curve #3078

y² = x³ − x² − 63259914160x + 5543489402376976

a-invariants: [0, -1, 0, -63259914160, 5543489402376976]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ
conductor (N): 72554931074823247344
discriminant (Δ): 2926515641564652838858448953869312
Faltings height: 5.1566
naive height: 86.2252
primes of bad reduction: 2, 3, 7, 47, 59, 71, 79, 139, 227, 239, 263
regulator: 4008772.0548969264540429412761373518095351495073651467537
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:36:15 UTC
last updated: 2026-09-26 20:36:15 UTC

Witness: 9 independent points log in to add more points →

(543941, 363414834)
(2781086901/5041, 133227380447714/357911)
(2929296, 4995583844)
(102212949/529, -281160683762/12167)
(100564, 14100528)
(-26021300/5329, 29759847934992/389017)
(-284564, 22402176)
(80880, 30920924)
(391788, -202231232)

Commentary

2-isogenous to curve#1246 via the rational 2-torsion point x=-360663. Independent witness points are the images of that curve's witnesses under 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 20:36:15 UTC · history