curve #734

y² + xy = x³ − 32247676839468690449257990x + 70605626370702287593291208184391240292

a-invariants: [1, 0, 0, -32247676839468690449257990, 70605626370702287593291208184391240292]
rank (lower bound): ≥ 18
torsion subgroup: trivial
conductor (N): 642884886366548148202012413927085430843957817030186813570
naive height: 187.8235
Faltings height: 13.4454
discriminant (Δ): -7361616441155997504222964414994197990171646052840844529598289788910080000000
primes of bad reduction: 2, 3, 5, 23, 41, 263, 16879, 46589, 547753, 51093698777909, 3926108614450504543793
regulator: 7994226767220038.8941231455108593818043346732675696419148006430438744235703
submitted by: Valery Asiryan
submitted at: 2026-09-16 17:19:35 UTC
last updated: 2026-09-16 17:19:35 UTC

Witness: 18 independent points log in to add more points →

(-4702387039396, -10874987972282036422)
(50149499482296821/21904, -9603302947727125786776379/3241792)
(2540581049174, -2253021992756651812)
(3139493965274, -555355557032431762)
(3237204281204, -370897539700387582)
(3293257404404, -350565505767537022)
(3293379750764, -350616135775315222)
(3337657734074, -394069144284338962)
(1001275123524386/289, -3359449989091188795056/4913)
(15109077930371/4, -13122583426646344421/8)
(4221242711084, -3114197866199302102)
(362716079587948/49, -5286094478696037899066/343)
(73516283211916/9, -506726620551246448274/27)
(39903955185311/4, -217875906904167853631/8)
(276956228354924, -4608144921260762687062)
(942259491950924, -28923305169622702431862)
(3268868141364524, -186894040249292665410262)
(3350906886958, -415428978238830044)

Commentary

Specialization T=1/4 of the family identified by wgxli in the commentary to curve #302. Starting from 17 reconstructed generic sections, minimized quartic searches produced a set of 18 points certified independent by exact quadratic-character certificates.

last edited by Valery Asiryan at 2026-09-16 17:19:35 UTC · history