Elliptic Curves of High Rank - Examples found using PARI
The
rank
is at least the number indicated. The generators were checked to be
independent. 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.
I have written a number of small programs in PARI, which look for points
and check if those are algebraically dependent. Also, in some cases,
I've used Tom Womack
's implementation of the Buhler-Gross algorithm, in order to find
the analytic rank of the curve.
| RANK |
CURVE
|
GENERATORS
|
CONDUCTOR | DATE |
| 10 |
[1951/164, 3537/164, -3222367/40344, -40302641/121032,
0] |
[[0, 0], [5123/246, 25615/492], [-5123/246,
-5123/246], [5123/492, 0], [-5123/492, -5123/246], [5123/164, -97337/246],
[-231945/3362, 188376893/413526], [-67717/984, 77577637/161376], [-347755/5043,
96049045/206763], [-1384417/20172, 543844735/1102736]] |
7482540863500421918989079881386 |
4/22/02 |
| 9 |
[3678/223, 3093/223, 947/223, 7718/223,
4848/223] |
[[0, 3], [-1, 2731/223], [1, 3], [-2, 6409/223],
[2, 840/223], [12, 4200/223], [-43, 615], [218, 440920/223], [-72257/892,
268966529/397832]] |
2329064581968829298382579563 |
3/24/02 |
| 8 |
[0, 0, 0, -100558717, 434345056405] |
[[5213, -227591], [211, 642757], [10215,
-687775], [-4791, 897859], [15217, -1558123], [-9793, -692777], [2753, 422341],
[2507, 444973]] |
40033961883083511212 |
11/25/2002 |
| 7(8) |
[-10827/221, -3624/221, 24703/221,6364/221,
0] |
[[3, 7557/221], [1, 47/221], [2, 0], [4,
18384/221], [0, 0], [113,95937/17],[-587, -234213/17]]
|
43807832595856317121370249558 |
3/6/2002 |
| 7(8) |
[14038/391, 227/391, -35959/391, -2018/391,
0] |
[[3, 1], [2, 7883/391], [4, 1], [1, 56],
[0, 35959/391], [-25, 974],[-312, 158727/23]] |
1291170212956351737656972844029 |
3/6/2002 |
| 7 |
[67/19, -112/19, -271/19, 148/19, 0] | [[3, 51/19], [4, 1],
[1, 11], [2, 137/19], [0, 271/19], [17, 747/19], [-53/19,371/19]]
|
284482315765346 |
3/6/2002 |
| 7 |
[49/4, -83/20, -176/5, 43/10, 0] |
[[4, 1], [2, 107/10], [3, 1],
[0, 176/5], [1, 23], [-1, 189/4], [13/2,27/10]] |
1236567695180150 |
3/6/2002 |
| 7 |
[-419/17, -196/17, 907/17, 324/17, 0] |
[[1, 5/17], [3, 333/17], [4, 752/17], [2,
0], [0, 0], [-143, -1595],[72, 1888]] |
734117670580272042 |
3/6/2002 |
| 7 |
[-450/31, -486/31,
983/31, 905/31, 0] |
[[1, 25/31], [2, 1],
[3, 274/31], [-10, -18], [4, 724/31], [0, 0],[133, 82954/31]] |
1813093458883013087 |
3/3/2002 |
| 7 |
[-798/29, -351/29, 1747/29, 586/29, 0] |
[[2, 0], [3, 618/29], [1, 8/29], [4, 1416/29],
[0, 0], [-14/9, -17/27],[421/9, 1007453/783]] |
744749604744411406837 |
3/6/2002 |
| 7 |
[959/43, -92/43, -2551/43, 12/43, 0] |
[[1, 37], [3, 1], [4, 1], [2, 633/43], [0,
2551/43], [28, 1456/43],[-115, 74265/43]] |
1160222589260858879718 |
3/6/2002 |
| 7 |
[-2843/109, -1288/109, 6191/109,2140/109,
0] |
[[1, 31/109], [3, 2229/109], [2, 0], [4,
5072/109], [0, 0], [-124,-93744/109],[-1379/9, -41783/27]] |
890996480530111379317102 |
3/6/2002 |
| 7 |
[3998/121, 1/121,
-10297/121, -486/121,0] |
[[3, 1], [4, 1], [1,
52], [2, 2301/121], [0, 10297/121], [-123,40016/11],[-38/9, 732944/3267]] |
1847187572186547694922973 |
3/6/2002 |
| 7 |
[-5352/211, -2463/211,11621/211,4082/211,
0] |
[[1, 61/211], [2, 0], [3, 4224/211],
[4, 9576/211], [0, 0], [167,1063623/211],[333, 2424906/211]] |
37453199271849584688329639
|
3/6/2002 |
|
|