curve #3126

y² + xy = x³ − 101634854745305789909593544356506871024975808975x + 12471286206505690638915431397948987770853556352343302542131897789660000

a-invariants: [1, 0, 0, -101634854745305789909593544356506871024975808975, 12471286206505690638915431397948987770853556352343302542131897789660000]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/4ℤ
conductor (N): 14131461780078584529422651011842782186617799735577501195
discriminant (Δ): 270356980638634919766539697028818833193340216949365998398398898998347077900500371667194279053453810888366979612519575115646868268687890625
Faltings height: 25.6135
naive height: 336.3268
primes of bad reduction: 3, 5, 7, 11, 29, 53, 103, 181, 193, 503, 641, 647, 1447, 1559, 2251, 2663, 7949, 14341, 298819, 23023237
regulator: 17620138000532.7086603324218320728263024518226025506298949561
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:38:31 UTC
last updated: 2026-09-26 20:38:31 UTC

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

(184273813157271150694825, -5800120553079609944828465423975)
(13540059657277308942891228095200/64947481, 9703933638564341763425152122984447169226416600/523411749379)
(10439427866750933876145376/25, 25873516969122592126593637840797101384/125)
(96164102596179060057910300/529, 20443284097289870153353460052539737550/12167)
(147695786024769367617907, 26116833670524048667487732980486591)
(601720560802859926798220117431825/2476953361, 5672638138756500060760467950762479369890380352350/123275491823609)
(486520786037421216051947965020025/2637130609, 37385262874649760446614058889165133907848671425/135424568163977)
(62843966891152815754954481851598764/318758480569, -1771182795506624093112488914428586373960777469699694/179966894269010003)
(1843846985114411609283367250604129989425/10030607985443929, 79753941216730576626745820381367024922370814619338633050/1004594709209350941954467)
(10450311344654113796382091700219599/23850169225, -840415618470099135217423352461674911178511257497327/3683300884262875)
(1578053822431536957304631565513264818537200/3361290514949472121, -1609750667644382187352038513122797303530701995993489381708885100/6162530404175917497955445581)

Commentary

2-isogenous image of D-z2z2old1011121314-049, rational kernel x=-1474189362843341123513601/4. Source: https://dujella.github.io/tors/z2z2old1011121314.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 20:38:31 UTC · history