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.
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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 ranktors(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) []
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