# 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 1 234446
5 1 19047851
6 1 5187563742
7 1 382623908456
8 1 457532830151317

AUTHOR: - William Stein (2007-10-07): initial version

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()
((0, 0, 1, -23737, 960366), 457532830151317)


#### Previous topic

Heegner points on elliptic curves over the rational numbers

#### Next topic

Elliptic curves over number fields