mordell-Weil group (modulo torsion), integral points which are not in the subgroup generated by the given points will almost certainly not be listed.

EXAMPLES: A curve of rank 3 with no torsion points

sage: E = EllipticCurve([0,0,1,-7,6])
sage: P1=E.point((2,0)); P2=E.point((-1,3)); P3=E.point((4,6))
sage: a=E.integral_points([P1,P2,P3]); a
[(-3 : 0 : 1), (-2 : 3 : 1), (-1 : 3 : 1), (0 : 2 : 1), (1 : 0 : 1), (2 : 0 : 1), (3 : 3 : 1), (4 : 6 : 1), (8 : 21 : 1), (11 : 35 : 1), (14 : 51 : 1), (21 : 95 : 1), (37 : 224 : 1), (52 : 374 : 1), (93 : 896 : 1), (342 : 6324 : 1), (406 : 8180 : 1), (816 : 23309 : 1)]
sage: a = E.integral_points([P1,P2,P3], verbose=True)
Using mw_basis  [(2 : 0 : 1), (3 : -4 : 1), (8 : -22 : 1)]
e1,e2,e3:  -3.0124303725933... 1.0658205476962... 1.94660982489710
Minimal and maximal eigenvalues of height pairing matrix: 0.637920814585005,2.31982967525725
x-coords of points on compact component with  -3 <=x<= 1
[-3, -2, -1, 0, 1]
x-coords of points on non-compact component with  2 <=x<= 6
[2, 3, 4]
starting search of remaining points using coefficient bound 5 and |x| bound 1.53897183921009e25
x-coords of extra integral points:
[2, 3, 4, 8, 11, 14, 21, 37, 52, 93, 342, 406, 816]
Total number of integral points: 18

It is not necessary to specify mw_base; if it is not provided, then the Mordell-Weil basis must be computed, which may take much longer.

sage: E = EllipticCurve([0,0,1,-7,6])
sage: a=E.integral_points(both_signs=True); a
[(-3 : -1 : 1), (-3 : 0 : 1), (-2 : -4 : 1), (-2 : 3 : 1), (-1 : -4 : 1), (-1 : 3 : 1), (0 : -3 : 1), (0 : 2 : 1), (1 : -1 : 1), (1 : 0 : 1), (2 : -1 : 1), (2 : 0 : 1), (3 : -4 : 1), (3 : 3 : 1), (4 : -7 : 1), (4 : 6 : 1), (8 : -22 : 1), (8 : 21 : 1), (11 : -36 : 1), (11 : 35 : 1), (14 : -52 : 1), (14 : 51 : 1), (21 : -96 : 1), (21 : 95 : 1), (37 : -225 : 1), (37 : 224 : 1), (52 : -375 : 1), (52 : 374 : 1), (93 : -897 : 1), (93 : 896 : 1), (342 : -6325 : 1), (342 : 6324 : 1), (406 : -8181 : 1), (406 : 8180 : 1), (816 : -23310 : 1), (816 : 23309 : 1)]

An example with negative discriminant:

sage: EllipticCurve('900d1').integral_points()
[(-11 : 27 : 1), (-4 : 34 : 1), (4 : 18 : 1), (16 : 54 : 1)]

Another example with rank 5 and no torsion points:

sage: E = EllipticCurve([-879984,319138704])
sage: P1=E.point((540,1188)); P2=E.point((576,1836))
sage: P3=E.point((468,3132)); P4=E.point((612,3132))
sage: P5=E.point((432,4428))
sage: a=E.integral_points([P1,P2,P3,P4,P5]); len(a)  # long time (18s on sage.math, 2011)
54
sage: [len(e.integral_points(both_signs=False)) for e in cremona_curves([11..100])]  # long time (15s on sage.math, 2011)
[2, 0, 2, 3, 2, 1, 3, 0, 2, 4, 2, 4, 3, 0, 0, 1, 2, 1, 2, 0, 2, 1, 0, 1, 3, 3, 1, 1, 4, 2, 3, 2, 0, 0, 5, 3, 2, 2, 1, 1, 1, 0, 1, 3, 0, 1, 0, 1, 1, 3, 6, 1, 2, 2, 2, 0, 0, 2, 3, 1, 2, 2, 1, 1, 0, 3, 2, 1, 0, 1, 0, 1, 3, 3, 1, 1, 5, 1, 0, 1, 1, 0, 1, 2, 0, 2, 0, 1, 1, 3, 1, 2, 2, 4, 4, 2, 1, 0, 0, 5, 1, 0, 1, 2, 0, 2, 2, 0, 0, 0, 1, 0, 3, 1, 5, 1, 2, 4, 1, 0, 1, 0, 1, 0, 1, 0, 2, 2, 0, 0, 1, 0, 1, 1, 4, 1, 0, 1, 1, 0, 4, 2, 0, 1, 1, 2, 3, 1, 1, 1, 1, 6, 2, 1, 1, 0, 2, 0, 6, 2, 0, 4, 2, 2, 0, 0, 1, 2, 0, 2, 1, 0, 3, 1, 2, 1, 4, 6, 3, 2, 1, 0, 2, 2, 0, 0, 5, 4, 1, 0, 0, 1, 0, 2, 2, 0, 0, 2, 3, 1, 3, 1, 1, 0, 1, 0, 0, 1, 2, 2, 0, 2, 0, 0, 1, 2, 0, 0, 4, 1, 0, 1, 1, 0, 1, 2, 0, 1, 4, 3, 1, 2, 2, 1, 1, 1, 1, 6, 3, 3, 3, 3, 1, 1, 1, 1, 1, 0, 7, 3, 0, 1, 3, 2, 1, 0, 3, 2, 1, 0, 2, 2, 6, 0, 0, 6, 2, 2, 3, 3, 5, 5, 1, 0, 6, 1, 0, 3, 1, 1, 2, 3, 1, 2, 1, 1, 0, 1, 0, 1, 0, 5, 5, 2, 2, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1]

The bug reported at trac ticket #4897 is now fixed:

sage: [P[0] for P in EllipticCurve([0,0,0,-468,2592]).integral_points()]
[-24, -18, -14, -6, -3, 4, 6, 18, 21, 24, 36, 46, 102, 168, 186, 381, 1476, 2034, 67246]

See trac ticket #22063:

sage: for n in [67,71,74,91]:
....:     assert 4*n^6+4*n^2 in [P[0] for P in EllipticCurve([0,0,0,2,n^2]).integral_points()]

Note

This function uses the algorithm given in [Coh2007I].

AUTHORS:

  • Michael Mardaus (2008-07)
  • Tobias Nagel (2008-07)
  • John Cremona (2008-07)
integral_short_weierstrass_model()

Return a model of the form \(y^2 = x^3 + ax + b\) for this curve with \(a,b\in\ZZ\).

EXAMPLES:

sage: E = EllipticCurve('17a1')
sage: E.integral_short_weierstrass_model()
Elliptic Curve defined by y^2  = x^3 - 11*x - 890 over Rational Field
integral_x_coords_in_interval(xmin, xmax)

Return the set of integers \(x\) with \(xmin\le x\le xmax\) which are \(x\)-coordinates of rational points on this curve.

INPUT:

  • xmin, xmax (integers) – two integers.

OUTPUT:

(set) The set of integers \(x\) with \(xmin\le x\le xmax\) which are \(x\)-coordinates of rational points on the elliptic curve.

EXAMPLES:

sage: E = EllipticCurve([0, 0, 1, -7, 6])
sage: xset = E.integral_x_coords_in_interval(-100,100)
sage: sorted(xset)
[-3, -2, -1, 0, 1, 2, 3, 4, 8, 11, 14, 21, 37, 52, 93]
sage: xset = E.integral_x_coords_in_interval(-100, 0)
sage: sorted(xset)
[-3, -2, -1, 0]
is_global_integral_model()

Return True iff self is integral at all primes.

EXAMPLES:

sage: E = EllipticCurve([1/2,1/5,1/5,1/5,1/5])
sage: E.is_global_integral_model()
False
sage: Emin=E.global_integral_model()
sage: Emin.is_global_integral_model()
True
is_good(p, check=True)

Return True if \(p\) is a prime of good reduction for \(E\).

INPUT:

  • p - a prime

OUTPUT: bool

EXAMPLES:

sage: e = EllipticCurve('11a')
sage: e.is_good(-8)
Traceback (most recent call last):
...
ValueError: p must be prime
sage: e.is_good(-8, check=False)
True
is_integral()

Return True if this elliptic curve has integral coefficients (in Z).

EXAMPLES:

sage: E = EllipticCurve(QQ,[1,1]); E
Elliptic Curve defined by y^2 = x^3 + x + 1 over Rational Field
sage: E.is_integral()
True
sage: E2=E.change_weierstrass_model(2,0,0,0); E2
Elliptic Curve defined by y^2 = x^3 + 1/16*x + 1/64 over Rational Field
sage: E2.is_integral()
False
is_isogenous(other, proof=True, maxp=200)

Return whether or not self is isogenous to other.

INPUT:

  • other – another elliptic curve.
  • proof (default True) – If False, the function will return True whenever the two curves have the same conductor and are isogenous modulo \(p\) for \(p\) up to maxp. If True, this test is followed by a rigorous test (which may be more time-consuming).
  • maxp (int, default 200) – The maximum prime \(p\) for which isogeny modulo \(p\) will be checked.

OUTPUT:

(bool) True if there is an isogeny from curve self to curve other.

METHOD:

First the conductors are compared as well as the Traces of Frobenius for good primes up to maxp. If any of these tests fail, False is returned. If they all pass and proof is False then True is returned, otherwise a complete set of curves isogenous to self is computed and other is checked for isomorphism with any of these,

EXAMPLES:

sage: E1 = EllipticCurve('14a1')
sage: E6 = EllipticCurve('14a6')
sage: E1.is_isogenous(E6)
True
sage: E1.is_isogenous(EllipticCurve('11a1'))
False
sage: EllipticCurve('37a1').is_isogenous(EllipticCurve('37b1'))
False
sage: E = EllipticCurve([2, 16])
sage: EE = EllipticCurve([87, 45])
sage: E.is_isogenous(EE)
False
is_local_integral_model(*p)

Tests if self is integral at the prime \(p\), or at all the primes if \(p\) is a list or tuple of primes

EXAMPLES:

sage: E = EllipticCurve([1/2,1/5,1/5,1/5,1/5])
sage: [E.is_local_integral_model(p) for p in (2,3,5)]
[False, True, False]
sage: E.is_local_integral_model(2,3,5)
False
sage: Eint2=E.local_integral_model(2)
sage: Eint2.is_local_integral_model(2)
True
is_minimal()

Return True iff this elliptic curve is a reduced minimal model.

The unique minimal Weierstrass equation for this elliptic curve. This is the model with minimal discriminant and \(a_1,a_2,a_3 \in \{0,\pm 1\}\).

Todo

This is not very efficient since it just computes the minimal model and compares. A better implementation using the Kraus conditions would be preferable.

EXAMPLES:

sage: E = EllipticCurve([10,100,1000,10000,1000000])
sage: E.is_minimal()
False
sage: E = E.minimal_model()
sage: E.is_minimal()
True
is_ordinary(p, ell=None)

Return True precisely when the mod-p representation attached to this elliptic curve is ordinary at ell.

INPUT:

  • p - a prime ell - a prime (default: p)

OUTPUT: bool

EXAMPLES:

sage: E = EllipticCurve('37a1')
sage: E.is_ordinary(37)
True
sage: E = EllipticCurve('32a1')
sage: E.is_ordinary(2)
False
sage: [p for p in prime_range(50) if E.is_ordinary(p)]
[5, 13, 17, 29, 37, 41]
is_p_integral(p)

Return True if this elliptic curve has \(p\)-integral coefficients.

INPUT:

  • p – a prime integer

EXAMPLES:

sage: E = EllipticCurve(QQ,[1,1]); E
Elliptic Curve defined by y^2 = x^3 + x + 1 over Rational Field
sage: E.is_p_integral(2)
True
sage: E2=E.change_weierstrass_model(2,0,0,0); E2
Elliptic Curve defined by y^2 = x^3 + 1/16*x + 1/64 over Rational Field
sage: E2.is_p_integral(2)
False
sage: E2.is_p_integral(3)
True
is_p_minimal(p)

Tests if curve is p-minimal at a given prime p.

INPUT: p – a prime

OUTPUT:

  • True – if curve is p-minimal
  • False – if curve is not p-minimal

EXAMPLES:

sage: E = EllipticCurve('441a2')
sage: E.is_p_minimal(7)
True
sage: E = EllipticCurve([0,0,0,0,(2*5*11)**10])
sage: [E.is_p_minimal(p) for p in prime_range(2,24)]
[False, True, False, True, False, True, True, True, True]
is_semistable()

Return True iff this elliptic curve is semi-stable at all primes.

EXAMPLES:

sage: E = EllipticCurve('37a1')
sage: E.is_semistable()
True
sage: E = EllipticCurve('90a1')
sage: E.is_semistable()
False
is_supersingular(p, ell=None)

Return True precisely when p is a prime of good reduction and the mod-p representation attached to this elliptic curve is supersingular at ell.

INPUT:

  • p - a prime ell - a prime (default: p)

OUTPUT: bool

EXAMPLES:

sage: E = EllipticCurve('37a1')
sage: E.is_supersingular(37)
False
sage: E = EllipticCurve('32a1')
sage: E.is_supersingular(2)
False
sage: E.is_supersingular(7)
True
sage: [p for p in prime_range(50) if E.is_supersingular(p)]
[3, 7, 11, 19, 23, 31, 43, 47]
isogenies_prime_degree(l=None)

Return a list of \(\ell\)-isogenies from self, where \(\ell\) is a prime.

INPUT:

  • l – either None or a prime or a list of primes.

OUTPUT:

(list) \(\ell\)-isogenies for the given \(\ell\) or if \(\ell\) is None, all \(\ell\)-isogenies.

Note

The codomains of the isogenies returned are standard minimal models. This is because the functions isogenies_prime_degree_genus_0() and isogenies_sporadic_Q() are implemented that way for curves defined over \(\QQ\).

EXAMPLES:

sage: E = EllipticCurve([45,32])
sage: E.isogenies_prime_degree()
[]
sage: E = EllipticCurve(j = -262537412640768000)
sage: E.isogenies_prime_degree()
[Isogeny of degree 163 from Elliptic Curve defined by y^2 + y = x^3 - 2174420*x + 1234136692 over Rational Field to Elliptic Curve defined by y^2 + y = x^3 - 57772164980*x - 5344733777551611 over Rational Field]
sage: E1 = E.quadratic_twist(6584935282)
sage: E1.isogenies_prime_degree()
[Isogeny of degree 163 from Elliptic Curve defined by y^2 = x^3 - 94285835957031797981376080*x + 352385311612420041387338054224547830898 over Rational Field to Elliptic Curve defined by y^2 = x^3 - 2505080375542377840567181069520*x - 1526091631109553256978090116318797845018020806 over Rational Field]

sage: E = EllipticCurve('14a1')
sage: E.isogenies_prime_degree(2)
[Isogeny of degree 2 from Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x - 6 over Rational Field to Elliptic Curve defined by y^2 + x*y + y = x^3 - 36*x - 70 over Rational Field]
sage: E.isogenies_prime_degree(3)
[Isogeny of degree 3 from Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x - 6 over Rational Field to Elliptic Curve defined by y^2 + x*y + y = x^3 - x over Rational Field, Isogeny of degree 3 from Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x - 6 over Rational Field to Elliptic Curve defined by y^2 + x*y + y = x^3 - 171*x - 874 over Rational Field]
sage: E.isogenies_prime_degree(5)
[]
sage: E.isogenies_prime_degree(11)
[]
sage: E.isogenies_prime_degree(29)
[]
sage: E.isogenies_prime_degree(4)
Traceback (most recent call last):
...
ValueError: 4 is not prime.
isogeny_class(algorithm='sage', order=None)

Return the \(\QQ\)-isogeny class of this elliptic curve.

INPUT:

  • algorithm - string: one of the following:
    • “database” - use the Cremona database (only works if curve is isomorphic to a curve in the database)
    • “sage” (default) - use the native Sage implementation.
  • order – None, string, or list of curves (default: None): If not None then the curves in the class are reordered after being computed. Note that if the order is None then the resulting order will depend on the algorithm.
    • if order is “database” or “sage”, then the reordering is so that the order of curves matches the order produced by that algorithm.
    • if order is “lmfdb” then the curves are sorted lexicographically by a-invariants, in the LMFDB database.
    • if order is a list of curves, then the curves in the class are reordered to be isomorphic with the specified list of curves.

OUTPUT:

An instance of the class sage.schemes.elliptic_curves.isogeny_class.IsogenyClass_EC_Rational. This object models a list of minimal models (with containment, index, etc based on isomorphism classes). It also has methods for computing the isogeny matrix and the list of isogenies between curves in this class.

Note

The curves in the isogeny class will all be standard minimal models.

