curve #3075

y² = x³ − 78803407375401821998417x + 8487575620694310325351848033268176

a-invariants: [0, 0, 0, -78803407375401821998417, 8487575620694310325351848033268176]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 16300273795523604802511289760
discriminant (Δ): 198688254992514938558434815141197956652364392083054211035484812840000
Faltings height: 11.9656
naive height: 169.7773
primes of bad reduction: 2, 5, 11, 19, 29, 59, 61, 67, 101, 137, 1031, 21799, 224149
regulator: 529601194.34034894500780973395009675181712090765888456652
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:36:05 UTC
last updated: 2026-09-26 20:36:05 UTC

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

(1438433583657, -1694515960717754460)
(-1627573450223/81, -73128907881316064800/729)
(7557225083647297/16641, -552993110657235100586800/2146689)
(1563134255947516033/4485924, -1451454132323078947396246375/9501187032)
(2449751767018316313/293764, -3832128779105024149540507125/159220088)
(810076876920318705/4656964, -66054810815412967684956897/10049728312)
(1433980006512873/9409, 3817463789679010480860/912673)
(1035722897888536833/5579044, -210927309619329556997951655/13177701928)
(8705983598325952/47961, -134599784916582348384340/10503459)

Commentary

2-isogenous to curve#2093 via the rational 2-torsion point x=648064835168. 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 20:36:06 UTC · history