Mordell Curves of High Rank - Examples found using PARI
The number indicated is a lower bound for the algebraic
rank
. The generators were checked to be independent ( either with
PARI
or
mwrank
). If the rank says 7(8) this means that I found 7 indenpendent
points, but the root number is 1 in the functional equation, so the rank
is expected to be even, and it would be at least 8.
The
Mordell Curves
are elliptic curves of the form y^2=x^3 + k
. Below I just give the appropriate parameter k
for each curve. These are just examples, they are not "best
possible" though. See
Womack's list
for the best possible k.
| RANK |
CURVE ( k )
|
GENERATORS
|
CONDUCTOR | DATE |
| 8 |
496837487681 |
[[-1478, 702573], [1732, 708543], [-5168, 599007],
[-7850, 114459], [7090, 923709], [2, 704867], [2060, 711041], [31210, 5558541]]] |
8886509609946041763279396 |
5/17/2003 |
| 7 |
157753592 |
[[502, 16860], [754, 24216], [2, 12560], [1138,
40392], [-526, 3496], [409, 15039], [1753, 74463]]] |
223975762100122176 |
5/16/2003 |
| 7 |
6920908856 |
[[-110, 83184], [-1658, 48612], [2, 83192], [710,
85316], [-55, 83191], [6182, 493132], [18985, 2617191]] |
431090814537533058624 |
5/16/2003 |
| 7 |
9892490513 |
[[2, 99461], [776, 101783], [-1384, 85097], [683,
101050], [23486, 3600637], [23627, 3633086], [66308, 17074825]] |
10569027803377860342252 |
5/16/2003 |
| 6 |
3193361 |
[[-128, 1047], [-146, 285], [79, 1920], [2, 1787],
[-40, 1769], [92, 1993]]] |
367111961147556 |
5/16/2003 |
|
|