curve #2763

y² + xy = x³ − 1382347325962111883164x − 19779238137137765055957749004016

a-invariants: [1, 0, 0, -1382347325962111883164, -19779238137137765055957749004016]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 469844835723354
discriminant (Δ): 50067399773461753067331094073953280187273881291103817331441664
Faltings height: 10.8454
naive height: 157.6478
primes of bad reduction: 2, 3, 7, 13, 23, 29, 31, 41, 1015057
regulator: 5728.686979714444768124772903659081243860046478
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:21:19 UTC
last updated: 2026-09-26 20:21:19 UTC

Witness: 4 independent points log in to add more points →

(118707447194, 38586037214605622)
(-3537014736635464/165649, 3092613327236321714812/67419143)
(-107701733042428/5041, 17085040151071940824/357911)
(1590229798882424, -63414684531373768759516)

Commentary

2-isogenous to curve#1638 via the rational 2-torsion point x=10839389624. 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 20:21:19 UTC · history