curve #5956

y² + y = x³ − 1328522356753416

a-invariants: [0, 0, 1, 0, -1328522356753416]
c-invariants: [0, 1147843316234951208]
rank (lower bound): ≥ 11
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 762467753834056830690424494294363
discriminant (Δ): -762467753834056830690424494294363
Faltings height: 4.9884
naive height: 83.1688
primes of bad reduction: 3, 20773, 255817138931
regulator: 22260056528.1118217825852394139906308031669550257528972788840
submitted by: jesper-petersen
submitted at: 2026-10-26 17:02:09 UTC
last updated: 2026-10-26 17:02:09 UTC

Witness: 11 independent points log in to add more points →

(442689/4, 41592115/8)
(138816, 36693944)
(642177/4, 424031571/8)
(306522, 165743711)
(650526, 523415439)
(1107888, 1165551687)
(1442412, 1731959928)
(1689846, 2196397295)
(7878657/64, 11865477889/512)
(4677816/25, 9033362136/125)
(5155746/25, 10783803456/125)

Commentary

Burkhardt quartic rational parametrization at (u,v,w)=(-1,-4,5), with homogeneous Burkhardt coordinates (-5,-4,24,-30,45); specialization t=506/755 of the corresponding generic rank-4 Mordell sextic. PARI/GP 2.17.4 ellrank bounds: [11,11].

last edited by jesper-petersen at 2026-10-26 17:02:10 UTC · history