curve #5532

y² = x³ + 15913951595784x

a-invariants: [0, 0, 0, 15913951595784, 0]
c-invariants: [-763869676597632, 0]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 389471519250989804116224 ☆ record for quartic twists, rank ≥ 8
discriminant (Δ): -257937254154827803026009084602688694419456
Faltings height: 6.6356
naive height: 102.8083
primes of bad reduction: 2, 3, 11, 17, 211, 433, 761
regulator: 91569081.2693439738484282805678373610493012542553387386
submitted by: Michael Rubinstein
submitted at: 2026-10-22 22:49:43 UTC
last updated: 2026-10-22 22:49:43 UTC

(271848225/4, 4489899653745/8)
(751393897/64, 21753392700491/512)
(809759553/49, 23704604901027/343)
(1356155025/49, 50457894632235/343)
(3047336224/169, 172288771764400/2197)
(40556401152/247009, 198609051184917696/122763473)
(709177878600/2401, 597272304232858680/117649)
(709177878600/121801, 723947358935299320/42508549)

Rank 8, proved: 8 independent points listed, matching the 2-descent upper bound. Torsion Z/2. Member of the family y^2 = x^3 + a x.

last edited by Michael Rubinstein at 2026-10-22 22:49:43 UTC · history