curve #5591

y² + y = x³ − 52658646638478

a-invariants: [0, 0, 1, 0, -52658646638478]
c-invariants: [0, 45497070695644776]
rank (lower bound): ≥ 10
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 2362735866713806045837778481
discriminant (Δ): -1197907084423899665239753689867
Faltings height: 4.4504
naive height: 76.7128
primes of bad reduction: 3, 13, 1367, 2843, 320699
regulator: 34122641193.39840290443630434003517716510156862111651954537
submitted by: jesper-petersen
submitted at: 2026-10-23 21:28:10 UTC
last updated: 2026-10-23 21:28:10 UTC

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

(47840, 7538641)
(383162, 237066550)
(365950/9, 103051682/27)
(8922394/225, 10511372647/3375)
(117492128/529, 1270477454903/12167)
(485216329/9216, 8545049892037/884736)
(313639664/1681, 5531957045897/68921)
(1148839224/19321, 33711586832163/2685619)
(4914669890/121801, 153479258527534/42508549)
(8838386388/27889, 830232864807207/4657463)

Commentary

Specialization t = -31/409 of a generic rank-4 Mordell family obtained from the Burkhardt quartic, with affine parameters (a,b,c,d) = (11/5,74/5,-8,26) and generating parameters (u,v,w) = (4,6,-4). Found using a sixth-power-residue Mestre–Nagao sieve across Burkhardt surface parameters, with 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 21:28:10 UTC · history