curve #2992

y² + xy = x³ − 108522197094174139587307940710359075x + 12172025558862407325217415271422830684021359195641857

a-invariants: [1, 0, 0, -108522197094174139587307940710359075, 12172025558862407325217415271422830684021359195641857]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 3236200724342912634084121282477610
discriminant (Δ): 17792340654479391758344901961217543732181119755783703740971135244704202046091794561714056709231614464000000
Faltings height: 19.1169
naive height: 253.6304
primes of bad reduction: 2, 5, 7, 19, 31, 37, 59, 127, 151, 163, 179, 181, 827, 1039, 7621, 54217
regulator: 235201498.679013008874681970223454604234327074754734719
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:32:21 UTC
last updated: 2026-09-26 20:32:21 UTC

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

(162141664482702488528084/130321, 63172696208493324537524820236503563/47045881)
(127683928492640830856/289, 1103696413388751454870644158049/4913)
(417061422988397414206/225, 8389940293661477614024704920359/3375)
(-292976632583584306, 137181464463385666783629513)
(294929844651866317754686/727609, -115750296510221044061813619248611979/620650477)
(6005527231598362898207806/5212089, 14163899325082280321706665421981304651/11899199187)
(7563938823219868567143601104526/6572337213649, -20018870208302140085973844946144088131178709631/16849218304131754393)
(72500225983722135520626895262/708070077961, 27522477908534608462653100692012795888936365/595819020431764709)

Commentary

2-isogenous image of D-z6old678-053, rational kernel x=-67332390755112274. 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 20:32:22 UTC · history