EXAMPLES:

sage: isocls = EllipticCurve('37b').isogeny_class(order="lmfdb")
sage: isocls
Elliptic curve isogeny class 37b
sage: isocls.curves
(Elliptic Curve defined by y^2 + y = x^3 + x^2 - 1873*x - 31833 over Rational Field,
 Elliptic Curve defined by y^2 + y = x^3 + x^2 - 23*x - 50 over Rational Field,
 Elliptic Curve defined by y^2 + y = x^3 + x^2 - 3*x + 1 over Rational Field)
sage: isocls.matrix()
[1 3 9]
[3 1 3]
[9 3 1]
sage: isocls = EllipticCurve('37b').isogeny_class('database', order="lmfdb"); isocls.curves
(Elliptic Curve defined by y^2 + y = x^3 + x^2 - 1873*x - 31833 over Rational Field,
 Elliptic Curve defined by y^2 + y = x^3 + x^2 - 23*x - 50 over Rational Field,
 Elliptic Curve defined by y^2 + y = x^3 + x^2 - 3*x + 1 over Rational Field)

This is an example of a curve with a \(37\)-isogeny:

sage: E = EllipticCurve([1,1,1,-8,6])
sage: isocls = E.isogeny_class(); isocls
Isogeny class of Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 - 8*x + 6 over Rational Field
sage: isocls.matrix()
[ 1 37]
[37  1]
sage: print("\n".join([repr(E) for E in isocls.curves]))
Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 - 8*x + 6 over Rational Field
Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 - 208083*x - 36621194 over Rational Field

This curve had numerous \(2\)-isogenies:

sage: e = EllipticCurve([1,0,0,-39,90])
sage: isocls = e.isogeny_class(); isocls.matrix()
[1 2 4 4 8 8]
[2 1 2 2 4 4]
[4 2 1 4 8 8]
[4 2 4 1 2 2]
[8 4 8 2 1 4]
[8 4 8 2 4 1]

See http://math.harvard.edu/~elkies/nature.html for more interesting examples of isogeny structures.

sage: E = EllipticCurve(j = -262537412640768000)
sage: isocls = E.isogeny_class(); isocls.matrix()
[  1 163]
[163   1]
sage: print("\n".join([repr(C) for C in isocls.curves]))
Elliptic Curve defined by y^2 + y = x^3 - 2174420*x + 1234136692 over Rational Field
Elliptic Curve defined by y^2 + y = x^3 - 57772164980*x - 5344733777551611 over Rational Field

The degrees of isogenies are invariant under twists:

sage: E = EllipticCurve(j = -262537412640768000)
sage: E1 = E.quadratic_twist(6584935282)
sage: isocls = E1.isogeny_class(); isocls.matrix()
[  1 163]
[163   1]
sage: E1.conductor()
18433092966712063653330496
sage: E = EllipticCurve('14a1')
sage: isocls = E.isogeny_class(); isocls.matrix()
[ 1  2  3  3  6  6]
[ 2  1  6  6  3  3]
[ 3  6  1  9  2 18]
[ 3  6  9  1 18  2]
[ 6  3  2 18  1  9]
[ 6  3 18  2  9  1]
sage: print("\n".join([repr(C) for C in isocls.curves]))
Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x - 6 over Rational Field
Elliptic Curve defined by y^2 + x*y + y = x^3 - 36*x - 70 over Rational Field
Elliptic Curve defined by y^2 + x*y + y = x^3 - x over Rational Field
Elliptic Curve defined by y^2 + x*y + y = x^3 - 171*x - 874 over Rational Field
Elliptic Curve defined by y^2 + x*y + y = x^3 - 11*x + 12 over Rational Field
Elliptic Curve defined by y^2 + x*y + y = x^3 - 2731*x - 55146 over Rational Field
sage: isocls2 = isocls.reorder('lmfdb'); isocls2.matrix()
[ 1  2  3  9 18  6]
[ 2  1  6 18  9  3]
[ 3  6  1  3  6  2]
[ 9 18  3  1  2  6]
[18  9  6  2  1  3]
[ 6  3  2  6  3  1]
sage: print("\n".join([repr(C) for C in isocls2.curves]))
Elliptic Curve defined by y^2 + x*y + y = x^3 - 2731*x - 55146 over Rational Field
Elliptic Curve defined by y^2 + x*y + y = x^3 - 171*x - 874 over Rational Field
Elliptic Curve defined by y^2 + x*y + y = x^3 - 36*x - 70 over Rational Field
Elliptic Curve defined by y^2 + x*y + y = x^3 - 11*x + 12 over Rational Field
Elliptic Curve defined by y^2 + x*y + y = x^3 - x over Rational Field
Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x - 6 over Rational Field
sage: E = EllipticCurve('11a1')
sage: isocls = E.isogeny_class(); isocls.matrix()
[ 1  5  5]
[ 5  1 25]
[ 5 25  1]
sage: f = isocls.isogenies()[0][1]; f.kernel_polynomial()
x^2 + x - 29/5
isogeny_degree(other)

Return the minimal degree of an isogeny between self and other.

INPUT:

  • other – another elliptic curve.

OUTPUT:

(int) The minimal degree of an isogeny from self to other, or 0 if the curves are not isogenous.

EXAMPLES:

sage: E = EllipticCurve([-1056, 13552])
sage: E2 = EllipticCurve([-127776, -18037712])
sage: E.isogeny_degree(E2)
11
sage: E1 = EllipticCurve('14a1')
sage: E2 = EllipticCurve('14a2')
sage: E3 = EllipticCurve('14a3')
sage: E4 = EllipticCurve('14a4')
sage: E5 = EllipticCurve('14a5')
sage: E6 = EllipticCurve('14a6')
sage: E3.isogeny_degree(E1)
3
sage: E3.isogeny_degree(E2)
6
sage: E3.isogeny_degree(E3)
1
sage: E3.isogeny_degree(E4)
9
sage: E3.isogeny_degree(E5)
2
sage: E3.isogeny_degree(E6)
18
sage: E1 = EllipticCurve('30a1')
sage: E2 = EllipticCurve('30a2')
sage: E3 = EllipticCurve('30a3')
sage: E4 = EllipticCurve('30a4')
sage: E5 = EllipticCurve('30a5')
sage: E6 = EllipticCurve('30a6')
sage: E7 = EllipticCurve('30a7')
sage: E8 = EllipticCurve('30a8')
sage: E1.isogeny_degree(E1)
1
sage: E1.isogeny_degree(E2)
2
sage: E1.isogeny_degree(E3)
3
sage: E1.isogeny_degree(E4)
4
sage: E1.isogeny_degree(E5)
4
sage: E1.isogeny_degree(E6)
6
sage: E1.isogeny_degree(E7)
12
sage: E1.isogeny_degree(E8)
12
sage: E1 = EllipticCurve('15a1')
sage: E2 = EllipticCurve('15a2')
sage: E3 = EllipticCurve('15a3')
sage: E4 = EllipticCurve('15a4')
sage: E5 = EllipticCurve('15a5')
sage: E6 = EllipticCurve('15a6')
sage: E7 = EllipticCurve('15a7')
sage: E8 = EllipticCurve('15a8')
sage: E1.isogeny_degree(E1)
1
sage: E7.isogeny_degree(E2)
8
sage: E7.isogeny_degree(E3)
2
sage: E7.isogeny_degree(E4)
8
sage: E7.isogeny_degree(E5)
16
sage: E7.isogeny_degree(E6)
16
sage: E7.isogeny_degree(E8)
4

0 is returned when the curves are not isogenous:

sage: A = EllipticCurve('37a1')
sage: B = EllipticCurve('37b1')
sage: A.isogeny_degree(B)
0
sage: A.is_isogenous(B)
False
isogeny_graph(order=None)

Return a graph representing the isogeny class of this elliptic curve, where the vertices are isogenous curves over \(\QQ\) and the edges are prime degree isogenies.

Note

The vertices are labeled 1 to n rather than 0 to n-1 to correspond to LMFDB and Cremona labels.

EXAMPLES:

sage: LL = []
sage: for e in cremona_optimal_curves(range(1, 38)):  # long time
....:  G = e.isogeny_graph()
....:  already = False
....:  for H in LL:
....:      if G.is_isomorphic(H):
....:          already = True
....:          break
....:  if not already:
....:      LL.append(G)
sage: graphs_list.show_graphs(LL)  # long time
sage: E = EllipticCurve('195a')
sage: G = E.isogeny_graph()
sage: for v in G: print("{} {}".format(v, G.get_vertex(v)))
1 Elliptic Curve defined by y^2 + x*y  = x^3 - 110*x + 435 over Rational Field
2 Elliptic Curve defined by y^2 + x*y  = x^3 - 115*x + 392 over Rational Field
3 Elliptic Curve defined by y^2 + x*y  = x^3 + 210*x + 2277 over Rational Field
4 Elliptic Curve defined by y^2 + x*y  = x^3 - 520*x - 4225 over Rational Field
5 Elliptic Curve defined by y^2 + x*y  = x^3 + 605*x - 19750 over Rational Field
6 Elliptic Curve defined by y^2 + x*y  = x^3 - 8125*x - 282568 over Rational Field
7 Elliptic Curve defined by y^2 + x*y  = x^3 - 7930*x - 296725 over Rational Field
8 Elliptic Curve defined by y^2 + x*y  = x^3 - 130000*x - 18051943 over Rational Field
sage: G.plot(edge_labels=True)
Graphics object consisting of 23 graphics primitives
kodaira_symbol(p)

Local Kodaira type of the elliptic curve at \(p\).

INPUT:

  • p – an integral prime

OUTPUT:

  • the Kodaira type of this elliptic curve at p, as a KodairaSymbol.

EXAMPLES:

sage: E = EllipticCurve('124a')
sage: E.kodaira_type(2)
IV
kodaira_type(p)

Local Kodaira type of the elliptic curve at \(p\).

INPUT:

  • p – an integral prime

OUTPUT:

  • the Kodaira type of this elliptic curve at p, as a KodairaSymbol.

EXAMPLES:

sage: E = EllipticCurve('124a')
sage: E.kodaira_type(2)
IV
kodaira_type_old(p)

Local Kodaira type of the elliptic curve at \(p\).

INPUT:

  • p, an integral prime

OUTPUT:

  • the Kodaira type of this elliptic curve at p, as a KodairaSymbol.

EXAMPLES:

sage: E = EllipticCurve('124a')
sage: E.kodaira_type_old(2)
IV
kolyvagin_point(D, c=1, check=True)

Return the Kolyvagin point on this curve associated to the quadratic imaginary field \(K=\QQ(\sqrt{D})\) and conductor \(c\).

INPUT:

  • \(D\) – a Heegner discriminant
  • \(c\) – (default: 1) conductor, must be coprime to \(DN\)
  • check – bool (default: True)

OUTPUT:

The Kolyvagin point \(P\) of conductor \(c\).

EXAMPLES:

sage: E = EllipticCurve('37a1')
sage: P = E.kolyvagin_point(-67); P
Kolyvagin point of discriminant -67 on elliptic curve of conductor 37
sage: P.numerical_approx()  # abs tol 1e-14
(6.00000000000000 : -15.0000000000000 : 1.00000000000000)
sage: P.index()
6
sage: g = E((0,-1,1)) # a generator
sage: E.regulator() == E.regulator_of_points([g])
True
sage: 6*g
(6 : -15 : 1)
label(space=False)

Return the Cremona label associated to (the minimal model) of this curve, if it is known. If not, raise a LookupError exception.

EXAMPLES:

sage: E = EllipticCurve('389a1')
sage: E.cremona_label()
'389a1'

The default database only contains conductors up to 10000, so any curve with conductor greater than that will cause an error to be raised. The optional package database_cremona_ellcurve contains many more curves.

sage: E = EllipticCurve([1, -1, 0, -79, 289])
sage: E.conductor()
234446
sage: E.cremona_label()  # optional - database_cremona_ellcurve
'234446a1'
sage: E = EllipticCurve((0, 0, 1, -79, 342))
sage: E.conductor()
19047851
sage: E.cremona_label()
Traceback (most recent call last):
...
LookupError: Cremona database does not contain entry for Elliptic Curve defined by y^2 + y = x^3 - 79*x + 342 over Rational Field
lmfdb_page()

Open the LMFDB web page of the elliptic curve in a browser.

See http://www.lmfdb.org

EXAMPLES:

sage: E = EllipticCurve('5077a1')
sage: E.lmfdb_page()  # optional -- webbrowser
local_integral_model(p)

Return a model of self which is integral at the prime \(p\).

EXAMPLES:

sage: E = EllipticCurve([0, 0, 1/216, -7/1296, 1/7776])
sage: E.local_integral_model(2)
Elliptic Curve defined by y^2 + 1/27*y = x^3 - 7/81*x + 2/243 over Rational Field
sage: E.local_integral_model(3)
Elliptic Curve defined by y^2 + 1/8*y = x^3 - 7/16*x + 3/32 over Rational Field
sage: E.local_integral_model(2).local_integral_model(3) == EllipticCurve('5077a1')
True
lseries()

Return the L-series of this elliptic curve.

Further documentation is available for the functions which apply to the L-series.

EXAMPLES:

sage: E = EllipticCurve('37a1')
sage: E.lseries()
Complex L-series of the Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
lseries_gross_zagier(A)

Return the Gross-Zagier L-series attached to self and an ideal class \(A\).

INPUT:

  • A – an ideal class in an imaginary quadratic number field \(K\)

This L-series \(L(E,A,s)\) is defined as the product of a shifted L-function of the quadratic character associated to \(K\) and the Dirichlet series whose \(n\)-th coefficient is the product of the \(n\)-th factor of the L-series of \(E\) and the number of integral ideal in \(A\) of norm \(n\). For any character \(\chi\) on the class group of \(K\), one gets \(L_K(E,\chi,s) = \sum_{A} \chi(A) L(E,A,s)\) where \(A\) runs through the class group of \(K\).

For the exact definition see section IV of [GrossZagier].

EXAMPLES:

sage: E = EllipticCurve('37a')
sage: K.<a> = QuadraticField(-40)
sage: A = K.class_group().gen(0); A
Fractional ideal class (2, 1/2*a)
sage: L = E.lseries_gross_zagier(A)  ; L
Gross Zagier L-series attached to Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field with ideal class Fractional ideal class (2, 1/2*a)
sage: L(1)
0.000000000000000
sage: L.taylor_series(1, 5)
0.000000000000000 - 5.51899839494458*z + 13.6297841350649*z^2 - 16.2292417817675*z^3 + 7.94788823722712*z^4 + O(z^5)

These should be equal:

sage: L(2) + E.lseries_gross_zagier(A^2)(2)
0.502803417587467
sage: E.lseries()(2) * E.quadratic_twist(-40).lseries()(2)
0.502803417587467

REFERENCES:

[GrossZagier]B. Gross and D. Zagier, Heegner points and derivatives of L-series. Invent. Math. 84 (1986), no. 2, 225-320.
manin_constant()

Return the Manin constant of this elliptic curve.

If \(\phi: X_0(N) \to E\) is the modular parametrization of minimal degree, then the Manin constant \(c\) is defined to be the rational number \(c\) such that \(\phi^*(\omega_E) = c\cdot \omega_f\) where \(\omega_E\) is a Néron differential and \(\omega_f = f(q) dq/q\) is the differential on \(X_0(N)\) corresponding to the newform \(f\) attached to the isogeny class of \(E\).

It is known that the Manin constant is an integer. It is conjectured that in each class there is at least one, more precisely the so-called strong Weil curve or \(X_0(N)\)-optimal curve, that has Manin constant \(1\).

OUTPUT:

an integer

This function only works if the curve is in the installed Cremona database. Sage includes by default a small database; for the full database you have to install an optional package.

EXAMPLES:

sage: EllipticCurve('11a1').manin_constant()
1
sage: EllipticCurve('11a2').manin_constant()
1
sage: EllipticCurve('11a3').manin_constant()
5

Check that it works even if the curve is non-minimal:

sage: EllipticCurve('11a3').change_weierstrass_model([1/35,0,0,0]).manin_constant()
5

Rather complicated examples (see trac ticket #12080)

sage: [ EllipticCurve('27a%s'%i).manin_constant() for i in [1,2,3,4]]
[1, 1, 3, 3]
sage: [ EllipticCurve('80b%s'%i).manin_constant() for i in [1,2,3,4]]
[1, 2, 1, 2]
matrix_of_frobenius(p, prec=20, check=False, check_hypotheses=True, algorithm='auto')

Returns the matrix of Frobenius on the Monsky Washnitzer cohomology of the short Weierstrass model of the minimal model of the elliptic curve.

INPUT:

  • p - prime (>= 3) for which \(E\) is good and ordinary

  • prec - (relative) \(p\)-adic precision for result (default 20)

  • check - boolean (default: False), whether to perform a consistency check. This will slow down the computation by a constant factor 2. (The consistency check is to verify that its trace is correct to the specified precision. Otherwise, the trace is used to compute one column from the other one (possibly after a change of basis).)

  • check_hypotheses - boolean, whether to check that this is a curve for which the \(p\)-adic sigma function makes sense

  • algorithm - one of “standard”, “sqrtp”, or “auto”. This selects which version of Kedlaya’s algorithm is used. The “standard” one is the one described in Kedlaya’s paper. The “sqrtp” one has better performance for large \(p\), but only works when \(p > 6N\) (\(N=\) prec). The “auto” option selects “sqrtp” whenever possible.

    Note that if the “sqrtp” algorithm is used, a consistency check will automatically be applied, regardless of the setting of the “check” flag.

OUTPUT: a matrix of \(p\)-adic number to precision prec

See also the documentation of padic_E2.

