curve #5575

y² + y = x³ + 5362476530907872

a-invariants: [0, 0, 1, 0, 5362476530907872]
c-invariants: [0, -4633179722704401624]
rank (lower bound): ≥ 10
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 115923021595516162928902260609
discriminant (Δ): -12422658763240298567949952953642267
Faltings height: 5.2210
naive height: 85.9596
primes of bad reduction: 3, 7, 16213080970243
regulator: 501907394.7623660791030964734334831496415002205320135758013
submitted by: jesper-petersen
submitted at: 2026-10-23 18:08:10 UTC
last updated: 2026-10-23 18:08:10 UTC

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

(-112728, 62689504)
(440601/4, 654776321/8)
(282660, 167170783)
(434994, 296094487)
(611412, 483656224)
(671622, 555261304)
(3030601/4, 5308284271/8)
(-2650746/25, 8072381867/125)
(1212225/16, 4872992449/64)
(16935468/49, 74082068320/343)

Commentary

Specialization at t = 501/565 of the rank-4 Mordell family associated with the Burkhardt quartic, using parameters (a,b,c,d) = (-43/36,-5/12,1/4,7/9). Found using a sixth-power-residue Mestre–Nagao sieve over rational specializations. PARI/GP 2.17.4 gives exact rank 10 (ellrank bounds [10,10]), with 10 independent rational points.

last edited by jesper-petersen at 2026-10-23 18:08:10 UTC · history