curve #5590

y² + y = x³ − 20326407091108

a-invariants: [0, 0, 1, 0, -20326407091108]
c-invariants: [0, 17562015726717096]
rank (lower bound): ≥ 10
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 244837229767961732153553843 ☆ record for Mordell curves, rank ≥ 10
discriminant (Δ): -178486340500844102739940751547
Faltings height: 4.2917
naive height: 74.8090
primes of bad reduction: 3, 283, 10640705191
regulator: 213623646.9386728417571897896031218446502518865636413635647
submitted by: jesper-petersen
submitted at: 2026-10-23 21:22:47 UTC
last updated: 2026-10-23 21:22:47 UTC

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

(113241/4, 12298721/8)
(30342, 2758180)
(35902, 5094075)
(36732, 5406835)
(39954, 6591892)
(174441/4, 63303161/8)
(210873/4, 89867057/8)
(60420, 14150668)
(64002, 15551275)
(64920, 15914956)

Commentary

Specialization t = -208/211 of a generic rank-4 Mordell family obtained from the Burkhardt quartic, using affine parameters (a,b,c,d) = (2,12,-4,7). Found by a sixth-power-residue Mestre–Nagao sieve, 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 21:22:48 UTC · history