EXAMPLES:

sage: E = EllipticCurve('37a1')
sage: E.matrix_of_frobenius(7)
[             2*7 + 4*7^2 + 5*7^4 + 6*7^5 + 6*7^6 + 7^8 + 4*7^9 + 3*7^10 + 2*7^11 + 5*7^12 + 4*7^14 + 7^16 + 2*7^17 + 3*7^18 + 4*7^19 + 3*7^20 + O(7^21)                                   2 + 3*7 + 6*7^2 + 7^3 + 3*7^4 + 5*7^5 + 3*7^7 + 7^8 + 3*7^9 + 6*7^13 + 7^14 + 7^16 + 5*7^17 + 4*7^18 + 7^19 + O(7^20)]
[    2*7 + 3*7^2 + 7^3 + 3*7^4 + 6*7^5 + 2*7^6 + 3*7^7 + 5*7^8 + 3*7^9 + 2*7^11 + 6*7^12 + 5*7^13 + 4*7^16 + 4*7^17 + 6*7^18 + 6*7^19 + 4*7^20 + O(7^21) 6 + 4*7 + 2*7^2 + 6*7^3 + 7^4 + 6*7^7 + 5*7^8 + 2*7^9 + 3*7^10 + 4*7^11 + 7^12 + 6*7^13 + 2*7^14 + 6*7^15 + 5*7^16 + 4*7^17 + 3*7^18 + 2*7^19 + O(7^20)]
sage: M = E.matrix_of_frobenius(11,prec=3); M
[   9*11 + 9*11^3 + O(11^4)          10 + 11 + O(11^3)]
[     2*11 + 11^2 + O(11^4) 6 + 11 + 10*11^2 + O(11^3)]
sage: M.det()
11 + O(11^4)
sage: M.trace()
6 + 10*11 + 10*11^2 + O(11^3)
sage: E.ap(11)
-5
sage: E = EllipticCurve('83a1')
sage: E.matrix_of_frobenius(3,6)
[                      2*3 + 3^5 + O(3^6)             2*3 + 2*3^2 + 2*3^3 + O(3^6)]
[              2*3 + 3^2 + 2*3^5 + O(3^6) 2 + 2*3^2 + 2*3^3 + 2*3^4 + 3^5 + O(3^6)]
minimal_model()

Return the unique minimal Weierstrass equation for this elliptic curve.

This is the model with minimal discriminant and \(a_1,a_2,a_3 \in \{0,\pm 1\}\).

EXAMPLES:

sage: E = EllipticCurve([10,100,1000,10000,1000000])
sage: E.minimal_model()
Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 + x + 1 over Rational Field
minimal_quadratic_twist()

Determine a quadratic twist with minimal conductor. Return a global minimal model of the twist and the fundamental discriminant of the quadratic field over which they are isomorphic.

Note

If there is more than one curve with minimal conductor, the one returned is the one with smallest label (if in the database), or the one with minimal \(a\)-invariant list (otherwise).

Note

For curves with \(j\)-invariant 0 or 1728 the curve returned is the minimal quadratic twist, not necessarily the minimal twist (which would have conductor 27 or 32 respectively).

EXAMPLES:

sage: E = EllipticCurve('121d1')
sage: E.minimal_quadratic_twist()
(Elliptic Curve defined by y^2 + y = x^3 - x^2 over Rational Field, -11)
sage: Et, D = EllipticCurve('32a1').minimal_quadratic_twist()
sage: D
1

sage: E = EllipticCurve('11a1')
sage: Et, D = E.quadratic_twist(-24).minimal_quadratic_twist()
sage: E == Et
True
sage: D
-24

sage: E = EllipticCurve([0,0,0,0,1000])
sage: E.minimal_quadratic_twist()
(Elliptic Curve defined by y^2 = x^3 + 1 over Rational Field, 40)
sage: E = EllipticCurve([0,0,0,1600,0])
sage: E.minimal_quadratic_twist()
(Elliptic Curve defined by y^2 = x^3 + 4*x over Rational Field, 5)

If the curve has square-free conductor then it is already minimal (see trac ticket #14060):

sage: E = next(cremona_optimal_curves([2*3*5*7*11]))
sage: (E, 1) == E.minimal_quadratic_twist()
True

An example where the minimal quadratic twist is not the minimal twist (which has conductor 27):

sage: E = EllipticCurve([0,0,0,0,7])
sage: E.j_invariant()
0
sage: E.minimal_quadratic_twist()[0].conductor()
5292
mod5family()

Return the family of all elliptic curves with the same mod-5 representation as self.

EXAMPLES:

sage: E = EllipticCurve('32a1')
sage: E.mod5family()
Elliptic Curve defined by y^2  = x^3 + 4*x over Fraction Field of Univariate Polynomial Ring in t over Rational Field
modular_degree(algorithm='sympow', M=1)

Return the modular degree at level \(MN\) of this elliptic curve. The case \(M==1\) corresponds to the classical definition of modular degree.

When \(M>1\), the function returns the degree of the map from \(X_0(MN) \to A\), where A is the abelian variety generated by embeddings of \(E\) into \(J_0(MN)\).

The result is cached. Subsequent calls, even with a different algorithm, just returned the cached result. The algorithm argument is ignored when \(M>1\).

INPUT:

  • algorithm - string:
  • 'sympow' - (default) use Mark Watkin’s (newer) C program sympow
  • 'magma' - requires that MAGMA be installed (also implemented by Mark Watkins)
  • M - Non-negative integer; the modular degree at level \(MN\) is returned
    (see above)

Note

On 64-bit computers ec does not work, so Sage uses sympow even if ec is selected on a 64-bit computer.

The correctness of this function when called with algorithm “sympow” is subject to the following three hypothesis:

  • Manin’s conjecture: the Manin constant is 1
  • Steven’s conjecture: the \(X_1(N)\)-optimal quotient is the curve with minimal Faltings height. (This is proved in most cases.)
  • The modular degree fits in a machine double, so it better be less than about 50-some bits. (If you use sympow this constraint does not apply.)

Moreover for all algorithms, computing a certain value of an \(L\)-function ‘uses a heuristic method that discerns when the real-number approximation to the modular degree is within epsilon [=0.01 for algorithm=’sympow’] of the same integer for 3 consecutive trials (which occur maybe every 25000 coefficients or so). Probably it could just round at some point. For rigour, you would need to bound the tail by assuming (essentially) that all the \(a_n\) are as large as possible, but in practice they exhibit significant (square root) cancellation. One difficulty is that it doesn’t do the sum in 1-2-3-4 order; it uses 1-2-4-8–3-6-12-24-9-18- (Euler product style) instead, and so you have to guess ahead of time at what point to curtail this expansion.’ (Quote from an email of Mark Watkins.)

Note

If the curve is loaded from the large Cremona database, then the modular degree is taken from the database.

EXAMPLES:

sage: E = EllipticCurve([0, -1, 1, -10, -20])
sage: E
Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
sage: E.modular_degree()
1
sage: E = EllipticCurve('5077a')
sage: E.modular_degree()
1984
sage: factor(1984)
2^6 * 31
sage: EllipticCurve([0, 0, 1, -7, 6]).modular_degree()
1984
sage: EllipticCurve([0, 0, 1, -7, 6]).modular_degree(algorithm='sympow')
1984
sage: EllipticCurve([0, 0, 1, -7, 6]).modular_degree(algorithm='magma')  # optional - magma
1984

We compute the modular degree of the curve with rank 4 having smallest (known) conductor:

sage: E = EllipticCurve([1, -1, 0, -79, 289])
sage: factor(E.conductor())  # conductor is 234446
2 * 117223
sage: factor(E.modular_degree())
2^7 * 2617

Higher level cases:

sage: E = EllipticCurve('11a')
sage: for M in range(1,11): print(E.modular_degree(M=M)) # long time (20s on 2009 MBP)
1
1
3
2
7
45
12
16
54
245
modular_form()

Return the cuspidal modular form associated to this elliptic curve.

EXAMPLES:

sage: E = EllipticCurve('37a')
sage: f = E.modular_form()
sage: f
q - 2*q^2 - 3*q^3 + 2*q^4 - 2*q^5 + O(q^6)

If you need to see more terms in the \(q\)-expansion:

sage: f.q_expansion(20)
q - 2*q^2 - 3*q^3 + 2*q^4 - 2*q^5 + 6*q^6 - q^7 + 6*q^9 + 4*q^10 - 5*q^11 - 6*q^12 - 2*q^13 + 2*q^14 + 6*q^15 - 4*q^16 - 12*q^18 + O(q^20)

Note

If you just want the \(q\)-expansion, use q_expansion().

modular_parametrization()

Return the modular parametrization of this elliptic curve, which is a map from \(X_0(N)\) to self, where \(N\) is the conductor of self.

EXAMPLES:

sage: E = EllipticCurve('15a')
sage: phi = E.modular_parametrization(); phi
Modular parameterization from the upper half plane to Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 - 10*x - 10 over Rational Field
sage: z = 0.1 + 0.2j
sage: phi(z)
(8.20822465478531 - 13.1562816054682*I : -8.79855099049364 + 69.4006129342200*I : 1.00000000000000)

This map is actually a map on \(X_0(N)\), so equivalent representatives in the upper half plane map to the same point:

sage: phi((-7*z-1)/(15*z+2))
(8.20822465478524 - 13.1562816054681*I : -8.79855099049... + 69.4006129342...*I : 1.00000000000000)

We can also get a series expansion of this modular parameterization:

sage: E = EllipticCurve('389a1')
sage: X,Y=E.modular_parametrization().power_series()
sage: X
q^-2 + 2*q^-1 + 4 + 7*q + 13*q^2 + 18*q^3 + 31*q^4 + 49*q^5 + 74*q^6 + 111*q^7 + 173*q^8 + 251*q^9 + 379*q^10 + 560*q^11 + 824*q^12 + 1199*q^13 + 1773*q^14 + 2548*q^15 + 3722*q^16 + 5374*q^17 + O(q^18)
sage: Y
-q^-3 - 3*q^-2 - 8*q^-1 - 17 - 33*q - 61*q^2 - 110*q^3 - 186*q^4 - 320*q^5 - 528*q^6 - 861*q^7 - 1383*q^8 - 2218*q^9 - 3472*q^10 - 5451*q^11 - 8447*q^12 - 13020*q^13 - 19923*q^14 - 30403*q^15 - 46003*q^16 + O(q^17)

The following should give 0, but only approximately:

sage: q = X.parent().gen()
sage: E.defining_polynomial()(X,Y,1) + O(q^11) == 0
True
modular_symbol(sign=1, normalize=None, implementation='eclib')

Return the modular symbol associated to this elliptic curve, with given sign.

INPUT:

  • sign - +1 (default) or -1.
  • normalize - (default: None); either ‘L_ratio’, ‘period’, or ‘none’ when implementation is ‘sage’; ignored if implementation is eclib. For ‘L_ratio’, the modular symbol tries to normalize correctly as explained below by comparing it to L_ratio for the curve and some small twists. The normalization ‘period’ uses the integral_period_map for modular symbols which is known to be equal to the desired normalization, up to the sign and a possible power of 2. With normalization ‘none’, the modular symbol is almost certainly not correctly normalized, i.e. all values will be a fixed scalar multiple of what they should be. However, the initial computation of the modular symbol is much faster when implementation sage is chosen, though evaluation of it after computing it is no faster.
  • implementation - either ‘eclib’ (default) or ‘sage’. Here ‘eclib’ uses John Cremona’s implementation in the eclib library, while ‘sage’ uses an implementation in Sage which is often quite a bit slower.

Note

The value at a rational number \(r\) is proportional to the real or imaginary part of the integral of \(2 \pi i f(z) dz\) from \(\infty\) to \(r\), where \(f\) is the newform attached to \(E\), suitably normalized so that all values of this map take values in \(\QQ\).

The normalization is such that for sign +1, the value at the cusp \(r\) is equal to the quotient of the real part of \(\int_{\infty}^{r}2\pi i f(z)dz\) by the least positive period of \(E\), where \(f\) is the newform attached to the isogeny class of \(E\). This is in contrast to the method L_ratio of lseries(), where the value is also divided by the number of connected components of \(E(\RR)\)). In particular the modular symbol depends on \(E\) and not only the isogeny class of \(E\). For negative modular symbols, the value is the quotient of the imaginary part of the above integral by the imaginary part of the smallest positive imaginary period.

EXAMPLES:

sage: E = EllipticCurve('37a1')
sage: M = E.modular_symbol(); M
Modular symbol with sign 1 over Rational Field attached to Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
sage: M(1/2)
0
sage: M(1/5)
1
sage: E = EllipticCurve('121b1')
sage: M = E.modular_symbol(implementation="sage")
Warning : Could not normalize the modular symbols, maybe all further results will be multiplied by -1 and a power of 2
sage: M(1/7)
-1/2

Different curves in an isogeny class have modular symbols which differ by a nonzero rational factor:

sage: E1 = EllipticCurve('11a1')
sage: M1 = E1.modular_symbol()
sage: M1(0)
1/5
sage: E2 = EllipticCurve('11a2')
sage: M2 = E2.modular_symbol()
sage: M2(0)
1
sage: E3 = EllipticCurve('11a3')
sage: M3 = E3.modular_symbol()
sage: M3(0)
1/25
sage: all(5*M1(r)==M2(r)==25*M3(r) for r in QQ.range_by_height(10))
True

With the default implementation using eclib, the symbols are correctly normalized automatically. With the Sage implementation we can choose to normalize using the L-ratio, unless that is 0 (for curves of positive rank) or using periods. Here is an example where the symbol is already normalized:

sage: E = EllipticCurve('11a2')
sage: E.modular_symbol(implementation = 'eclib')(0)
1
sage: E.modular_symbol(implementation = 'sage', normalize='L_ratio')(0)
1
sage: E.modular_symbol(implementation = 'sage', normalize='none')(0)
1
sage: E.modular_symbol(implementation = 'sage', normalize='period')(0)
1

Here is an example where both normalization methods work, while the non-normalized symbol is incorrect:

sage: E = EllipticCurve('11a3')
sage: E.modular_symbol(implementation = 'eclib')(0)
1/25
sage: E.modular_symbol(implementation = 'sage', normalize='none')(0)
1
sage: E.modular_symbol(implementation = 'sage', normalize='L_ratio')(0)
1/25
sage: E.modular_symbol(implementation = 'sage', normalize='period')(0)
1/25

Since trac ticket #10256, the interface for negative modular symbols in eclib is available:

sage: E = EllipticCurve('11a1')
sage: Mplus = E.modular_symbol(+1); Mplus
Modular symbol with sign 1 over Rational Field attached to Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
sage: [Mplus(1/i) for i in [1..11]]
[1/5, -4/5, -3/10, 7/10, 6/5, 6/5, 7/10, -3/10, -4/5, 1/5, 0]
sage: Mminus = E.modular_symbol(-1); Mminus
Modular symbol with sign -1 over Rational Field attached to Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
sage: [Mminus(1/i) for i in [1..11]]
[0, 0, 1/2, 1/2, 0, 0, -1/2, -1/2, 0, 0, 0]
modular_symbol_numerical(sign=1, prec=53)

Return the modular symbol as a numerical function.

Note

This method does not compute spaces of modular symbols, so it is suitable for curves of larger conductor than can be handled by modular_symbol().

EXAMPLES:

sage: E = EllipticCurve('19a1')
sage: f = E.modular_symbol_numerical(1)
sage: g = E.modular_symbol()
sage: f(0), g(0)  # abs tol 1e-14
(0.333333333333330, 1/3)
sage: f(oo), g(oo)
(-0.000000000000000, 0)

sage: E = EllipticCurve('79a1')
sage: f = E.modular_symbol_numerical(-1)
sage: g = E.modular_symbol(-1, implementation="sage")
sage: f(1/3), g(1/3)  # abs tol 1e-13
(1.00000000000001, 1)
sage: f(oo), g(oo)
(0.000000000000000, 0)
modular_symbol_space(sign=1, base_ring=Rational Field, bound=None)

Return the space of cuspidal modular symbols associated to this elliptic curve, with given sign and base ring.

INPUT:

  • sign - 0, -1, or 1
  • base_ring - a ring

EXAMPLES:

sage: f = EllipticCurve('37b')
sage: f.modular_symbol_space()
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 3 for Gamma_0(37) of weight 2 with sign 1 over Rational Field
sage: f.modular_symbol_space(-1)
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 2 for Gamma_0(37) of weight 2 with sign -1 over Rational Field
sage: f.modular_symbol_space(0, bound=3)
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 5 for Gamma_0(37) of weight 2 with sign 0 over Rational Field

Note

If you just want the \(q\)-expansion, use q_expansion().

mwrank(options='')

Run Cremona’s mwrank program on this elliptic curve and return the result as a string.

INPUT:

  • options (string) – run-time options passed when starting mwrank. The format is as follows (see below for examples of usage):
    • -v n (verbosity level) sets verbosity to n (default=1)
    • -o (PARI/GP style output flag) turns ON extra PARI/GP short output (default is OFF)
    • -p n (precision) sets precision to \(n\) decimals (default=15)
    • -b n (quartic bound) bound on quartic point search (default=10)
    • -x n (n_aux) number of aux primes used for sieving (default=6)
    • -l (generator list flag) turns ON listing of points (default ON unless v=0)
    • -s (selmer_only flag) if set, computes Selmer rank only (default: not set)
    • -d (skip_2nd_descent flag) if set, skips the second descent for curves with 2-torsion (default: not set)
    • -S n (sat_bd) upper bound on saturation primes (default=100, -1 for automatic)

OUTPUT:

  • string - output of mwrank on this curve

Note

The output is a raw string and completely illegible using automatic display, so it is recommended to use print for legible output.

EXAMPLES:

sage: E = EllipticCurve('37a1')
sage: E.mwrank() #random
...
sage: print(E.mwrank())
Curve [0,0,1,-1,0] :        Basic pair: I=48, J=-432
disc=255744
...
Generator 1 is [0:-1:1]; height 0.05111...

Regulator = 0.05111...

The rank and full Mordell-Weil basis have been determined unconditionally.
...

