curve #5587

y² + y = x³ + 18499046903878700

a-invariants: [0, 0, 1, 0, 18499046903878700]
c-invariants: [0, -15983176524951197016]
rank (lower bound): ≥ 10
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 8163020482154546866896024441
discriminant (Δ): -147836766104022582296623194751879227
Faltings height: 5.4273
naive height: 88.4361
primes of bad reduction: 3, 7, 13, 71, 20389, 228617
regulator: 170652592.6661126163857766366533065157083085456658302022949
submitted by: jesper-petersen
submitted at: 2026-10-23 21:00:31 UTC
last updated: 2026-10-23 21:00:31 UTC

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

(-264432, 2971363)
(-73164, 134563747)
(-41090, 135755924)
(-19292, 135984803)
(455217/4, 1130606123/8)
(174048, 154179868)
(6946836, 18310177372)
(-5425446/25, 11373085967/125)
(-4084626/25, 14862671807/125)
(-31850/9, 3672298025/27)

Commentary

Specialization t = 89/82 of a generic rank-4 Mordell family obtained from the Burkhardt quartic, with rational parametrization parameters (u,v,w) = (-4,7,8). 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 21:00:31 UTC · history