curve #5586

y² = x³ − 4933709942719895064

a-invariants: [0, 0, 0, 0, -4933709942719895064]
c-invariants: [0, 4262725390509989335296]
rank (lower bound): ≥ 11
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 36289369535338444294503260444736
discriminant (Δ): -10515525321121840900839730239002115689472
Faltings height: 6.3584
naive height: 99.6084
primes of bad reduction: 2, 3, 7, 13, 978917, 2817677
regulator: 1245416267175.26743200259561479901524722022600943124899265858
submitted by: jesper-petersen
submitted at: 2026-10-23 20:59:36 UTC
last updated: 2026-10-23 20:59:36 UTC

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

(1871142, 1271802168)
(4363450, 8839968956)
(5559450, 12918781044)
(21183162, 97470556092)
(62608105/4, 495069606773/8)
(687852321/25, 18038124040131/125)
(29917275738/16129, 2464962455336724/2048383)
(142302305929/63001, 40594284485393345/15813251)
(991356220009/215296, 961798472950984885/99897344)
(8054421821754/1311025, 22614215619219746508/1501123625)
(35865760043841/20025625, 80712039033424301139/89614671875)

Commentary

Specialization t = 11/9 of a generic rank-4 Mordell family obtained from the Burkhardt quartic, with rational parametrization parameters (u,v,w) = (5,4,-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 11, with bounds [11,11] and 11 independent rational points.

last edited by jesper-petersen at 2026-10-23 20:59:36 UTC · history