Options to mwrank can be passed:

sage: E = EllipticCurve([0,0,0,877,0])

Run mwrank with ‘verbose’ flag set to 0 but list generators if found

sage: print(E.mwrank('-v0 -l'))
Curve [0,0,0,877,0] :   0 <= rank <= 1
Regulator = 1

Run mwrank again, this time with a higher bound for point searching on homogeneous spaces:

sage: print(E.mwrank('-v0 -l -b11'))
Curve [0,0,0,877,0] :   Rank = 1
Generator 1 is [29604565304828237474403861024284371796799791624792913256602210:-256256267988926809388776834045513089648669153204356603464786949:490078023219787588959802933995928925096061616470779979261000]; height 95.98037...
Regulator = 95.98037...
mwrank_curve(verbose=False)

Construct an mwrank_EllipticCurve from this elliptic curve

The resulting mwrank_EllipticCurve has available methods from John Cremona’s eclib library.

EXAMPLES:

sage: E = EllipticCurve('11a1')
sage: EE = E.mwrank_curve()
sage: EE
y^2+ y = x^3 - x^2 - 10*x - 20
sage: type(EE)
<class 'sage.libs.eclib.interface.mwrank_EllipticCurve'>
sage: EE.isogeny_class()
([[0, -1, 1, -10, -20], [0, -1, 1, -7820, -263580], [0, -1, 1, 0, 0]],
[[0, 5, 5], [5, 0, 0], [5, 0, 0]])
newform()

Same as self.modular_form().

EXAMPLES:

sage: E = EllipticCurve('37a1')
sage: E.newform()
q - 2*q^2 - 3*q^3 + 2*q^4 - 2*q^5 + O(q^6)
sage: E.newform() == E.modular_form()
True
ngens(proof=None)

Return the number of generators of this elliptic curve.

Note

See gens() for further documentation. The function ngens() calls gens() if not already done, but only with default parameters. Better results may be obtained by calling mwrank() with carefully chosen parameters.

EXAMPLES:

sage: E = EllipticCurve('37a1')
sage: E.ngens()
1

sage: E = EllipticCurve([0,0,0,877,0])
sage: E.ngens()
1

sage: print(E.mwrank('-v0 -b12 -l'))
Curve [0,0,0,877,0] :   Rank = 1
Generator 1 is [29604565304828237474403861024284371796799791624792913256602210:-256256267988926809388776834045513089648669153204356603464786949:490078023219787588959802933995928925096061616470779979261000]; height 95.98037...
Regulator = 95.980...
optimal_curve()

Given an elliptic curve that is in the installed Cremona database, return the optimal curve isogenous to it.

EXAMPLES:

The following curve is not optimal:

sage: E = EllipticCurve('11a2'); E
Elliptic Curve defined by y^2 + y = x^3 - x^2 - 7820*x - 263580 over Rational Field
sage: E.optimal_curve()
Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
sage: E.optimal_curve().cremona_label()
'11a1'

Note that 990h is the special case where the optimal curve isn’t the first in the Cremona labeling:

sage: E = EllipticCurve('990h4'); E
Elliptic Curve defined by y^2 + x*y + y = x^3 - x^2 + 6112*x - 41533 over Rational Field
sage: F = E.optimal_curve(); F
Elliptic Curve defined by y^2 + x*y + y = x^3 - x^2 - 1568*x - 4669 over Rational Field
sage: F.cremona_label()
'990h3'
sage: EllipticCurve('990a1').optimal_curve().cremona_label()   # a isn't h.
'990a1'

If the input curve is optimal, this function returns that curve (not just a copy of it or a curve isomorphic to it!):

sage: E = EllipticCurve('37a1')
sage: E.optimal_curve() is E
True

Also, if this curve is optimal but not given by a minimal model, this curve will still be returned, so this function need not return a minimal model in general.

sage: F = E.short_weierstrass_model(); F
Elliptic Curve defined by y^2  = x^3 - 16*x + 16 over Rational Field
sage: F.optimal_curve()
Elliptic Curve defined by y^2  = x^3 - 16*x + 16 over Rational Field
ordinary_primes(B)

Return a list of all ordinary primes for this elliptic curve up to and possibly including B.

EXAMPLES:

sage: e = EllipticCurve('11a')
sage: e.aplist(20)
[-2, -1, 1, -2, 1, 4, -2, 0]
sage: e.ordinary_primes(97)
[3, 5, 7, 11, 13, 17, 23, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]
sage: e = EllipticCurve('49a')
sage: e.aplist(20)
[1, 0, 0, 0, 4, 0, 0, 0]
sage: e.supersingular_primes(97)
[3, 5, 13, 17, 19, 31, 41, 47, 59, 61, 73, 83, 89, 97]
sage: e.ordinary_primes(97)
[2, 11, 23, 29, 37, 43, 53, 67, 71, 79]
sage: e.ordinary_primes(3)
[2]
sage: e.ordinary_primes(2)
[2]
sage: e.ordinary_primes(1)
[]
padic_E2(p, prec=20, check=False, check_hypotheses=True, algorithm='auto')

Returns the value of the \(p\)-adic modular form \(E2\) for \((E, \omega)\) where \(\omega\) is the usual invariant differential \(dx/(2y + a_1 x + a_3)\).

INPUT:

  • p - prime (= 5) for which \(E\) is good and ordinary

  • prec - (relative) p-adic precision (= 1) for result

  • check - boolean, whether to perform a consistency check. This will slow down the computation by a constant factor 2. (The consistency check is to compute the whole matrix of frobenius on Monsky-Washnitzer cohomology, and verify that its trace is correct to the specified precision. Otherwise, the trace is used to compute one column from the other one (possibly after a change of basis).)

  • check_hypotheses - boolean, whether to check that this is a curve for which the p-adic sigma function makes sense

  • algorithm - one of “standard”, “sqrtp”, or “auto”. This selects which version of Kedlaya’s algorithm is used. The “standard” one is the one described in Kedlaya’s paper. The “sqrtp” one has better performance for large \(p\), but only works when \(p > 6N\) (\(N=\) prec). The “auto” option selects “sqrtp” whenever possible.

    Note that if the “sqrtp” algorithm is used, a consistency check will automatically be applied, regardless of the setting of the “check” flag.

OUTPUT: p-adic number to precision prec

Note

If the discriminant of the curve has nonzero valuation at p, then the result will not be returned mod \(p^\text{prec}\), but it still will have prec digits of precision.

Todo

Once we have a better implementation of the “standard” algorithm, the algorithm selection strategy for “auto” needs to be revisited.

AUTHORS:

  • David Harvey (2006-09-01): partly based on code written by Robert Bradshaw at the MSRI 2006 modular forms workshop

ACKNOWLEDGMENT: - discussion with Eyal Goren that led to the trace trick.

EXAMPLES: Here is the example discussed in the paper “Computation of p-adic Heights and Log Convergence” (Mazur, Stein, Tate):

sage: EllipticCurve([-1, 1/4]).padic_E2(5)
2 + 4*5 + 2*5^3 + 5^4 + 3*5^5 + 2*5^6 + 5^8 + 3*5^9 + 4*5^10 + 2*5^11 + 2*5^12 + 2*5^14 + 3*5^15 + 3*5^16 + 3*5^17 + 4*5^18 + 2*5^19 + O(5^20)

Let’s try to higher precision (this is the same answer the MAGMA implementation gives):

sage: EllipticCurve([-1, 1/4]).padic_E2(5, 100)
2 + 4*5 + 2*5^3 + 5^4 + 3*5^5 + 2*5^6 + 5^8 + 3*5^9 + 4*5^10 + 2*5^11 + 2*5^12 + 2*5^14 + 3*5^15 + 3*5^16 + 3*5^17 + 4*5^18 + 2*5^19 + 4*5^20 + 5^21 + 4*5^22 + 2*5^23 + 3*5^24 + 3*5^26 + 2*5^27 + 3*5^28 + 2*5^30 + 5^31 + 4*5^33 + 3*5^34 + 4*5^35 + 5^36 + 4*5^37 + 4*5^38 + 3*5^39 + 4*5^41 + 2*5^42 + 3*5^43 + 2*5^44 + 2*5^48 + 3*5^49 + 4*5^50 + 2*5^51 + 5^52 + 4*5^53 + 4*5^54 + 3*5^55 + 2*5^56 + 3*5^57 + 4*5^58 + 4*5^59 + 5^60 + 3*5^61 + 5^62 + 4*5^63 + 5^65 + 3*5^66 + 2*5^67 + 5^69 + 2*5^70 + 3*5^71 + 3*5^72 + 5^74 + 5^75 + 5^76 + 3*5^77 + 4*5^78 + 4*5^79 + 2*5^80 + 3*5^81 + 5^82 + 5^83 + 4*5^84 + 3*5^85 + 2*5^86 + 3*5^87 + 5^88 + 2*5^89 + 4*5^90 + 4*5^92 + 3*5^93 + 4*5^94 + 3*5^95 + 2*5^96 + 4*5^97 + 4*5^98 + 2*5^99 + O(5^100)

Check it works at low precision too:

sage: EllipticCurve([-1, 1/4]).padic_E2(5, 1)
2 + O(5)
sage: EllipticCurve([-1, 1/4]).padic_E2(5, 2)
2 + 4*5 + O(5^2)
sage: EllipticCurve([-1, 1/4]).padic_E2(5, 3)
2 + 4*5 + O(5^3)

TODO: With the old(-er), i.e., = sage-2.4 p-adics we got \(5 + O(5^2)\) as output, i.e., relative precision 1, but with the newer p-adics we get relative precision 0 and absolute precision 1.

sage: EllipticCurve([1, 1, 1, 1, 1]).padic_E2(5, 1)
O(5)

Check it works for different models of the same curve (37a), even when the discriminant changes by a power of p (note that E2 depends on the differential too, which is why it gets scaled in some of the examples below):

sage: X1 = EllipticCurve([-1, 1/4])
sage: X1.j_invariant(), X1.discriminant()
 (110592/37, 37)
sage: X1.padic_E2(5, 10)
 2 + 4*5 + 2*5^3 + 5^4 + 3*5^5 + 2*5^6 + 5^8 + 3*5^9 + O(5^10)
sage: X2 = EllipticCurve([0, 0, 1, -1, 0])
sage: X2.j_invariant(), X2.discriminant()
 (110592/37, 37)
sage: X2.padic_E2(5, 10)
 2 + 4*5 + 2*5^3 + 5^4 + 3*5^5 + 2*5^6 + 5^8 + 3*5^9 + O(5^10)
sage: X3 = EllipticCurve([-1*(2**4), 1/4*(2**6)])
sage: X3.j_invariant(), X3.discriminant() / 2**12
 (110592/37, 37)
sage: 2**(-2) * X3.padic_E2(5, 10)
 2 + 4*5 + 2*5^3 + 5^4 + 3*5^5 + 2*5^6 + 5^8 + 3*5^9 + O(5^10)
sage: X4 = EllipticCurve([-1*(7**4), 1/4*(7**6)])
sage: X4.j_invariant(), X4.discriminant() / 7**12
 (110592/37, 37)
sage: 7**(-2) * X4.padic_E2(5, 10)
 2 + 4*5 + 2*5^3 + 5^4 + 3*5^5 + 2*5^6 + 5^8 + 3*5^9 + O(5^10)
sage: X5 = EllipticCurve([-1*(5**4), 1/4*(5**6)])
sage: X5.j_invariant(), X5.discriminant() / 5**12
 (110592/37, 37)
sage: 5**(-2) * X5.padic_E2(5, 10)
 2 + 4*5 + 2*5^3 + 5^4 + 3*5^5 + 2*5^6 + 5^8 + 3*5^9 + O(5^10)
sage: X6 = EllipticCurve([-1/(5**4), 1/4/(5**6)])
sage: X6.j_invariant(), X6.discriminant() * 5**12
 (110592/37, 37)
sage: 5**2 * X6.padic_E2(5, 10)
 2 + 4*5 + 2*5^3 + 5^4 + 3*5^5 + 2*5^6 + 5^8 + 3*5^9 + O(5^10)

Test check=True vs check=False:

sage: EllipticCurve([-1, 1/4]).padic_E2(5, 1, check=False)
2 + O(5)
sage: EllipticCurve([-1, 1/4]).padic_E2(5, 1, check=True)
2 + O(5)
sage: EllipticCurve([-1, 1/4]).padic_E2(5, 30, check=False)
2 + 4*5 + 2*5^3 + 5^4 + 3*5^5 + 2*5^6 + 5^8 + 3*5^9 + 4*5^10 + 2*5^11 + 2*5^12 + 2*5^14 + 3*5^15 + 3*5^16 + 3*5^17 + 4*5^18 + 2*5^19 + 4*5^20 + 5^21 + 4*5^22 + 2*5^23 + 3*5^24 + 3*5^26 + 2*5^27 + 3*5^28 + O(5^30)
sage: EllipticCurve([-1, 1/4]).padic_E2(5, 30, check=True)
2 + 4*5 + 2*5^3 + 5^4 + 3*5^5 + 2*5^6 + 5^8 + 3*5^9 + 4*5^10 + 2*5^11 + 2*5^12 + 2*5^14 + 3*5^15 + 3*5^16 + 3*5^17 + 4*5^18 + 2*5^19 + 4*5^20 + 5^21 + 4*5^22 + 2*5^23 + 3*5^24 + 3*5^26 + 2*5^27 + 3*5^28 + O(5^30)

Here’s one using the \(p^{1/2}\) algorithm:

sage: EllipticCurve([-1, 1/4]).padic_E2(3001, 3, algorithm="sqrtp")
1907 + 2819*3001 + 1124*3001^2 + O(3001^3)
padic_height(p, prec=20, sigma=None, check_hypotheses=True)

Compute the cyclotomic p-adic height.

The equation of the curve must be minimal at \(p\).

INPUT:

  • p - prime >= 5 for which the curve has semi-stable reduction
  • prec - integer >= 1 (default 20), desired precision of result
  • sigma - precomputed value of sigma. If not supplied, this function will call padic_sigma to compute it.
  • check_hypotheses - boolean, whether to check that this is a curve for which the p-adic height makes sense

OUTPUT: A function that accepts two parameters:

  • a Q-rational point on the curve whose height should be computed
  • optional boolean flag ‘check’: if False, it skips some input checking, and returns the p-adic height of that point to the desired precision.
  • The normalization (sign and a factor 1/2 with respect to some other normalizations that appear in the literature) is chosen in such a way as to make the p-adic Birch Swinnerton-Dyer conjecture hold as stated in [Mazur-Tate-Teitelbaum].

AUTHORS:

  • Jennifer Balakrishnan: original code developed at the 2006 MSRI graduate workshop on modular forms
  • David Harvey (2006-09-13): integrated into Sage, optimised to speed up repeated evaluations of the returned height function, addressed some thorny precision questions
  • David Harvey (2006-09-30): rewrote to use division polynomials for computing denominator of \(nP\).
  • David Harvey (2007-02): cleaned up according to algorithms in “Efficient Computation of p-adic Heights”
  • Chris Wuthrich (2007-05): added supersingular and multiplicative heights

EXAMPLES:

sage: E = EllipticCurve("37a")
sage: P = E.gens()[0]
sage: h = E.padic_height(5, 10)
sage: h(P)
5 + 5^2 + 5^3 + 3*5^6 + 4*5^7 + 5^9 + O(5^10)

An anomalous case:

sage: h = E.padic_height(53, 10)
sage: h(P)
26*53^-1 + 30 + 20*53 + 47*53^2 + 10*53^3 + 32*53^4 + 9*53^5 + 22*53^6 + 35*53^7 + 30*53^8 + 17*53^9 + O(53^10)

Boundary case:

sage: E.padic_height(5, 3)(P)
5 + 5^2 + O(5^3)

A case that works the division polynomial code a little harder:

sage: E.padic_height(5, 10)(5*P)
5^3 + 5^4 + 5^5 + 3*5^8 + 4*5^9 + O(5^10)

Check that answers agree over a range of precisions:

sage: max_prec = 30    # make sure we get past p^2    # long time
sage: full = E.padic_height(5, max_prec)(P)           # long time
sage: for prec in range(1, max_prec):                 # long time
....:     assert E.padic_height(5, prec)(P) == full   # long time

A supersingular prime for a curve:

sage: E = EllipticCurve('37a')
sage: E.is_supersingular(3)
True
sage: h = E.padic_height(3, 5)
sage: h(E.gens()[0])
(3 + 3^3 + O(3^6), 2*3^2 + 3^3 + 3^4 + 3^5 + 2*3^6 + O(3^7))
sage: E.padic_regulator(5)
5 + 5^2 + 5^3 + 3*5^6 + 4*5^7 + 5^9 + 5^10 + 3*5^11 + 3*5^12 + 5^13 + 4*5^14 + 5^15 + 2*5^16 + 5^17 + 2*5^18 + 4*5^19 + O(5^20)
sage: E.padic_regulator(3, 5)
(3 + 2*3^2 + 3^3 + O(3^4), 3^2 + 2*3^3 + 3^4 + O(3^5))

A torsion point in both the good and supersingular cases:

sage: E = EllipticCurve('11a')
sage: P = E.torsion_subgroup().gen(0).element(); P
(5 : 5 : 1)
sage: h = E.padic_height(19, 5)
sage: h(P)
0
sage: h = E.padic_height(5, 5)
sage: h(P)
0

The result is not dependent on the model for the curve:

sage: E = EllipticCurve([0,0,0,0,2^12*17])
sage: Em = E.minimal_model()
sage: P = E.gens()[0]
sage: Pm = Em.gens()[0]
sage: h = E.padic_height(7)
sage: hm = Em.padic_height(7)
sage: h(P) == hm(Pm)
True
padic_height_pairing_matrix(p, prec=20, height=None, check_hypotheses=True)

Computes the cyclotomic \(p\)-adic height pairing matrix of this curve with respect to the basis self.gens() for the Mordell-Weil group for a given odd prime p of good ordinary reduction.

