curve #2982

y² + xy = x³ + 146742076645244304640470683176613999x + 171252811866641246597010492560493589669538011580337065

a-invariants: [1, 0, 0, 146742076645244304640470683176613999, 171252811866641246597010492560493589669538011580337065]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 57726875537107193277056539769346690
discriminant (Δ): -12871720290263634455969455343510682632604566330477917509582790341664816855244070837762797640831190318285946880
Faltings height: 19.6161
naive height: 258.6731
primes of bad reduction: 2, 3, 5, 7, 11, 17, 19, 23, 29, 53, 67, 167, 2333, 17257811, 4858141109
regulator: 3933080732.56272951717968151581607049424469672567597957
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:31:46 UTC
last updated: 2026-09-26 20:31:46 UTC

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

(20246611501315634474/49, 188628636373967199227078072237/343)
(67364116443960568085992714/132687361, 937972134948179671815826725756402730469/1528425711359)
(299848941253355186, -492150894549056720640386965)
(-3725859169263175334/9, 5369809936111231178308088857/27)
(772432896085059194, 863409113895994772051526011)
(2782658504078870117308187174/366301321, 147002906747320354044358067005539209194909/7010640982619)
(175144501334329855586/121, -2460149622899655462904680351791/1331)
(3201605622908104410994/3969, 225666961851886083984574947407005/250047)

Commentary

2-isogenous image of D-z6old678-026, rational kernel x=234046202231251658. 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 20:31:46 UTC · history