curve #3071

y² = x³ + x² + 53972314975x + 545444967133759

a-invariants: [0, 1, 0, 53972314975, 545444967133759]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ
conductor (N): 733406508076595441216
discriminant (Δ): -10190719808189110783371071777210368
Faltings height: 5.2150
naive height: 85.7488
primes of bad reduction: 2, 17, 89, 223, 257, 2503, 34897
regulator: 2225382.8597962923078481868897283023205775986800566415020
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:35:54 UTC
last updated: 2026-09-26 20:35:54 UTC

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

(78717, 72675644)
(167154, 119321083)
(305757, 213617324)
(191013/4, 454784057/8)
(302394, 210992873)
(111494947/121, -1214528504944/1331)
(6201579, -15454625008)
(50869739, -362822154768)
(2577005083/6241, -150545191776080/493039)

Commentary

2-isogenous to curve#1372 via the rational 2-torsion point x=5043. 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 20:35:54 UTC · history