curve #3165

y² + y = x³ + 44182596082121121317135170025680399046545625711306

a-invariants: [0, 0, 1, 0, 44182596082121121317135170025680399046545625711306]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/3ℤ
conductor (N): 4771720376869081102250598362773483097026927576821075
discriminant (Δ): -843307976112133532769138566443187160732218487040617714467468347345320317932872818745629810948243116875
Faltings height: 18.2367
naive height: 242.1480
primes of bad reduction: 3, 5, 7, 13, 19, 23, 31, 43, 59, 61, 73, 79, 103, 109, 157, 457
regulator: 404372782109.950291826338422910606269442708978244360624946463
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:40:22 UTC
last updated: 2026-09-26 20:40:22 UTC

Witness: 11 independent points log in to add more points →

(-30156002278649820, 4093799681127459731025817)
(11364087102067560, 6756491872572362690626342)
(-20835788771691894, 5927660006237675713476241)
(1134264920569989390, 1208031685828825118221478017)
(8907565209691176834, 26585114133655761890666064910)
(111849199886121334, 37992674604901443769570910)
(11724873521668020, 6767159346634715672034457)
(-138658831412368575/4, 12719819443574268333325811/8)
(165971060901522240, 67941788876402816577138982)
(994768217796990, 6647073075327662243966017)
(532896351059436225/16, 576457310785324883248677823/64)

Commentary

Elkies - Rogers (2004). Source: https://dujella.github.io/tors/z3old891011121314.html (curve entry 42). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:40:22 UTC · history