curve #3008

y² = x³ + x² − 86722816844364885148516741654596x + 310838184737326323107371505278260668623533663780

a-invariants: [0, 1, 0, -86722816844364885148516741654596, 310838184737326323107371505278260668623533663780]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 13260845337120940680039952567380
discriminant (Δ): 2655243965401917160766341002763671876064796000909864343890276036071902494340665910826417478400
Faltings height: 17.0139
naive height: 232.2344
primes of bad reduction: 2, 3, 5, 7, 11, 17, 19, 23, 31, 43, 47, 53, 61, 79, 257, 631, 10861, 13709
regulator: 14385630.3769752887910293525879663204214280205065687288
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:33:05 UTC
last updated: 2026-09-26 20:33:05 UTC

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

(2585749904949138242424/477481, -1255426617235037848426105118454/329939371)
(29732884949891274720/5329, 10011435206474365021932726390/389017)
(36315257685071045/4, -4170892928546548081347855/8)
(8254810966628784, -396807588639680488547574)
(68372388845627144788172868/12590410849, -8615221114747317592207174493488487850/1412732230133743)
(5350566321656256, 1005178224707961592134)
(5131059697446577880767099669334093318367765273912/609942635079647122985874187347481, -6330713866726629037346818008487379445507679697071759726113592616950922010/15063763459416929189759159086893604838729733194371)
(5293514459761536, 10041504928525830200826)

Commentary

2-isogenous image of D-z6old678-094, rational kernel x=-2675901289983228. 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 20:33:06 UTC · history