curve #2959

y² + xy + y = x³ − x² − 15083314391992894266752759738555371803887x + 1111149683752660581018463019579555583110801348696315616689399

a-invariants: [1, -1, 1, -15083314391992894266752759738555371803887, 1111149683752660581018463019579555583110801348696315616689399]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 2487091744114190027859845240035171290
discriminant (Δ): -313751154056900904331587271195142929622844082602615161034254118972326404596543980472870071663616277129979756544000000000000
Faltings height: 22.2028
naive height: 290.0441
primes of bad reduction: 2, 3, 5, 11, 13, 17, 23, 41, 59, 73, 83, 97, 103, 109, 127, 151, 1327, 7823, 155539
regulator: 228939734.934822854988518085377795657111981502541567107
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:30:16 UTC
last updated: 2026-09-26 20:30:16 UTC

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

(865841141442688857163/9, 20032288537164037358396295203842/27)
(11078899071388059922243/49, 1043718918559427129528228575705578/343)
(110201848120231385005197509448523/838171377289, -908227847977799168842665082304338827855666422402/767360144821493413)
(6487057577624565465212480563/199809, 522478626021906410934003600496379442151258/89314623)
(-60212137725683728549953/3721, -263824414981936285581175383152310674/226981)
(22618129532002434797386192203/138603529, -2824129351876003621827199558035363786076418/1631779346917)
(-5777706787156060429973/361, -7964874449543668830378924464638686/6859)
(-206556124286467882981211677413/21910216441, -3629612571757029358155790590052981145302731794/3243172147813261)

Commentary

2-isogenous image of D-z6old678-086, rational kernel x=1199793431050579939947/4. 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 20:30:16 UTC · history