curve #3115

y² + xy + y = x³ − x² − 39974143692225071353019320368277463579304047x − 93485599610388421872677319402859407612112625383551208581674399881

a-invariants: [1, -1, 1, -39974143692225071353019320368277463579304047, -93485599610388421872677319402859407612112625383551208581674399881]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 2329609314357493409271842921820896872147183012290
discriminant (Δ): 312573307088884863253575885734840703026547488517541812958733508484047136504307334765262472281788273863176812166584057334480896000000
Faltings height: 23.9944
naive height: 312.8040
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 61, 71, 79, 199, 601, 619, 859, 1123, 7207, 191413, 3560785447
regulator: 3786698286834.34068489090647325297972506327160175878010871107
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:37:59 UTC
last updated: 2026-09-26 20:37:59 UTC

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

(-3820960803782437245293, -58898189468648535408490003509594)
(-4036260076497007842573, -45870816456455952722736719565914)
(7351306746245758740627, -99646384490227694728068088250714)
(-3717081800981354887173, -61186346688090184471896574410914)
(-5780806276364674591550237/1369, -215618147625050560212265599905169842/50653)
(1406786692009528931645727, -1668545229120370129145845825488613814)
(5035043284564019992315172227153814061/91542261043984, -11219997428118984006861610980366224289045202116974240441/875855482033320883648)
(-3414513869223011776181, -56543573819851229319322226977242)
(188573855300136391141742271925454263441773/25552860783533835049, -14964847423934676186475483967531732524308926763298016870825752/129169296205645829232612299093)
(-7833415626001196166442727853/2343961, -189246719535780898498030973258581129068374/3588604291)
(55017757660089931182454869601258473/575569547569, -12875938673891333736830894226118755977180499851016658/436663319667340247)

Commentary

2-isogenous image of D-z4old9101112-007, rational kernel x=2111993798863171867827. 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 20:37:59 UTC · history