curve #3013

y² + xy = x³ − 9838570603528332493896135471085980x + 130192516098079949159739773657150735222919929264400

a-invariants: [1, 0, 0, -9838570603528332493896135471085980, 130192516098079949159739773657150735222919929264400]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 1778372045405097567630514914990
discriminant (Δ): 53627881079994132777818047919012861197732643143663563850076779341830589978361477982298875528477184000000
Faltings height: 18.5962
naive height: 246.4285
primes of bad reduction: 2, 3, 5, 7, 11, 23, 43, 47, 71, 127, 191, 239, 359, 1223, 5881, 15583
regulator: 87300900.4056206373622225057964331284695448897811261185
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:33:18 UTC
last updated: 2026-09-26 20:33:18 UTC

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

(-27704998359007890, -19532144207274497684660430)
(91962779652885960, -1775809393498642586499180)
(-13043679945785640, -16009511602178088065673180)
(79709536522896745272, -711647764880967953005226198364)
(133162924839344481818190/326041, -47186175712246858147150599647299680/186169411)
(89649709477742633352/961, -146624548933986115178632674084/29791)
(9314642436058095580760/1681, -898834754570075044134250780874780/68921)
(56119614469952386461550230313584993806863494975729845712/4185254789639178937591551350782941578881, -223128374124914590114424379851524074640966368171815315210488124056152780800134841868/270758969877756342231168104328857519276414071693019244625471)

Commentary

2-isogenous image of D-z6old678-073, rational kernel x=52620481624914360. 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 20:33:18 UTC · history