curve #5585

y² + y = x³ − 853489718247042

a-invariants: [0, 0, 1, 0, -853489718247042]
c-invariants: [0, 737415116565444072]
rank (lower bound): ≥ 11
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 6422206327230105208457479480347
discriminant (Δ): -314688110034275155214416494537003
Faltings height: 4.9146
naive height: 82.2839
primes of bad reduction: 3, 7, 69672630060983
regulator: 712152600065.339881260184878103204424225167968586779112935344
submitted by: jesper-petersen
submitted at: 2026-10-23 20:55:23 UTC
last updated: 2026-10-23 20:55:23 UTC

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

(95484, 4129946)
(125244, 33333093)
(375858, 228568770)
(1225722, 1356710877)
(7755174, 21596676461)
(11980794/25, 41308358697/125)
(147291081/64, 1787514072571/512)
(118685436/289, 1285001750394/4913)
(3179254932/1681, 179250606250237/68921)
(21490187992/31329, 3146192923859317/5545233)
(4684002947694/1062961, 10137329749327540203/1095912791)

Commentary

Specialization t = -93/13 of a generic rank-4 Mordell family obtained from the Burkhardt quartic, with affine parameters (a,b,c,d) = (-10,-32,-9,-4). Found by varying the Burkhardt parameters using the rational parametrization of Bruin–Nasserden, followed by a sixth-power-residue Mestre–Nagao sieve and conductor-aware candidate selection. PARI/GP 2.17.4 establishes exact rank 11, with bounds [11,11] and 11 independent rational points.

last edited by jesper-petersen at 2026-10-23 20:55:23 UTC · history