INPUT:

  • p - prime >= 5
  • prec - answer will be returned modulo \(p^{\mathrm{prec}}\)
  • height - precomputed height function. If not supplied, this function will call padic_height to compute it.
  • check_hypotheses - boolean, whether to check that this is a curve for which the p-adic height makes sense

OUTPUT: The p-adic cyclotomic height pairing matrix of this curve to the given precision.

Todo

remove restriction that curve must be in minimal Weierstrass form. This is currently required for E.gens().

AUTHORS:

  • David Harvey, Liang Xiao, Robert Bradshaw, Jennifer Balakrishnan: original implementation at the 2006 MSRI graduate workshop on modular forms
  • David Harvey (2006-09-13): cleaned up and integrated into Sage, removed some redundant height computations

EXAMPLES:

sage: E = EllipticCurve("37a")
sage: E.padic_height_pairing_matrix(5, 10)
[5 + 5^2 + 5^3 + 3*5^6 + 4*5^7 + 5^9 + O(5^10)]

A rank two example:

sage: e =EllipticCurve('389a')
sage: e._set_gens([e(-1, 1), e(1,0)])  # avoid platform dependent gens
sage: e.padic_height_pairing_matrix(5,10)
[                      3*5 + 2*5^2 + 5^4 + 5^5 + 5^7 + 4*5^9 + O(5^10) 5 + 4*5^2 + 5^3 + 2*5^4 + 3*5^5 + 4*5^6 + 5^7 + 5^8 + 2*5^9 + O(5^10)]
[5 + 4*5^2 + 5^3 + 2*5^4 + 3*5^5 + 4*5^6 + 5^7 + 5^8 + 2*5^9 + O(5^10)                         4*5 + 2*5^4 + 3*5^6 + 4*5^7 + 4*5^8 + O(5^10)]

An anomalous rank 3 example:

sage: e = EllipticCurve("5077a")
sage: e._set_gens([e(-1,3), e(2,0), e(4,6)])
sage: e.padic_height_pairing_matrix(5,4)
[4 + 3*5 + 4*5^2 + 4*5^3 + O(5^4)       4 + 4*5^2 + 2*5^3 + O(5^4)       3*5 + 4*5^2 + 5^3 + O(5^4)]
[      4 + 4*5^2 + 2*5^3 + O(5^4)   3 + 4*5 + 3*5^2 + 5^3 + O(5^4)                 2 + 4*5 + O(5^4)]
[      3*5 + 4*5^2 + 5^3 + O(5^4)                 2 + 4*5 + O(5^4)     1 + 3*5 + 5^2 + 5^3 + O(5^4)]
padic_height_via_multiply(p, prec=20, E2=None, check_hypotheses=True)

Computes the cyclotomic p-adic height.

The equation of the curve must be minimal at \(p\).

INPUT:

  • p - prime >= 5 for which the curve has good ordinary reduction
  • prec - integer >= 2 (default 20), desired precision of result
  • E2 - precomputed value of E2. If not supplied, this function will call padic_E2 to compute it. The value supplied must be correct mod \(p^(prec-2)\) (or slightly higher in the anomalous case; see the code for details).
  • check_hypotheses - boolean, whether to check that this is a curve for which the p-adic height makes sense

OUTPUT: A function that accepts two parameters:

  • a Q-rational point on the curve whose height should be computed
  • optional boolean flag ‘check’: if False, it skips some input checking, and returns the p-adic height of that point to the desired precision.
  • The normalization (sign and a factor 1/2 with respect to some other normalizations that appear in the literature) is chosen in such a way as to make the p-adic Birch Swinnerton-Dyer conjecture hold as stated in [Mazur-Tate-Teitelbaum].

AUTHORS:

  • David Harvey (2008-01): based on the padic_height() function, using the algorithm of”Computing p-adic heights via point multiplication”

EXAMPLES:

sage: E = EllipticCurve("37a")
sage: P = E.gens()[0]
sage: h = E.padic_height_via_multiply(5, 10)
sage: h(P)
5 + 5^2 + 5^3 + 3*5^6 + 4*5^7 + 5^9 + O(5^10)

An anomalous case:

sage: h = E.padic_height_via_multiply(53, 10)
sage: h(P)
26*53^-1 + 30 + 20*53 + 47*53^2 + 10*53^3 + 32*53^4 + 9*53^5 + 22*53^6 + 35*53^7 + 30*53^8 + 17*53^9 + O(53^10)

Supply the value of E2 manually:

sage: E2 = E.padic_E2(5, 8)
sage: E2
2 + 4*5 + 2*5^3 + 5^4 + 3*5^5 + 2*5^6 + O(5^8)
sage: h = E.padic_height_via_multiply(5, 10, E2=E2)
sage: h(P)
5 + 5^2 + 5^3 + 3*5^6 + 4*5^7 + 5^9 + O(5^10)

Boundary case:

sage: E.padic_height_via_multiply(5, 3)(P)
5 + 5^2 + O(5^3)

Check that answers agree over a range of precisions:

sage: max_prec = 30    # make sure we get past p^2    # long time
sage: full = E.padic_height(5, max_prec)(P)           # long time
sage: for prec in range(2, max_prec):                 # long time
....:     assert E.padic_height_via_multiply(5, prec)(P) == full   # long time
padic_lseries(p, normalize=None, implementation='eclib', precision=None)

Return the \(p\)-adic \(L\)-series of self at \(p\), which is an object whose approx method computes approximation to the true \(p\)-adic \(L\)-series to any desired precision.

INPUT:

  • p – prime
  • normalize – ‘L_ratio’ (default), ‘period’ or ‘none’; this is describes the way the modular symbols are normalized. See modular_symbol for more details.
  • implementation – ‘eclib’ (default), ‘sage’, ‘pollackstevens’; Whether to use John Cremona’s eclib, the Sage implementation, or Pollack-Stevens’ implementation of overconvergent modular symbols.

EXAMPLES:

sage: E = EllipticCurve('37a')
sage: L = E.padic_lseries(5); L
5-adic L-series of Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
sage: type(L)
<class 'sage.schemes.elliptic_curves.padic_lseries.pAdicLseriesOrdinary'>

We compute the \(3\)-adic \(L\)-series of two curves of rank \(0\) and in each case verify the interpolation property for their leading coefficient (i.e., value at 0):

sage: e = EllipticCurve('11a')
sage: ms = e.modular_symbol()
sage: [ms(1/11), ms(1/3), ms(0), ms(oo)]
[0, -3/10, 1/5, 0]
sage: ms(0)
1/5
sage: L = e.padic_lseries(3)
sage: P = L.series(5)
sage: P(0)
2 + 3 + 3^2 + 2*3^3 + 2*3^5 + 3^6 + O(3^7)
sage: alpha = L.alpha(9); alpha
2 + 3^2 + 2*3^3 + 2*3^4 + 2*3^6 + 3^8 + O(3^9)
sage: R.<x> = QQ[]
sage: f = x^2 - e.ap(3)*x + 3
sage: f(alpha)
O(3^9)
sage: r = e.lseries().L_ratio(); r
1/5
sage: (1 - alpha^(-1))^2 * r
2 + 3 + 3^2 + 2*3^3 + 2*3^5 + 3^6 + 3^7 + O(3^9)
sage: P(0)
2 + 3 + 3^2 + 2*3^3 + 2*3^5 + 3^6 + O(3^7)

Next consider the curve 37b:

sage: e = EllipticCurve('37b')
sage: L = e.padic_lseries(3)
sage: P = L.series(5)
sage: alpha = L.alpha(9); alpha
1 + 2*3 + 3^2 + 2*3^5 + 2*3^7 + 3^8 + O(3^9)
sage: r = e.lseries().L_ratio(); r
1/3
sage: (1 - alpha^(-1))^2 * r
3 + 3^2 + 2*3^4 + 2*3^5 + 2*3^6 + 3^7 + O(3^9)
sage: P(0)
3 + 3^2 + 2*3^4 + 2*3^5 + O(3^6)

We can use Sage modular symbols instead to compute the \(L\)-series:

sage: e = EllipticCurve('11a')
sage: L = e.padic_lseries(3, implementation = 'sage')
sage: L.series(5,prec=10)
2 + 3 + 3^2 + 2*3^3 + 2*3^5 + 3^6 + O(3^7) + (1 + 3 + 2*3^2 + 3^3 + O(3^4))*T + (1 + 2*3 + O(3^4))*T^2 + (3 + 2*3^2 + O(3^3))*T^3 + (2*3 + 3^2 + O(3^3))*T^4 + (2 + 2*3 + 2*3^2 + O(3^3))*T^5 + (1 + 3^2 + O(3^3))*T^6 + (2 + 3^2 + O(3^3))*T^7 + (2 + 2*3 + 2*3^2 + O(3^3))*T^8 + (2 + O(3^2))*T^9 + O(T^10)

Finally, we can use the overconvergent method of Pollack-Stevens.:

sage: e = EllipticCurve('11a')
sage: L = e.padic_lseries(3, implementation = 'pollackstevens', precision = 6)
sage: L.series(5)
2 + 3 + 3^2 + 2*3^3 + 2*3^5 + O(3^6) + (1 + 3 + 2*3^2 + 3^3 + O(3^4))*T + (1 + 2*3 + O(3^2))*T^2 + (3 + O(3^2))*T^3 + O(3^0)*T^4 + O(T^5)
sage: L[3]
3 + O(3^2)

Another example with a semistable prime.:

sage: E = EllipticCurve("11a1")
sage: L = E.padic_lseries(11, implementation = 'pollackstevens', precision=3)
sage: L[1]
10 + 3*11 + O(11^2)
sage: L[3]
O(11^0)
padic_regulator(p, prec=20, height=None, check_hypotheses=True)

Compute the cyclotomic \(p\)-adic regulator of this curve.

INPUT:

  • p – prime >= 5
  • prec – answer will be returned modulo \(p^{\mathrm{prec}}\)
  • height – precomputed height function. If not supplied, this function will call padic_height to compute it.
  • check_hypotheses – boolean, whether to check that this is a curve for which the p-adic height makes sense

OUTPUT: The p-adic cyclotomic regulator of this curve, to the requested precision.

If the rank is 0, we output 1.

Todo

Remove restriction that curve must be in minimal Weierstrass form. This is currently required for E.gens().

AUTHORS:

  • Liang Xiao: original implementation at the 2006 MSRI graduate workshop on modular forms
  • David Harvey (2006-09-13): cleaned up and integrated into Sage, removed some redundant height computations
  • Chris Wuthrich (2007-05-22): added multiplicative and supersingular cases
  • David Harvey (2007-09-20): fixed some precision loss that was occurring

EXAMPLES:

sage: E = EllipticCurve("37a")
sage: E.padic_regulator(5, 10)
5 + 5^2 + 5^3 + 3*5^6 + 4*5^7 + 5^9 + O(5^10)

An anomalous case:

sage: E.padic_regulator(53, 10)
26*53^-1 + 30 + 20*53 + 47*53^2 + 10*53^3 + 32*53^4 + 9*53^5 + 22*53^6 + 35*53^7 + 30*53^8 + O(53^9)

An anomalous case where the precision drops some:

sage: E = EllipticCurve("5077a")
sage: E.padic_regulator(5, 10)
5 + 5^2 + 4*5^3 + 2*5^4 + 2*5^5 + 2*5^6 + 4*5^7 + 2*5^8 + 5^9 + O(5^10)

Check that answers agree over a range of precisions:

sage: max_prec = 30    # make sure we get past p^2    # long time
sage: full = E.padic_regulator(5, max_prec)           # long time
sage: for prec in range(1, max_prec):                 # long time
....:     assert E.padic_regulator(5, prec) == full   # long time

A case where the generator belongs to the formal group already (trac ticket #3632):

sage: E = EllipticCurve([37,0])
sage: E.padic_regulator(5,10)
2*5^2 + 2*5^3 + 5^4 + 5^5 + 4*5^6 + 3*5^8 + 4*5^9 + O(5^10)

The result is not dependent on the model for the curve:

sage: E = EllipticCurve([0,0,0,0,2^12*17])
sage: Em = E.minimal_model()
sage: E.padic_regulator(7) == Em.padic_regulator(7)
True

Allow a Python int as input:

sage: E = EllipticCurve('37a')
sage: E.padic_regulator(int(5))
5 + 5^2 + 5^3 + 3*5^6 + 4*5^7 + 5^9 + 5^10 + 3*5^11 + 3*5^12 + 5^13 + 4*5^14 + 5^15 + 2*5^16 + 5^17 + 2*5^18 + 4*5^19 + O(5^20)
padic_sigma(p, N=20, E2=None, check=False, check_hypotheses=True)

Computes the p-adic sigma function with respect to the standard invariant differential \(dx/(2y + a_1 x + a_3)\), as defined by Mazur and Tate, as a power series in the usual uniformiser \(t\) at the origin.

The equation of the curve must be minimal at \(p\).

INPUT:

  • p - prime >= 5 for which the curve has good ordinary reduction
  • N - integer >= 1 (default 20), indicates precision of result; see OUTPUT section for description
  • E2 - precomputed value of E2. If not supplied, this function will call padic_E2 to compute it. The value supplied must be correct mod \(p^{N-2}\).
  • check - boolean, whether to perform a consistency check (i.e. verify that the computed sigma satisfies the defining
  • differential equation - note that this does NOT guarantee correctness of all the returned digits, but it comes pretty close :-))
  • check_hypotheses - boolean, whether to check that this is a curve for which the p-adic sigma function makes sense

OUTPUT: A power series \(t + \cdots\) with coefficients in \(\ZZ_p\).

The output series will be truncated at \(O(t^{N+1})\), and the coefficient of \(t^n\) for \(n \geq 1\) will be correct to precision \(O(p^{N-n+1})\).

In practice this means the following. If \(t_0 = p^k u\), where \(u\) is a \(p\)-adic unit with at least \(N\) digits of precision, and \(k \geq 1\), then the returned series may be used to compute \(\sigma(t_0)\) correctly modulo \(p^{N+k}\) (i.e. with \(N\) correct \(p\)-adic digits).

ALGORITHM: Described in “Efficient Computation of p-adic Heights” (David Harvey), which is basically an optimised version of the algorithm from “p-adic Heights and Log Convergence” (Mazur, Stein, Tate).

Running time is soft-\(O(N^2 \log p)\), plus whatever time is necessary to compute \(E_2\).

AUTHORS:

  • David Harvey (2006-09-12)
  • David Harvey (2007-02): rewrote

EXAMPLES:

sage: EllipticCurve([-1, 1/4]).padic_sigma(5, 10)
O(5^11) + (1 + O(5^10))*t + O(5^9)*t^2 + (3 + 2*5^2 + 3*5^3 + 3*5^6 + 4*5^7 + O(5^8))*t^3 + O(5^7)*t^4 + (2 + 4*5^2 + 4*5^3 + 5^4 + 5^5 + O(5^6))*t^5 + O(5^5)*t^6 + (2 + 2*5 + 5^2 + 4*5^3 + O(5^4))*t^7 + O(5^3)*t^8 + (1 + 2*5 + O(5^2))*t^9 + O(5)*t^10 + O(t^11)

Run it with a consistency check:

sage: EllipticCurve("37a").padic_sigma(5, 10, check=True)
O(5^11) + (1 + O(5^10))*t + O(5^9)*t^2 + (3 + 2*5^2 + 3*5^3 + 3*5^6 + 4*5^7 + O(5^8))*t^3 + (3 + 2*5 + 2*5^2 + 2*5^3 + 2*5^4 + 2*5^5 + 2*5^6 + O(5^7))*t^4 + (2 + 4*5^2 + 4*5^3 + 5^4 + 5^5 + O(5^6))*t^5 + (2 + 3*5 + 5^4 + O(5^5))*t^6 + (4 + 3*5 + 2*5^2 + O(5^4))*t^7 + (2 + 3*5 + 2*5^2 + O(5^3))*t^8 + (4*5 + O(5^2))*t^9 + (1 + O(5))*t^10 + O(t^11)

Boundary cases:

sage: EllipticCurve([1, 1, 1, 1, 1]).padic_sigma(5, 1)
 (1 + O(5))*t + O(t^2)
sage: EllipticCurve([1, 1, 1, 1, 1]).padic_sigma(5, 2)
 (1 + O(5^2))*t + (3 + O(5))*t^2 + O(t^3)

Supply your very own value of E2:

sage: X = EllipticCurve("37a")
sage: my_E2 = X.padic_E2(5, 8)
sage: my_E2 = my_E2 + 5**5    # oops!!!
sage: X.padic_sigma(5, 10, E2=my_E2)
O(5^11) + (1 + O(5^10))*t + O(5^9)*t^2 + (3 + 2*5^2 + 3*5^3 + 4*5^5 + 2*5^6 + 3*5^7 + O(5^8))*t^3 + (3 + 2*5 + 2*5^2 + 2*5^3 + 2*5^4 + 2*5^5 + 2*5^6 + O(5^7))*t^4 + (2 + 4*5^2 + 4*5^3 + 5^4 + 3*5^5 + O(5^6))*t^5 + (2 + 3*5 + 5^4 + O(5^5))*t^6 + (4 + 3*5 + 2*5^2 + O(5^4))*t^7 + (2 + 3*5 + 2*5^2 + O(5^3))*t^8 + (4*5 + O(5^2))*t^9 + (1 + O(5))*t^10 + O(t^11)

Check that sigma is “weight 1”.

sage: f = EllipticCurve([-1, 3]).padic_sigma(5, 10)
sage: g = EllipticCurve([-1*(2**4), 3*(2**6)]).padic_sigma(5, 10)
sage: t = f.parent().gen()
sage: f(2*t)/2
(1 + O(5^10))*t + (4 + 3*5 + 3*5^2 + 3*5^3 + 4*5^4 + 4*5^5 + 3*5^6 + 5^7 + O(5^8))*t^3 + (3 + 3*5^2 + 5^4 + 2*5^5 + O(5^6))*t^5 + (4 + 5 + 3*5^3 + O(5^4))*t^7 + (4 + 2*5 + O(5^2))*t^9 + O(5)*t^10 + O(t^11)
sage: g
O(5^11) + (1 + O(5^10))*t + O(5^9)*t^2 + (4 + 3*5 + 3*5^2 + 3*5^3 + 4*5^4 + 4*5^5 + 3*5^6 + 5^7 + O(5^8))*t^3 + O(5^7)*t^4 + (3 + 3*5^2 + 5^4 + 2*5^5 + O(5^6))*t^5 + O(5^5)*t^6 + (4 + 5 + 3*5^3 + O(5^4))*t^7 + O(5^3)*t^8 + (4 + 2*5 + O(5^2))*t^9 + O(5)*t^10 + O(t^11)
sage: f(2*t)/2 -g
O(t^11)

Test that it returns consistent results over a range of precision:

sage: max_N = 30   # get up to at least p^2         # long time
sage: E = EllipticCurve([1, 1, 1, 1, 1])            # long time
sage: p = 5                                         # long time
sage: E2 = E.padic_E2(5, max_N)                     # long time
sage: max_sigma = E.padic_sigma(p, max_N, E2=E2)    # long time
sage: for N in range(3, max_N):                     # long time
....:    sigma = E.padic_sigma(p, N, E2=E2)         # long time
....:    assert sigma == max_sigma
padic_sigma_truncated(p, N=20, lamb=0, E2=None, check_hypotheses=True)

Compute the p-adic sigma function with respect to the standard invariant differential \(dx/(2y + a_1 x + a_3)\), as defined by Mazur and Tate, as a power series in the usual uniformiser \(t\) at the origin.

