# Tables of elliptic curves of given rank¶

The default database of curves contains the following data:

Rank Number of curves Maximal conductor
0 30427 9999
1 31871 9999
2 2388 9999
3 836 119888
4 10 1175648
5 5 37396136
6 5 6663562874
7 5 896913586322
8 6 457532830151317
9 7 ~9.612839e+21
10 6 ~1.971057e+21
11 6 ~1.803406e+24
12 1 ~2.696017e+29
14 1 ~3.627533e+37
15 1 ~1.640078e+56
17 1 ~2.750021e+56
19 1 ~1.373776e+65
20 1 ~7.381324e+73
21 1 ~2.611208e+85
22 1 ~2.272064e+79
23 1 ~1.139647e+89
24 1 ~3.257638e+95
28 1 ~3.455601e+141

Note that lists for r>=4 are not exhaustive; there may well be curves of the given rank with conductor less than the listed maximal conductor, which are not included in the tables.

AUTHORS: - William Stein (2007-10-07): initial version - Simon Spicer (2014-10-24): Added examples of more high-rank curves

See also the functions cremona_curves() and cremona_optimal_curves() which enable easy looping through the Cremona elliptic curve database.

class sage.schemes.elliptic_curves.ec_database.EllipticCurves
rank(rank, tors=0, n=10, labels=False)

Return a list of at most $$n$$ non-isogenous curves with given rank and torsion order.

INPUT:

• rank (int) – the desired rank
• tors (int, default 0) – the desired torsion order (ignored if 0)
• n (int, default 10) – the maximum number of curves returned.
• labels (bool, default False) – if True, return Cremona labels instead of curves.

OUTPUT:

(list) A list at most $$n$$ of elliptic curves of required rank.

EXAMPLES:

sage: elliptic_curves.rank(n=5, rank=3, tors=2, labels=True)
['59450i1', '59450i2', '61376c1', '61376c2', '65481c1']

sage: elliptic_curves.rank(n=5, rank=0, tors=5, labels=True)
['11a1', '11a3', '38b1', '50b1', '50b2']

sage: elliptic_curves.rank(n=5, rank=1, tors=7, labels=True)
['574i1', '4730k1', '6378c1']

sage: e = elliptic_curves.rank(6)[0]; e.ainvs(), e.conductor()
((1, 1, 0, -2582, 48720), 5187563742)
sage: e = elliptic_curves.rank(7)[0]; e.ainvs(), e.conductor()
((0, 0, 0, -10012, 346900), 382623908456)
sage: e = elliptic_curves.rank(8)[0]; e.ainvs(), e.conductor()
((1, -1, 0, -106384, 13075804), 249649566346838)


For large conductors, the labels are not known:

sage: L = elliptic_curves.rank(6, n=3); L
[Elliptic Curve defined by y^2 + x*y = x^3 + x^2 - 2582*x + 48720 over Rational Field,
Elliptic Curve defined by y^2 + y = x^3 - 7077*x + 235516 over Rational Field,
Elliptic Curve defined by y^2 + x*y = x^3 - x^2 - 2326*x + 43456 over Rational Field]
sage: L[0].cremona_label()
Traceback (most recent call last):
...
LookupError: Cremona database does not contain entry for Elliptic Curve defined by y^2 + x*y = x^3 + x^2 - 2582*x + 48720 over Rational Field
sage: elliptic_curves.rank(6, n=3, labels=True)
[]