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

Bases: object

rank(rank, tors=0, n=10, labels=False)

Return a list of at most \(n\) 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)
[]