The equation of the curve must be minimal at \(p\).

This function differs from padic_sigma() in the precision profile of the returned power series; see OUTPUT below.

INPUT:

  • p - prime >= 5 for which the curve has good ordinary reduction
  • N - integer >= 2 (default 20), indicates precision of result; see OUTPUT section for description
  • lamb - integer >= 0, see OUTPUT section for description
  • E2 - precomputed value of E2. If not supplied, this function will call padic_E2 to compute it. The value supplied must be correct mod \(p^{N-2}\).
  • check_hypotheses - boolean, whether to check that this is a curve for which the p-adic sigma function makes sense

OUTPUT: A power series \(t + \cdots\) with coefficients in \(\ZZ_p\).

The coefficient of \(t^j\) for \(j \geq 1\) will be correct to precision \(O(p^{N - 2 + (3 - j)(lamb + 1)})\).

ALGORITHM: Described in “Efficient Computation of p-adic Heights” (David Harvey, to appear in LMS JCM), which is basically an optimised version of the algorithm from “p-adic Heights and Log Convergence” (Mazur, Stein, Tate), and “Computing p-adic heights via point multiplication” (David Harvey, still draft form).

Running time is soft-\(O(N^2 \lambda^{-1} \log p)\), plus whatever time is necessary to compute \(E_2\).

AUTHOR:

  • David Harvey (2008-01): wrote based on previous padic_sigma function()

EXAMPLES:

sage: E = EllipticCurve([-1, 1/4])
sage: E.padic_sigma_truncated(5, 10)
O(5^11) + (1 + O(5^10))*t + O(5^9)*t^2 + (3 + 2*5^2 + 3*5^3 + 3*5^6 + 4*5^7 + O(5^8))*t^3 + O(5^7)*t^4 + (2 + 4*5^2 + 4*5^3 + 5^4 + 5^5 + O(5^6))*t^5 + O(5^5)*t^6 + (2 + 2*5 + 5^2 + 4*5^3 + O(5^4))*t^7 + O(5^3)*t^8 + (1 + 2*5 + O(5^2))*t^9 + O(5)*t^10 + O(t^11)

Note the precision of the \(t^3\) coefficient depends only on \(N\), not on lamb:

sage: E.padic_sigma_truncated(5, 10, lamb=2)
O(5^17) + (1 + O(5^14))*t + O(5^11)*t^2 + (3 + 2*5^2 + 3*5^3 + 3*5^6 + 4*5^7 + O(5^8))*t^3 + O(5^5)*t^4 + (2 + O(5^2))*t^5 + O(t^6)

Compare against plain padic_sigma() function over a dense range of N and lamb

sage: E = EllipticCurve([1, 2, 3, 4, 7])                            # long time
sage: E2 = E.padic_E2(5, 50)                                        # long time
sage: for N in range(2, 10):                                        # long time
....:    for lamb in range(10):                                     # long time
....:       correct = E.padic_sigma(5, N + 3*lamb, E2=E2)           # long time
....:       compare = E.padic_sigma_truncated(5, N=N, lamb=lamb, E2=E2)    # long time
....:       assert compare == correct                               # long time
pari_curve()

Return the PARI curve corresponding to this elliptic curve.

EXAMPLES:

sage: E = EllipticCurve([0, 0, 1, -1, 0])
sage: e = E.pari_curve()
sage: type(e)
<... 'cypari2.gen.Gen'>
sage: e.type()
't_VEC'
sage: e.ellan(10)
[1, -2, -3, 2, -2, 6, -1, 0, 6, 4]
sage: E = EllipticCurve(RationalField(), ['1/3', '2/3'])
sage: e = E.pari_curve()
sage: e[:5]
[0, 0, 0, 1/3, 2/3]

When doing certain computations, PARI caches the results:

sage: E = EllipticCurve('37a1')
sage: _ = E.__dict__.pop('_pari_curve', None)  # clear cached data
sage: Epari = E.pari_curve()
sage: Epari
[0, 0, 1, -1, 0, 0, -2, 1, -1, 48, -216, 37, 110592/37, Vecsmall([1]), [Vecsmall([64, 1])], [0, 0, 0, 0, 0, 0, 0, 0]]
sage: Epari.omega()
[2.99345864623196, -2.45138938198679*I]
sage: Epari
[0, 0, 1, -1, 0, 0, -2, 1, -1, 48, -216, 37, 110592/37, Vecsmall([1]), [Vecsmall([64, 1])], [[2.99345864623196, -2.45138938198679*I], 0, [0.837565435283323, 0.269594436405445, -1.10715987168877, 1.37675430809421, 1.94472530697209, 0.567970998877878]~, 0, 0, 0, 0, 0]]

This shows that the bug uncovered by trac ticket #4715 is fixed:

sage: Ep = EllipticCurve('903b3').pari_curve()

This still works, even when the curve coefficients are large (see trac ticket #13163):

sage: E = EllipticCurve([4382696457564794691603442338788106497, 28, 3992, 16777216, 298])
sage: E.pari_curve()
[4382696457564794691603442338788106497, 28, 3992, 16777216, 298, ...]
sage: E.minimal_model()
Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 - 7686423934083797390675981169229171907674183588326184511391146727143672423167091484392497987721106542488224058921302964259990799229848935835464702*x + 8202280443553761483773108648734271851215988504820214784899752662100459663011709992446860978259617135893103951840830254045837355547141096270521198994389833928471736723050112419004202643591202131091441454709193394358885 over Rational Field
pari_mincurve()

Return the PARI curve corresponding to a minimal model for this elliptic curve.

EXAMPLES:

sage: E = EllipticCurve(RationalField(), ['1/3', '2/3'])
sage: e = E.pari_mincurve()
sage: e[:5]
[0, 0, 0, 27, 486]
sage: E.conductor()
47232
sage: e.ellglobalred()
[47232, [1, 0, 0, 0], 2, [2, 7; 3, 2; 41, 1], [[7, 2, 0, 1], [2, -3, 0, 2], [1, 5, 0, 1]]]
period_lattice(embedding=None)

Return the period lattice of the elliptic curve with respect to the differential \(dx/(2y + a_1x + a_3)\).

INPUT:

  • embedding - ignored (for compatibility with the period_lattice function for elliptic_curve_number_field)

OUTPUT:

(period lattice) The PeriodLattice_ell object associated to this elliptic curve (with respect to the natural embedding of \(\QQ\) into \(\RR\)).

EXAMPLES:

sage: E = EllipticCurve('37a')
sage: E.period_lattice()
Period lattice associated to Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field

Search for points on a curve up to an input bound on the naive logarithmic height.

INPUT:

  • height_limit (float) - bound on naive height

  • verbose (bool) - (default: False)

    If True, report on the saturation process.

    If False, just return the result.

  • rank_bound (bool) - (default: None)

    If provided, stop saturating once we find this many independent nontorsion points.

OUTPUT: points (list) - list of independent points which generate the subgroup of the Mordell-Weil group generated by the points found and then saturated.

Warning

height_limit is logarithmic, so increasing by 1 will cause the running time to increase by a factor of approximately 4.5 (=exp(1.5)).

IMPLEMENTATION: Uses Michael Stoll’s ratpoints module in PARI/GP.

EXAMPLES:

sage: E = EllipticCurve('389a1')
sage: E.point_search(5, verbose=False)
[(-1 : 1 : 1), (0 : 0 : 1)]

Increasing the height_limit takes longer, but finds no more points:

sage: E.point_search(10, verbose=False)
[(-1 : 1 : 1), (0 : 0 : 1)]

In fact this curve has rank 2 so no more than 2 points will ever be output, but we are not using this fact.

sage: E.saturation(_)
([(-1 : 1 : 1), (0 : 0 : 1)], 1, 0.152460177943144)

What this shows is that if the rank is 2 then the points listed do generate the Mordell-Weil group (mod torsion). Finally,

sage: E.rank()
2

If we only need one independent generator:

sage: E.point_search(5, verbose=False, rank_bound=1)
[(-2 : 0 : 1)]
pollack_stevens_modular_symbol(sign=0, implementation='eclib')

Create the modular symbol attached to the elliptic curve, suitable for overconvergent calculations.

INPUT:

  • sign – +1 or -1 or 0 (default), in which case this it is the sum of the two
  • implementation – either ‘eclib’ (default) or ‘sage’. This determines classical modular symbols which implementation of the underlying classical modular symbols is used

EXAMPLES:

sage: E = EllipticCurve('113a1')
sage: symb = E.pollack_stevens_modular_symbol()
sage: symb
Modular symbol of level 113 with values in Sym^0 Q^2
sage: symb.values()
[-1/2, 1, -1, 0, 0, 1, 1, -1, 0, -1, 0, 0, 0, 1, -1, 0, 0, 0, 1, 0, 0]

sage: E = EllipticCurve([0,1])
sage: symb = E.pollack_stevens_modular_symbol(+1)
sage: symb.values()
[-1/6, 1/12, 0, 1/6, 1/12, 1/3, -1/12, 0, -1/6, -1/12, -1/4, -1/6, 1/12]
prove_BSD(E, verbosity=0, two_desc='mwrank', proof=None, secs_hi=5, return_BSD=False)

Attempts to prove the Birch and Swinnerton-Dyer conjectural formula for \(E\), returning a list of primes \(p\) for which this function fails to prove BSD(E,p). Here, BSD(E,p) is the statement: “the Birch and Swinnerton-Dyer formula holds up to a rational number coprime to \(p\).”

INPUT:

  • E - an elliptic curve

  • verbosity - int, how much information about the proof to print.

    • 0 - print nothing
    • 1 - print sketch of proof
    • 2 - print information about remaining primes
  • two_desc - string (default 'mwrank'), what to use for the two-descent. Options are 'mwrank', 'simon', 'sage'

  • proof - bool or None (default: None, see proof.elliptic_curve or sage.structure.proof). If False, this function just immediately returns the empty list.

  • secs_hi - maximum number of seconds to try to compute the Heegner index before switching over to trying to compute the Heegner index bound. (Rank 0 only!)

  • return_BSD - bool (default: False) whether to return an object which contains information to reconstruct a proof

NOTE:

When printing verbose output, phrases such as “by Mazur” are referring to the following list of papers:

REFERENCES:

[Cha]B. Cha. Vanishing of some cohomology goups and bounds for the Shafarevich-Tate groups of elliptic curves. J. Number Theory, 111:154- 178, 2005.
[Jetchev]D. Jetchev. Global divisibility of Heegner points and Tamagawa numbers. Compos. Math. 144 (2008), no. 4, 811–826.
[Kato]K. Kato. p-adic Hodge theory and values of zeta functions of modular forms. Astérisque, (295):ix, 117-290, 2004.
[Kolyvagin]V. A. Kolyvagin. On the structure of Shafarevich-Tate groups. Algebraic geometry, 94–121, Lecture Notes in Math., 1479, Springer, Berlin, 1991.
[LawsonWuthrich]T. Lawson and C. Wuthrich, Vanishing of some Galois cohomology groups for elliptic curves, Arxiv 1505.02940
[LumStein]A. Lum, W. Stein. Verification of the Birch and Swinnerton-Dyer Conjecture for Elliptic Curves with Complex Multiplication (unpublished)
[Mazur]B. Mazur. Modular curves and the Eisenstein ideal. Inst. Hautes Études Sci. Publ. Math. No. 47 (1977), 33–186 (1978).
[Rubin]K. Rubin. The “main conjectures” of Iwasawa theory for imaginary quadratic fields. Invent. Math. 103 (1991), no. 1, 25–68.
[SteinWuthrich]W. Stein and C. Wuthrich, Algorithms for the Arithmetic of Elliptic Curves using Iwasawa Theory Mathematics of Computation 82 (2013), 1757-1792.
[SteinEtAl]G. Grigorov, A. Jorza, S. Patrikis, W. Stein, C. Tarniţǎ. Computational verification of the Birch and Swinnerton-Dyer conjecture for individual elliptic curves. Math. Comp. 78 (2009), no. 268, 2397–2425.

EXAMPLES:

sage: EllipticCurve('11a').prove_BSD(verbosity=2)
p = 2: True by 2-descent
True for p not in {2, 5} by Kolyvagin.
Kolyvagin's bound for p = 5 applies by Lawson-Wuthrich
True for p = 5 by Kolyvagin bound
[]

sage: EllipticCurve('14a').prove_BSD(verbosity=2)
p = 2: True by 2-descent
True for p not in {2, 3} by Kolyvagin.
Kolyvagin's bound for p = 3 applies by Lawson-Wuthrich
True for p = 3 by Kolyvagin bound
[]

sage: E = EllipticCurve("20a1")
sage: E.prove_BSD(verbosity=2)
p = 2: True by 2-descent
True for p not in {2, 3} by Kolyvagin.
Kato further implies that #Sha[3] is trivial.
[]

sage: E = EllipticCurve("50b1")
sage: E.prove_BSD(verbosity=2)
p = 2: True by 2-descent
True for p not in {2, 3, 5} by Kolyvagin.
Kolyvagin's bound for p = 3 applies by Lawson-Wuthrich
Kolyvagin's bound for p = 5 applies by Lawson-Wuthrich
True for p = 3 by Kolyvagin bound
True for p = 5 by Kolyvagin bound
[]
sage: E.prove_BSD(two_desc='simon')
[]

A rank two curve:

sage: E = EllipticCurve('389a')

We know nothing with proof=True:

sage: E.prove_BSD()
Set of all prime numbers: 2, 3, 5, 7, ...

We (think we) know everything with proof=False:

sage: E.prove_BSD(proof=False)
[]

A curve of rank 0 and prime conductor:

sage: E = EllipticCurve('19a')
sage: E.prove_BSD(verbosity=2)
p = 2: True by 2-descent
True for p not in {2, 3} by Kolyvagin.
Kolyvagin's bound for p = 3 applies by Lawson-Wuthrich
True for p = 3 by Kolyvagin bound
[]

sage: E = EllipticCurve('37a')
sage: E.rank()
1
sage: E._EllipticCurve_rational_field__rank
(1, True)
sage: E.analytic_rank = lambda : 0
sage: E.prove_BSD()
Traceback (most recent call last):
...
RuntimeError: It seems that the rank conjecture does not hold for this curve (Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field)! This may be a counterexample to BSD, but is more likely a bug.

We test the consistency check for the 2-part of Sha:

sage: E = EllipticCurve('37a')
sage: S = E.sha(); S
Tate-Shafarevich group for the Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
sage: def foo(use_database):
....:  return 4
sage: S.an = foo
sage: E.prove_BSD()
Traceback (most recent call last):
...
RuntimeError: Apparent contradiction: 0 <= rank(sha[2]) <= 0, but ord_2(sha_an) = 2

An example with a Tamagawa number at 5:

sage: E = EllipticCurve('123a1')
sage: E.prove_BSD(verbosity=2)
p = 2: True by 2-descent
True for p not in {2, 5} by Kolyvagin.
Kolyvagin's bound for p = 5 applies by Lawson-Wuthrich
True for p = 5 by Kolyvagin bound
[]

A curve for which 3 divides the order of the Tate-Shafarevich group:

sage: E = EllipticCurve('681b')
sage: E.prove_BSD(verbosity=2)               # long time
p = 2: True by 2-descent...
True for p not in {2, 3} by Kolyvagin....
Remaining primes:
p = 3: irreducible, surjective, non-split multiplicative
    (0 <= ord_p <= 2)
    ord_p(#Sha_an) = 2
[3]

A curve for which we need to use heegner_index_bound:

sage: E = EllipticCurve('198b')
sage: E.prove_BSD(verbosity=1, secs_hi=1)
p = 2: True by 2-descent
True for p not in {2, 3} by Kolyvagin.
[3]

The return_BSD option gives an object with detailed information about the proof:

sage: E = EllipticCurve('26b')
sage: B = E.prove_BSD(return_BSD=True)
sage: B.two_tor_rk
0
sage: B.N
26
sage: B.gens
[]
sage: B.primes
[]
sage: B.heegner_indexes
{-23: 2}
sage: E = EllipticCurve('960d1')
sage: E.prove_BSD(verbosity=1)  # long time (4s on sage.math, 2011)
p = 2: True by 2-descent
True for p not in {2} by Kolyvagin.
[]
q_eigenform(prec)

Synonym for self.q_expansion(prec).

