curve #2964

y² + xy = x³ − x² − 41395996089895542703288843593449012919x + 102417183132054092141880504360873611415315276934580577933

a-invariants: [1, -1, 0, -41395996089895542703288843593449012919, 102417183132054092141880504360873611415315276934580577933]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 77775552131973229100114648668568825370
discriminant (Δ): 8622234302599899825576507504372051098109700262434430191379942000308011224963304671601542617832701513342603626000
Faltings height: 20.3934
naive height: 271.4623
primes of bad reduction: 2, 3, 5, 7, 13, 19, 29, 31, 101, 167, 229, 397, 431, 1613, 7907, 142789, 461917
regulator: 5230589820.10840507419805257549556834099888755662275828
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:30:36 UTC
last updated: 2026-09-26 20:30:36 UTC

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

(19353325195586704474, 80932552835479910330648974925)
(75886220115783959346351381199/19647348561, 1063534845585928042467239571306011635962610/2753949200446809)
(6807626962769186263824233976357409/1601064496996416, 115930180567897533960158052532520184369967592228989/64063880441924680318464)
(24524543716861722824038866836514/26503747016041, 1099256988375361052245267746336211557611292599955/136446033809294939339)
(55550368404099364187822619797242/9608208283521, -223549235783061855562373144143207202523581778221/29782668906721162431)
(7151108077874057906230888882856854/2103384591254241, 95305741701978525771447628047661672453179141914555/96466835495162732337711)
(119018464818001018571206/24649, -15043457025823494530073509781823335/3869893)
(1050662633415068877306381100287/135511598161, -784853584611500388198108086141779691450529879/49884394003429159)

Commentary

2-isogenous image of D-z6old678-016, rational kernel x=-30462022090411145901/4. 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 20:30:36 UTC · history