curve #3070

y² = x³ − x² + 39921087400x + 2783918068969116

a-invariants: [0, -1, 0, 39921087400, 2783918068969116]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ
conductor (N): 1115505417782881151832
discriminant (Δ): -7419924170119692834064048189344768
Faltings height: 5.1858
naive height: 84.8441
primes of bad reduction: 2, 3, 13, 19, 23, 29, 61, 89, 180241, 288313
regulator: 25806625.593764877805131272669363733051811811245901010005
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:35:50 UTC
last updated: 2026-09-26 20:35:50 UTC

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

(-60762, 11571288)
(481281, 365344686)
(1079398, 1141700552)
(-61154, 10671256)
(418273, 304400198)
(766558, 695574008)
(-22559, -43264966)
(-725495/16, -1899117731/64)
(223933, 151502008)

Commentary

2-isogenous to curve#1364 via the rational 2-torsion point x=31682. 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 20:35:50 UTC · history