EXAMPLES:

sage: E = EllipticCurve('37a1')
sage: E.q_eigenform(10)
q - 2*q^2 - 3*q^3 + 2*q^4 - 2*q^5 + 6*q^6 - q^7 + 6*q^9 + O(q^10)
sage: E.q_eigenform(10) == E.q_expansion(10)
True
q_expansion(prec)

Return the \(q\)-expansion to precision prec of the newform attached to this elliptic curve.

INPUT:

  • prec - an integer

OUTPUT:

a power series (in the variable ‘q’)

Note

If you want the output to be a modular form and not just a \(q\)-expansion, use modular_form().

EXAMPLES:

sage: E = EllipticCurve('37a1')
sage: E.q_expansion(20)
q - 2*q^2 - 3*q^3 + 2*q^4 - 2*q^5 + 6*q^6 - q^7 + 6*q^9 + 4*q^10 - 5*q^11 - 6*q^12 - 2*q^13 + 2*q^14 + 6*q^15 - 4*q^16 - 12*q^18 + O(q^20)
quadratic_twist(D)

Return the global minimal model of the quadratic twist of this curve by D.

EXAMPLES:

sage: E = EllipticCurve('37a1')
sage: E7=E.quadratic_twist(7); E7
Elliptic Curve defined by y^2  = x^3 - 784*x + 5488 over Rational Field
sage: E7.conductor()
29008
sage: E7.quadratic_twist(7) == E
True
rank(use_database=True, verbose=False, only_use_mwrank=True, algorithm='mwrank_lib', proof=None)

Return the rank of this elliptic curve, assuming no conjectures.

If we fail to provably compute the rank, raises a RuntimeError exception.

INPUT:

  • use_database (bool) – (default: True), if True, try to look up the rank in the Cremona database.

  • verbose - (default: False), if specified changes the verbosity of mwrank computations.

  • algorithm - (default: ‘mwrank_lib’), one of:

    • 'mwrank_shell' - call mwrank shell command
    • 'mwrank_lib' - call mwrank c library
  • only_use_mwrank - (default: True) if False try using analytic rank methods first.

  • proof - bool or None (default: None, see proof.elliptic_curve or sage.structure.proof). Note that results obtained from databases are considered proof = True

OUTPUT: the rank of the elliptic curve as Integer

IMPLEMENTATION: Uses L-functions, mwrank, and databases.

EXAMPLES:

sage: EllipticCurve('11a').rank()
0
sage: EllipticCurve('37a').rank()
1
sage: EllipticCurve('389a').rank()
2
sage: EllipticCurve('5077a').rank()
3
sage: EllipticCurve([1, -1, 0, -79, 289]).rank()   # This will use the default proof behavior of True
4
sage: EllipticCurve([0, 0, 1, -79, 342]).rank(proof=False)
5
sage: EllipticCurve([0, 0, 1, -79, 342]).simon_two_descent()[0]  # long time (7s on sage.math, 2012)
5

Examples with denominators in defining equations:

sage: E = EllipticCurve([0, 0, 0, 0, -675/4])
sage: E.rank()
0
sage: E = EllipticCurve([0, 0, 1/2, 0, -1/5])
sage: E.rank()
1
sage: E.minimal_model().rank()
1

A large example where mwrank doesn’t determine the result with certainty:

sage: EllipticCurve([1,0,0,0,37455]).rank(proof=False)
0
sage: EllipticCurve([1,0,0,0,37455]).rank(proof=True)
Traceback (most recent call last):
...
RuntimeError: rank not provably correct (lower bound: 0)
rank_bound()

Upper bound on the rank of the curve, computed using 2-descent. In many cases, this is the actual rank of the curve. If the curve has no 2-torsion it is the same as the 2-selmer rank.

EXAMPLES: The following is the curve 960D1, which has rank 0, but Sha of order 4.

sage: E = EllipticCurve([0, -1, 0, -900, -10098])
sage: E.rank_bound()
0

It gives 0 instead of 2, because it knows Sha is nontrivial. In contrast, for the curve 571A, also with rank 0 and Sha of order 4, we get a worse bound:

sage: E = EllipticCurve([0, -1, 1, -929, -10595])
sage: E.rank_bound()
2
sage: E.rank(only_use_mwrank=False)   # uses L-function
0
real_components()

Return 1 if there is 1 real component and 2 if there are 2.

EXAMPLES:

sage: E = EllipticCurve('37a')
sage: E.real_components ()
2
sage: E = EllipticCurve('37b')
sage: E.real_components ()
2
sage: E = EllipticCurve('11a')
sage: E.real_components ()
1
reduction(p)

Return the reduction of the elliptic curve at a prime of good reduction.

Note

The actual reduction is done in self.change_ring(GF(p)); the reduction is performed after changing to a model which is minimal at p.

INPUT:

  • p – a (positive) prime number

OUTPUT: an elliptic curve over the finite field GF(p)

EXAMPLES:

sage: E = EllipticCurve('389a1')
sage: E.reduction(2)
Elliptic Curve defined by y^2 + y = x^3 + x^2 over Finite Field of size 2
sage: E.reduction(3)
Elliptic Curve defined by y^2 + y = x^3 + x^2 + x over Finite Field of size 3
sage: E.reduction(5)
Elliptic Curve defined by y^2 + y = x^3 + x^2 + 3*x over Finite Field of size 5
sage: E.reduction(38)
Traceback (most recent call last):
...
AttributeError: p must be prime.
sage: E.reduction(389)
Traceback (most recent call last):
...
AttributeError: The curve must have good reduction at p.
sage: E = EllipticCurve([5^4,5^6])
sage: E.reduction(5)
Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field of size 5
regulator(proof=None, precision=53, **kwds)

Return the regulator of this curve, which must be defined over \(\QQ\).

INPUT:

  • proof – bool or None (default: None, see proof.[tab] or sage.structure.proof). Note that results from databases are considered proof = True
  • precision – (int, default 53): the precision in bits of the result
  • **kwds – passed to gens() method

EXAMPLES:

sage: E = EllipticCurve([0, 0, 1, -1, 0])
sage: E.regulator()
0.0511114082399688
sage: EllipticCurve('11a').regulator()
1.00000000000000
sage: EllipticCurve('37a').regulator()
0.0511114082399688
sage: EllipticCurve('389a').regulator()
0.152460177943144
sage: EllipticCurve('5077a').regulator()
0.41714355875838...
sage: EllipticCurve([1, -1, 0, -79, 289]).regulator()
1.50434488827528
sage: EllipticCurve([0, 0, 1, -79, 342]).regulator(proof=False)  # long time (6s on sage.math, 2011)
14.790527570131...
root_number(p=None)

Return the root number of this elliptic curve.

This is 1 if the order of vanishing of the L-function L(E,s) at 1 is even, and -1 if it is odd.

INPUT:

  • \(p\) – optional, default (None); if given, return the local root number at \(p\)

EXAMPLES:

sage: EllipticCurve('11a1').root_number()
1
sage: EllipticCurve('37a1').root_number()
-1
sage: EllipticCurve('389a1').root_number()
1
sage: type(EllipticCurve('389a1').root_number())
<... 'sage.rings.integer.Integer'>

sage: E = EllipticCurve('100a1')
sage: E.root_number(2)
-1
sage: E.root_number(5)
1
sage: E.root_number(7)
1

The root number is cached:

sage: E.root_number(2) is E.root_number(2)
True
sage: E.root_number()
1
satisfies_heegner_hypothesis(D)

Returns True precisely when \(D\) is a fundamental discriminant that satisfies the Heegner hypothesis for this elliptic curve.

EXAMPLES:

sage: E = EllipticCurve('11a1')
sage: E.satisfies_heegner_hypothesis(-7)
True
sage: E.satisfies_heegner_hypothesis(-11)
False
saturation(points, verbose=False, max_prime=0, odd_primes_only=False)

Given a list of rational points on E, compute the saturation in E(Q) of the subgroup they generate.

INPUT:

  • points (list) - list of points on E
  • verbose (bool) - (default: False), if True, give verbose output
  • max_prime (int) - (default: 0), saturation is performed for all primes up to max_prime. If max_prime==0, perform saturation at all primes, i.e., compute the true saturation.
  • odd_primes_only (bool) - only do saturation at odd primes

OUTPUT:

  • saturation (list) - points that form a basis for the saturation
  • index (int) - the index of the group generated by points in their saturation
  • regulator (real with default precision) - regulator of saturated points.

ALGORITHM: Uses Cremona’s mwrank package. With max_prime=0, we call mwrank with successively larger prime bounds until the full saturation is provably found. The results of saturation at the previous primes is stored in each case, so this should be reasonably fast.

EXAMPLES:

sage: E = EllipticCurve('37a1')
sage: P=E(0,0)
sage: Q=5*P; Q
(1/4 : -5/8 : 1)
sage: E.saturation([Q])
([(0 : 0 : 1)], 5, 0.0511114082399688)
selmer_rank()

The rank of the 2-Selmer group of the curve.

EXAMPLES: The following is the curve 960D1, which has rank 0, but Sha of order 4.

sage: E = EllipticCurve([0, -1, 0, -900, -10098])
sage: E.selmer_rank()
3

Here the Selmer rank is equal to the 2-torsion rank (=1) plus the 2-rank of Sha (=2), and the rank itself is zero:

sage: E.rank()
0

In contrast, for the curve 571A, also with rank 0 and Sha of order 4, we get a worse bound:

sage: E = EllipticCurve([0, -1, 1, -929, -10595])
sage: E.selmer_rank()
2
sage: E.rank_bound()
2

To establish that the rank is in fact 0 in this case, we would need to carry out a higher descent:

sage: E.three_selmer_rank() # optional: magma
0

Or use the L-function to compute the analytic rank:

sage: E.rank(only_use_mwrank=False)
0
sha()

Return an object of class ‘sage.schemes.elliptic_curves.sha_tate.Sha’ attached to this elliptic curve.

This can be used in functions related to bounding the order of Sha (The Tate-Shafarevich group of the curve).

EXAMPLES:

sage: E = EllipticCurve('37a1')
sage: S=E.sha()
sage: S
Tate-Shafarevich group for the Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
sage: S.bound_kolyvagin()
([2], 1)
silverman_height_bound(algorithm='default')

Return the Silverman height bound. This is a positive real (floating point) number B such that for all points \(P\) on the curve over any number field, \(|h(P) - \hat{h}(P)| \leq B\), where \(h(P)\) is the naive logarithmic height of \(P\) and \(\hat{h}(P)\) is the canonical height.

INPUT:

  • algorithm

    • ‘default’ (default) – compute using a Python implementation in Sage
    • ‘mwrank’ – use a C++ implementation in the mwrank library

NOTES:

  • The CPS_height_bound is often better (i.e. smaller) than the Silverman bound, but it only applies for points over the base field, whereas the Silverman bound works over all number fields.
  • The Silverman bound is also fairly straightforward to compute over number fields, but isn’t implemented here.
  • Silverman’s paper is ‘The Difference Between the Weil Height and the Canonical Height on Elliptic Curves’, Math. Comp., Volume 55, Number 192, pages 723-743. We use a correction by Bremner with 0.973 replaced by 0.961, as explained in the source code to mwrank (htconst.cc).

EXAMPLES:

sage: E = EllipticCurve('37a1')
sage: E.silverman_height_bound()
4.825400758180918
sage: E.silverman_height_bound(algorithm='mwrank')
4.825400758180918
sage: E.CPS_height_bound()
0.16397076103046915
simon_two_descent(verbose=0, lim1=5, lim3=50, limtriv=3, maxprob=20, limbigprime=30, known_points=None)

Return lower and upper bounds on the rank of the Mordell-Weil group \(E(\QQ)\) and a list of points of infinite order.

INPUT:

  • self – an elliptic curve \(E\) over \(\QQ\)
  • verbose – 0, 1, 2, or 3 (default: 0), the verbosity level
  • lim1 – (default: 5) limit on trivial points on quartics
  • lim3 – (default: 50) limit on points on ELS quartics
  • limtriv – (default: 3) limit on trivial points on \(E\)
  • maxprob – (default: 20)
  • limbigprime - (default: 30) to distinguish between small
    and large prime numbers. Use probabilistic tests for large primes. If 0, don’t any probabilistic tests.
  • known_points – (default: None) list of known points on the curve

OUTPUT: a triple (lower, upper, list) consisting of

  • lower (integer) – lower bound on the rank
  • upper (integer) – upper bound on the rank
  • list – list of points of infinite order in \(E(\QQ)\)

The integer upper is in fact an upper bound on the dimension of the 2-Selmer group, hence on the dimension of \(E(\QQ)/2E(\QQ)\). It is equal to the dimension of the 2-Selmer group except possibly if \(E(\QQ)[2]\) has dimension 1. In that case, upper may exceed the dimension of the 2-Selmer group by an even number, due to the fact that the algorithm does not perform a second descent.

To obtain a list of generators, use E.gens().

IMPLEMENTATION: Uses Denis Simon’s PARI/GP scripts from http://www.math.unicaen.fr/~simon/

EXAMPLES:

We compute the ranks of the curves of lowest known conductor up to rank \(8\). Amazingly, each of these computations finishes almost instantly!

sage: E = EllipticCurve('11a1')
sage: E.simon_two_descent()
(0, 0, [])
sage: E = EllipticCurve('37a1')
sage: E.simon_two_descent()
(1, 1, [(0 : 0 : 1)])
sage: E = EllipticCurve('389a1')
sage: E._known_points = []  # clear cached points
sage: E.simon_two_descent()
(2, 2, [(1 : 0 : 1), (-11/9 : 28/27 : 1)])
sage: E = EllipticCurve('5077a1')
sage: E.simon_two_descent()
(3, 3, [(1 : 0 : 1), (2 : 0 : 1), (0 : 2 : 1)])

In this example Simon’s program does not find any points, though it does correctly compute the rank of the 2-Selmer group.

sage: E = EllipticCurve([1, -1, 0, -751055859, -7922219731979])
sage: E.simon_two_descent()
(1, 1, [])

The rest of these entries were taken from Tom Womack’s page http://tom.womack.net/maths/conductors.htm

sage: E = EllipticCurve([1, -1, 0, -79, 289])
sage: E.simon_two_descent()
(4, 4, [(6 : -1 : 1), (4 : 3 : 1), (5 : -2 : 1), (8 : 7 : 1)])
sage: E = EllipticCurve([0, 0, 1, -79, 342])
sage: E.simon_two_descent()  # long time (9s on sage.math, 2011)
(5, 5, [(5 : 8 : 1), (10 : 23 : 1), (3 : 11 : 1), (-3 : 23 : 1), (0 : 18 : 1)])
sage: E = EllipticCurve([1, 1, 0, -2582, 48720])
sage: r, s, G = E.simon_two_descent(); r,s
(6, 6)
sage: E = EllipticCurve([0, 0, 0, -10012, 346900])
sage: r, s, G = E.simon_two_descent(); r,s
(7, 7)
sage: E = EllipticCurve([0, 0, 1, -23737, 960366])
sage: r, s, G = E.simon_two_descent(); r,s
(8, 8)

Example from trac ticket #10832:

sage: E = EllipticCurve([1,0,0,-6664,86543])
sage: E.simon_two_descent()
(2, 3, [(-1/4 : 2377/8 : 1), (323/4 : 1891/8 : 1)])
sage: E.rank()
2
sage: E.gens()
[(-1/4 : 2377/8 : 1), (323/4 : 1891/8 : 1)]

Example where the lower bound is known to be 1 despite that the algorithm has not found any points of infinite order

sage: E = EllipticCurve([1, 1, 0, -23611790086, 1396491910863060])
sage: E.simon_two_descent()
(1, 2, [])
sage: E.rank()
1
sage: E.gens()     # uses mwrank
[(4311692542083/48594841 : -13035144436525227/338754636611 : 1)]

Example for trac ticket #5153:

sage: E = EllipticCurve([3,0])
sage: E.simon_two_descent()
(1, 2, [(1 : 2 : 1)])

The upper bound on the 2-Selmer rank returned by this method need not be sharp. In following example, the upper bound equals the actual 2-Selmer rank plus 2 (see trac ticket #10735):

sage: E = EllipticCurve('438e1')
sage: E.simon_two_descent()
(0, 3, [])
sage: E.selmer_rank()  # uses mwrank
1
supersingular_primes(B)

Return a list of all supersingular primes for this elliptic curve up to and possibly including B.

EXAMPLES:

sage: e = EllipticCurve('11a')
sage: e.aplist(20)
[-2, -1, 1, -2, 1, 4, -2, 0]
sage: e.supersingular_primes(1000)
[2, 19, 29, 199, 569, 809]
sage: e = EllipticCurve('27a')
sage: e.aplist(20)
[0, 0, 0, -1, 0, 5, 0, -7]
sage: e.supersingular_primes(97)
[2, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89]
sage: e.ordinary_primes(97)
[7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97]
sage: e.supersingular_primes(3)
[2]
sage: e.supersingular_primes(2)
[2]
sage: e.supersingular_primes(1)
[]
tamagawa_exponent(p)

The Tamagawa index of the elliptic curve at \(p\).

