curve #3118

y² + xy = x³ − 1459970401640676352579376906257934718034044x + 320987301586854733330789085776409688960966170493687125337640464

a-invariants: [1, 0, 0, -1459970401640676352579376906257934718034044, 320987301586854733330789085776409688960966170493687125337640464]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 5300423313342790652288129860235660003742350867406
discriminant (Δ): 154654400366966794418098748571230938665625480794206374985557209664330968501993800518362480955123419359380212703337968300489703424
Faltings height: 23.2926
naive height: 302.8746
primes of bad reduction: 2, 3, 7, 13, 29, 31, 41, 59, 61, 79, 163, 601, 607, 1231, 1327, 2652103, 4368649, 823091551
regulator: 9397427525292.03567514102025986377346817220489321033811411609
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:38:09 UTC
last updated: 2026-09-26 20:38:09 UTC

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

(2867062906556364017179136376/744769, 146212873126244780502580002302709651500604/642735647)
(646728688219721393656164243594/586366225, 102817264565527037323299906645417597907083498/14198858138375)
(5480996156873503165116386424/95277121, -14322909273385233997766737694284872369842748/929999978081)
(186719146294659345200877020071980/26863489, -2551427740006342224885349948609991970048439339956/139233463487)
(42338629215192051833116629048/33953929, -4146915051642691449535654394846912441142228/197849544283)
(285663722724101353664903955761269700664/1960476195880225, 4828005182116638655191130930366095643474961334920503440708/86804524051180630584625)
(8900241949697427913200280/6561, -15373022865136613195758411448321048348/531441)
(777687582068898806019580399229489646/437697990240289, -528904881948689787115700525780717579726481043738228686/9157181637448812156337)
(-388095910742608519409736970989649607437694/509874778343733999849, -362485932827031028831874780675472761489082762924129327367255594/11513179992111684147547796549493)
(82021428715354816101559409804784401009106/53101966424431694449, -16192413635241542316214493051244144812711126820189640717596048/386959849682067640134596471543)
(4977497474745332074646730135237383243185464/4427087761620405282025, -2957076939130131937439112705697247261710021652513715012032609772/294562323674673506804521139915125)

Commentary

2-isogenous image of D-z4old9101112-013, rational kernel x=653020238752531025784. Source: https://dujella.github.io/tors/z4old9101112.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 20:38:09 UTC · history