curve #3711

y² + xy + y = x³ − x² − 69859130374350575918888x + 7106943719495552619386843632295147

a-invariants: [1, -1, 1, -69859130374350575918888, 7106943719495552619386843632295147]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/6ℤ
conductor (N): 65902957246370790
discriminant (Δ): 2269159638517652028241474546319295218162667867820064043171840
Faltings height: 11.4912
naive height: 169.4159
primes of bad reduction: 2, 3, 5, 7, 13, 19, 37, 53, 101, 113, 127, 149
regulator: 64736.466076055482767657813583662272375728797564088
submitted by: Michael Rubinstein
submitted at: 2026-09-30 01:47:51 UTC
last updated: 2026-09-30 01:47:51 UTC

Witness: 6 independent points log in to add more points →

(152593944685, 3113797511961)
(-21921722998875/121, 156452483647300190171/1331)
(3163884318595/36, 8763699918162167143/216)
(18457879218597/121, 48778698207704027/1331)
(34394447011561/225, 606114969229344571/3375)
(80791543011933/529, 1038838519202111375/12167)

Commentary

Specialization at t = 2707062/16837 of the universal curve with a 6-torsion point, y^2 + txy + (t+2)y = x^3 (as searched in arXiv:2003.00077, Sec. 13), found by searching small-height t = a/b for small conductor, with points from a 2-descent (PARI ellrank).

last edited by Michael Rubinstein at 2026-09-30 01:47:51 UTC · history