This is the index of the component group \(E(\QQ_p)/E^0(\QQ_p)\). It equals the Tamagawa number (as the component group is cyclic) except for types \(I_m^*\) (\(m\) even) when the group can be \(C_2 \times C_2\).

EXAMPLES:

sage: E = EllipticCurve('816a1')
sage: E.tamagawa_number(2)
4
sage: E.tamagawa_exponent(2)
2
sage: E.kodaira_symbol(2)
I2*
sage: E = EllipticCurve('200c4')
sage: E.kodaira_symbol(5)
I4*
sage: E.tamagawa_number(5)
4
sage: E.tamagawa_exponent(5)
2

See trac ticket #4715:

sage: E = EllipticCurve('117a3')
sage: E.tamagawa_exponent(13)
4
tamagawa_number(p)

The Tamagawa number of the elliptic curve at \(p\).

This is the order of the component group \(E(\QQ_p)/E^0(\QQ_p)\).

EXAMPLES:

sage: E = EllipticCurve('11a')
sage: E.tamagawa_number(11)
5
sage: E = EllipticCurve('37b')
sage: E.tamagawa_number(37)
3
tamagawa_number_old(p)

The Tamagawa number of the elliptic curve at \(p\).

This is the order of the component group \(E(\QQ_p)/E^0(\QQ_p)\).

EXAMPLES:

sage: E = EllipticCurve('11a')
sage: E.tamagawa_number_old(11)
5
sage: E = EllipticCurve('37b')
sage: E.tamagawa_number_old(37)
3
tamagawa_product()

Return the product of the Tamagawa numbers.

EXAMPLES:

sage: E = EllipticCurve('54a')
sage: E.tamagawa_product ()
3
tate_curve(p)

Create the Tate curve over the \(p\)-adics associated to this elliptic curve.

This Tate curve is a \(p\)-adic curve with split multiplicative reduction of the form \(y^2+xy=x^3+s_4 x+s_6\) which is isomorphic to the given curve over the algebraic closure of \(\QQ_p\). Its points over \(\QQ_p\) are isomorphic to \(\QQ_p^{\times}/q^{\ZZ}\) for a certain parameter \(q \in \ZZ_p\).

INPUT:

  • \(p\) – a prime where the curve has split multiplicative reduction

EXAMPLES:

sage: e = EllipticCurve('130a1')
sage: e.tate_curve(2)
2-adic Tate curve associated to the Elliptic Curve defined by y^2 + x*y + y = x^3 - 33*x + 68 over Rational Field

The input curve must have multiplicative reduction at the prime.

sage: e.tate_curve(3)
Traceback (most recent call last):
...
ValueError: The elliptic curve must have multiplicative reduction at 3

We compute with \(p=5\):

sage: T = e.tate_curve(5); T
5-adic Tate curve associated to the Elliptic Curve defined by y^2 + x*y + y = x^3 - 33*x + 68 over Rational Field

We find the Tate parameter \(q\):

sage: T.parameter(prec=5)
3*5^3 + 3*5^4 + 2*5^5 + 2*5^6 + 3*5^7 + O(5^8)

We compute the \(\mathcal{L}\)-invariant of the curve:

sage: T.L_invariant(prec=10)
5^3 + 4*5^4 + 2*5^5 + 2*5^6 + 2*5^7 + 3*5^8 + 5^9 + O(5^10)
three_selmer_rank(algorithm='UseSUnits')

Return the 3-selmer rank of this elliptic curve, computed using Magma.

INPUT:

  • algorithm - ‘Heuristic’ (which is usually much faster in large examples), ‘FindCubeRoots’, or ‘UseSUnits’ (default)

OUTPUT: nonnegative integer

EXAMPLES: A rank 0 curve:

sage: EllipticCurve('11a').three_selmer_rank()       # optional - magma
0

A rank 0 curve with rational 3-isogeny but no 3-torsion

sage: EllipticCurve('14a3').three_selmer_rank()      # optional - magma
0

A rank 0 curve with rational 3-torsion:

sage: EllipticCurve('14a1').three_selmer_rank()      # optional - magma
1

A rank 1 curve with rational 3-isogeny:

sage: EllipticCurve('91b').three_selmer_rank()       # optional - magma
2

A rank 0 curve with nontrivial 3-Sha. The Heuristic option makes this about twice as fast as without it.

sage: EllipticCurve('681b').three_selmer_rank(algorithm='Heuristic')   # long time (10 seconds); optional - magma
2
torsion_order()

Return the order of the torsion subgroup.

EXAMPLES:

sage: e = EllipticCurve('11a')
sage: e.torsion_order()
5
sage: type(e.torsion_order())
<... 'sage.rings.integer.Integer'>
sage: e = EllipticCurve([1,2,3,4,5])
sage: e.torsion_order()
1
sage: type(e.torsion_order())
<... 'sage.rings.integer.Integer'>
torsion_points()

Return the torsion points of this elliptic curve as a sorted list.

OUTPUT: A list of all the torsion points on this elliptic curve.

EXAMPLES:

sage: EllipticCurve('11a').torsion_points()
[(0 : 1 : 0), (5 : -6 : 1), (5 : 5 : 1), (16 : -61 : 1), (16 : 60 : 1)]
sage: EllipticCurve('37b').torsion_points()
[(0 : 1 : 0), (8 : -19 : 1), (8 : 18 : 1)]

Some curves with large torsion groups:

sage: E = EllipticCurve([-1386747, 368636886])
sage: T = E.torsion_subgroup(); T
Torsion Subgroup isomorphic to Z/8 + Z/2 associated to the
 Elliptic Curve defined by y^2 = x^3 - 1386747*x + 368636886 over
 Rational Field
sage: E.torsion_points()
[(-1293 : 0 : 1),
 (-933 : -29160 : 1),
 (-933 : 29160 : 1),
 (-285 : -27216 : 1),
 (-285 : 27216 : 1),
 (0 : 1 : 0),
 (147 : -12960 : 1),
 (147 : 12960 : 1),
 (282 : 0 : 1),
 (1011 : 0 : 1),
 (1227 : -22680 : 1),
 (1227 : 22680 : 1),
 (2307 : -97200 : 1),
 (2307 : 97200 : 1),
 (8787 : -816480 : 1),
 (8787 : 816480 : 1)]
sage: EllipticCurve('210b5').torsion_points()
[(-41/4 : 37/8 : 1),
 (-5 : -103 : 1),
 (-5 : 107 : 1),
 (0 : 1 : 0),
 (10 : -208 : 1),
 (10 : 197 : 1),
 (37 : -397 : 1),
 (37 : 359 : 1),
 (100 : -1153 : 1),
 (100 : 1052 : 1),
 (415 : -8713 : 1),
 (415 : 8297 : 1)]
sage: EllipticCurve('210e2').torsion_points()
[(-36 : 18 : 1),
 (-26 : -122 : 1),
 (-26 : 148 : 1),
 (-8 : -122 : 1),
 (-8 : 130 : 1),
 (0 : 1 : 0),
 (4 : -62 : 1),
 (4 : 58 : 1),
 (31/4 : -31/8 : 1),
 (28 : -14 : 1),
 (34 : -122 : 1),
 (34 : 88 : 1),
 (64 : -482 : 1),
 (64 : 418 : 1),
 (244 : -3902 : 1),
 (244 : 3658 : 1)]
torsion_subgroup()

Return the torsion subgroup of this elliptic curve.

OUTPUT: The EllipticCurveTorsionSubgroup instance associated to this elliptic curve.

Note

To see the torsion points as a list, use torsion_points().

EXAMPLES:

sage: EllipticCurve('11a').torsion_subgroup()
Torsion Subgroup isomorphic to Z/5 associated to the Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
sage: EllipticCurve('37b').torsion_subgroup()
Torsion Subgroup isomorphic to Z/3 associated to the Elliptic Curve defined by y^2 + y = x^3 + x^2 - 23*x - 50 over Rational Field
sage: e = EllipticCurve([-1386747,368636886]);e
Elliptic Curve defined by y^2  = x^3 - 1386747*x + 368636886 over Rational Field
sage: G = e.torsion_subgroup(); G
Torsion Subgroup isomorphic to Z/8 + Z/2 associated to the
 Elliptic Curve defined by y^2 = x^3 - 1386747*x + 368636886 over
 Rational Field
sage: G.0*3 + G.1
(1227 : 22680 : 1)
sage: G.1
(282 : 0 : 1)
sage: list(G)
[(0 : 1 : 0), (147 : 12960 : 1), (2307 : 97200 : 1), (-933 : 29160 : 1), (1011 : 0 : 1), (-933 : -29160 : 1), (2307 : -97200 : 1), (147 : -12960 : 1), (282 : 0 : 1), (8787 : 816480 : 1), (-285 : 27216 : 1), (1227 : 22680 : 1), (-1293 : 0 : 1), (1227 : -22680 : 1), (-285 : -27216 : 1), (8787 : -816480 : 1)]
two_descent(verbose=True, selmer_only=False, first_limit=20, second_limit=8, n_aux=-1, second_descent=1)

Compute 2-descent data for this curve.

INPUT:

  • verbose - (default: True) print what mwrank is doing. If False, no output is printed.
  • selmer_only - (default: False) selmer_only switch
  • first_limit - (default: 20) firstlim is bound on x+z second_limit- (default: 8) secondlim is bound on log max x,z , i.e. logarithmic
  • n_aux - (default: -1) n_aux only relevant for general 2-descent when 2-torsion trivial; n_aux=-1 causes default to be used (depends on method)
  • second_descent - (default: True) second_descent only relevant for descent via 2-isogeny

OUTPUT:

Return True if the descent succeeded, i.e. if the lower bound and the upper bound for the rank are the same. In this case, generators and the rank are cached. A return value of False indicates that either rational points were not found, or that Sha[2] is nontrivial and mwrank was unable to determine this for sure.

EXAMPLES:

sage: E = EllipticCurve('37a1')
sage: E.two_descent(verbose=False)
True
two_descent_simon(verbose=0, lim1=5, lim3=50, limtriv=3, maxprob=20, limbigprime=30, known_points=None)

Return lower and upper bounds on the rank of the Mordell-Weil group \(E(\QQ)\) and a list of points of infinite order.

INPUT:

  • self – an elliptic curve \(E\) over \(\QQ\)
  • verbose – 0, 1, 2, or 3 (default: 0), the verbosity level
  • lim1 – (default: 5) limit on trivial points on quartics
  • lim3 – (default: 50) limit on points on ELS quartics
  • limtriv – (default: 3) limit on trivial points on \(E\)
  • maxprob – (default: 20)
  • limbigprime - (default: 30) to distinguish between small
    and large prime numbers. Use probabilistic tests for large primes. If 0, don’t any probabilistic tests.
  • known_points – (default: None) list of known points on the curve

OUTPUT: a triple (lower, upper, list) consisting of

  • lower (integer) – lower bound on the rank
  • upper (integer) – upper bound on the rank
  • list – list of points of infinite order in \(E(\QQ)\)

The integer upper is in fact an upper bound on the dimension of the 2-Selmer group, hence on the dimension of \(E(\QQ)/2E(\QQ)\). It is equal to the dimension of the 2-Selmer group except possibly if \(E(\QQ)[2]\) has dimension 1. In that case, upper may exceed the dimension of the 2-Selmer group by an even number, due to the fact that the algorithm does not perform a second descent.

To obtain a list of generators, use E.gens().

IMPLEMENTATION: Uses Denis Simon’s PARI/GP scripts from http://www.math.unicaen.fr/~simon/

EXAMPLES:

We compute the ranks of the curves of lowest known conductor up to rank \(8\). Amazingly, each of these computations finishes almost instantly!

sage: E = EllipticCurve('11a1')
sage: E.simon_two_descent()
(0, 0, [])
sage: E = EllipticCurve('37a1')
sage: E.simon_two_descent()
(1, 1, [(0 : 0 : 1)])
sage: E = EllipticCurve('389a1')
sage: E._known_points = []  # clear cached points
sage: E.simon_two_descent()
(2, 2, [(1 : 0 : 1), (-11/9 : 28/27 : 1)])
sage: E = EllipticCurve('5077a1')
sage: E.simon_two_descent()
(3, 3, [(1 : 0 : 1), (2 : 0 : 1), (0 : 2 : 1)])

In this example Simon’s program does not find any points, though it does correctly compute the rank of the 2-Selmer group.

sage: E = EllipticCurve([1, -1, 0, -751055859, -7922219731979])
sage: E.simon_two_descent()
(1, 1, [])

The rest of these entries were taken from Tom Womack’s page http://tom.womack.net/maths/conductors.htm

sage: E = EllipticCurve([1, -1, 0, -79, 289])
sage: E.simon_two_descent()
(4, 4, [(6 : -1 : 1), (4 : 3 : 1), (5 : -2 : 1), (8 : 7 : 1)])
sage: E = EllipticCurve([0, 0, 1, -79, 342])
sage: E.simon_two_descent()  # long time (9s on sage.math, 2011)
(5, 5, [(5 : 8 : 1), (10 : 23 : 1), (3 : 11 : 1), (-3 : 23 : 1), (0 : 18 : 1)])
sage: E = EllipticCurve([1, 1, 0, -2582, 48720])
sage: r, s, G = E.simon_two_descent(); r,s
(6, 6)
sage: E = EllipticCurve([0, 0, 0, -10012, 346900])
sage: r, s, G = E.simon_two_descent(); r,s
(7, 7)
sage: E = EllipticCurve([0, 0, 1, -23737, 960366])
sage: r, s, G = E.simon_two_descent(); r,s
(8, 8)

Example from trac ticket #10832:

sage: E = EllipticCurve([1,0,0,-6664,86543])
sage: E.simon_two_descent()
(2, 3, [(-1/4 : 2377/8 : 1), (323/4 : 1891/8 : 1)])
sage: E.rank()
2
sage: E.gens()
[(-1/4 : 2377/8 : 1), (323/4 : 1891/8 : 1)]

Example where the lower bound is known to be 1 despite that the algorithm has not found any points of infinite order

sage: E = EllipticCurve([1, 1, 0, -23611790086, 1396491910863060])
sage: E.simon_two_descent()
(1, 2, [])
sage: E.rank()
1
sage: E.gens()     # uses mwrank
[(4311692542083/48594841 : -13035144436525227/338754636611 : 1)]

Example for trac ticket #5153:

sage: E = EllipticCurve([3,0])
sage: E.simon_two_descent()
(1, 2, [(1 : 2 : 1)])

The upper bound on the 2-Selmer rank returned by this method need not be sharp. In following example, the upper bound equals the actual 2-Selmer rank plus 2 (see trac ticket #10735):

sage: E = EllipticCurve('438e1')
sage: E.simon_two_descent()
(0, 3, [])
sage: E.selmer_rank()  # uses mwrank
1
sage.schemes.elliptic_curves.ell_rational_field.cremona_curves(conductors)

Return iterator over all known curves (in database) with conductor in the list of conductors.

EXAMPLES:

sage: [(E.label(), E.rank()) for E in cremona_curves(srange(35,40))]
[('35a1', 0),
('35a2', 0),
('35a3', 0),
('36a1', 0),
('36a2', 0),
('36a3', 0),
('36a4', 0),
('37a1', 1),
('37b1', 0),
('37b2', 0),
('37b3', 0),
('38a1', 0),
('38a2', 0),
('38a3', 0),
('38b1', 0),
('38b2', 0),
('39a1', 0),
('39a2', 0),
('39a3', 0),
('39a4', 0)]
sage.schemes.elliptic_curves.ell_rational_field.cremona_optimal_curves(conductors)

Return iterator over all known optimal curves (in database) with conductor in the list of conductors.

EXAMPLES:

sage: [(E.label(), E.rank()) for E in cremona_optimal_curves(srange(35,40))]
[('35a1', 0),
('36a1', 0),
('37a1', 1),
('37b1', 0),
('38a1', 0),
('38b1', 0),
('39a1', 0)]

There is one case – 990h3 – when the optimal curve isn’t labeled with a 1:

sage: [e.cremona_label() for e in cremona_optimal_curves([990])]
['990a1', '990b1', '990c1', '990d1', '990e1', '990f1', '990g1', '990h3', '990i1', '990j1', '990k1', '990l1']
sage.schemes.elliptic_curves.ell_rational_field.elliptic_curve_congruence_graph(curves)

Return the congruence graph for this set of elliptic curves.

INPUT:

  • curves – a list of elliptic curves

OUTPUT:

The graph with each curve as a vertex (labelled by its Cremona label) and an edge from \(E\) to \(F\) labelled \(p\) if and only if \(E\) is congruent to \(F\) mod \(p\)

EXAMPLES:

sage: from sage.schemes.elliptic_curves.ell_rational_field import elliptic_curve_congruence_graph
sage: curves = list(cremona_optimal_curves([11..30]))
sage: G = elliptic_curve_congruence_graph(curves)
sage: G
Graph on 12 vertices
sage.schemes.elliptic_curves.ell_rational_field.integral_points_with_bounded_mw_coeffs(E, mw_base, N, x_bound)

Return the set of integers \(x\) which are \(x\)-coordinates of points on the curve \(E\) which are linear combinations of the generators (basis and torsion points) with coefficients bounded by \(N\).

INPUT:

  • E - an elliptic curve
  • mw_base - a list of points on \(E\) (generators)
  • N - a positive integer (bound on coefficients)
  • x_bound - a positive real number (upper bound on size of x-coordinates)

OUTPUT:

(list) list of integral points on \(E\) which are linear combinations of the given points with coefficients bounded by \(N\) in absolute value.