curve #5594

y² + y = x³ − 521694011521261294

a-invariants: [0, 0, 1, 0, -521694011521261294]
c-invariants: [0, 450743625954369757800]
rank (lower bound): ≥ 11
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 131659388254401650311201474884075
discriminant (Δ): -117575125195887033769160697108351076875
Faltings height: 5.9839
naive height: 95.1149
primes of bad reduction: 3, 5, 7, 63092246291309
regulator: 1471397397895.85944939922347928621226750808182419741998392445
submitted by: jesper-petersen
submitted at: 2026-10-23 21:30:22 UTC
last updated: 2026-10-23 21:30:22 UTC

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

(832776, 236324686)
(1371454, 1434522470)
(2521036, 3937138446)
(3087924, 5377965169)
(13816761/4, 51032059655/8)
(160954809/4, 2041992754673/8)
(69690516/25, 574734025814/125)
(334298281/324, 4428955257913/5832)
(9333162636/10609, 435961576209415/1092727)
(22967096964/625, 3480622096632313/15625)
(1128383910574/383161, 1186325572334271890/237176659)

Commentary

Specialization t = 67/404 of a generic rank-4 Mordell family obtained from the Burkhardt quartic, with affine parameters (a,b,c,d) = (-13/5,56/5,-2,-32/5) and generating parameters (u,v,w) = (-2,6,-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 11, with bounds [11,11] and 11 independent rational points.

last edited by jesper-petersen at 2026-10-23 21:30:22 UTC · history