curve #5592

y² = x³ + 565594198271964

a-invariants: [0, 0, 0, 0, 565594198271964]
c-invariants: [0, -488673387306976896]
rank (lower bound): ≥ 10
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 3949343174307478086465536016
discriminant (Δ): -138195416355367273201602036271872
Faltings height: 4.8461
naive height: 81.4609
primes of bad reduction: 2, 3, 5236983317333
regulator: 310632628.9965224954508018031716480980649852247998680536753
submitted by: jesper-petersen
submitted at: 2026-10-23 21:28:54 UTC
last updated: 2026-10-23 21:28:54 UTC

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

(-82662, 874206)
(-67515, 16057467)
(-60630, 18512658)
(-10979, 23754385)
(45858, 25729974)
(75450, 31545342)
(153021, 64409985)
(473538, 326727594)
(-25791/4, 190212705/8)
(484617/4, 387313947/8)

Commentary

Specialization t = 247/229 of a generic rank-4 Mordell family obtained from the Burkhardt quartic, with affine parameters (a,b,c,d) = (4/5,-24/5,6,-9) and generating parameters (u,v,w) = (-1,-4,5). 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 21:28:54 UTC · history