curve #2968

y² + xy = x³ + 8841661702319829376066200695956010610x + 16461748163381777180547691579850244647369719638524060100

a-invariants: [1, 0, 0, 8841661702319829376066200695956010610, 16461748163381777180547691579850244647369719638524060100]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 16294472568573972597369514526808180930
discriminant (Δ): -161303905370946951287855010660150719860320248928516361633174611351098937666689992135584958145318366186083047000000
Faltings height: 20.4068
naive height: 267.8044
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 23, 41, 89, 127, 167, 257, 457, 1459, 1709, 4603, 13302293
regulator: 223989878.968200193584283163279879806885307372889028653
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:30:51 UTC
last updated: 2026-09-26 20:30:51 UTC

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

(14795452455216380138160/4489, -2713669942071976282334258376064470/300763)
(11609560813282984040976/3481, 1871615497946263765499486642279958/205379)
(405600337069486380, -4484936466204128690208299190)
(-3581659666796422474185972120/2837373289, -274113493576416341270960190698974820993370/151138362985163)
(122660630226193880, -4189047322128548297631402190)
(196764991383625507505790/3721, -87424140908395577993918191610149890/226981)
(3570839345837157351570/24649, -16302154726469824365940328082993570/3869893)
(-1497560222835004702680/1849, -235444377215651261521468480377930/79507)

Commentary

2-isogenous image of D-z6old678-069, rational kernel x=11909531944555196831/4. 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 20:30:51 UTC · history