curve #5588

y² + y = x³ − 32569099722251

a-invariants: [0, 0, 1, 0, -32569099722251]
c-invariants: [0, 28139702160024648]
rank (lower bound): ≥ 10
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 152747460967379598834429006081
discriminant (Δ): -458242382902138796503287018243
Faltings height: 4.3703
naive height: 75.7519
primes of bad reduction: 3, 139, 2777, 337501001
regulator: 825494747.5152503954580245217412297504216342040799660024150
submitted by: jesper-petersen
submitted at: 2026-10-23 21:01:22 UTC
last updated: 2026-10-23 21:01:22 UTC

Witness: 10 independent points log in to add more points →

(31943, 155267)
(32077, 660313)
(32347, 1129816)
(131297/4, 13378851/8)
(34691, 3029904)
(43543, 7070227)
(117551, 39897099)
(199693, 89054182)
(208567, 95079628)
(1138873/4, 1214524583/8)

Commentary

Specialization t = -8/137 of a generic rank-4 Mordell family obtained from the Burkhardt quartic, with rational parametrization parameters (u,v,w) = (7,-4,2). Found by varying the Burkhardt parameters using the rational parametrization of Bruin–Nasserden, followed by a sixth-power-residue Mestre–Nagao sieve, 3-isogeny deduplication, and conductor-aware candidate selection. PARI/GP 2.17.4 establishes exact rank 10, with bounds [10,10] and 10 independent rational points.

last edited by jesper-petersen at 2026-10-23 21:01:22 UTC · history