curve #2998

y² + xy = x³ − 5522640347965041637165334795034270x + 13947222719120123321051443094950791427181459592900

a-invariants: [1, 0, 0, -5522640347965041637165334795034270, 13947222719120123321051443094950791427181459592900]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 61889652460509651794748110211690
discriminant (Δ): 10696002364522815410297854752573974822626731931872945518249774902061211125119163361090551144759763456000
Faltings height: 18.4591
naive height: 244.6961
primes of bad reduction: 2, 3, 5, 11, 13, 17, 19, 23, 29, 103, 269, 1063, 2461489, 923659129
regulator: 407967375.187311270560385614486490174950472955359581581
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:32:39 UTC
last updated: 2026-09-26 20:32:39 UTC

Witness: 8 independent points log in to add more points →

(61473486585611004/25, 77283175942284802749615882/125)
(157304891881646340540/49, -1972409301924710939573041242210/343)
(1619279926715429100, -2058381035784516976321998270)
(18332699889101287547280/73441, -2370833275762318224856675155072210/19902511)
(148193842085982440122380/923521, -50670582911989761687340779426191070/887503681)
(-95642736115144957730556780/1268998129, -64787979088070272072835183686480794570/45205520349367)
(4662144333423838780/9, -9962885593989017688498640090/27)
(2386751825253379514443461750300/5420734093009, -3634726157674499948702646643486778589638094210/12620807889845925223)

Commentary

2-isogenous image of D-z6old678-001, rational kernel x=37773199222238940. Source: https://dujella.github.io/tors/z6old678.html. Independent witness points transported by the degree-2 isogeny. Witnesses saturated by exact rational halves of published and transported relations; quadratic-character independence is certified.

last edited by Natanael Wildner Fraga at 2026-09-26 20:32:40 UTC · history