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# 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)

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