curve #5576

y² + y = x³ + 3525373767292382

a-invariants: [0, 0, 1, 0, 3525373767292382]
c-invariants: [0, -3045922934940618264]
rank (lower bound): ≥ 10
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 50101326073522937635710189489
discriminant (Δ): -5369008406016938565855611036209707
Faltings height: 5.1510
naive height: 85.1207
primes of bad reduction: 3, 7, 10658726431723
regulator: 422747866.7028079263142718337482877559238035635963798210718
submitted by: jesper-petersen
submitted at: 2026-10-23 18:40:08 UTC
last updated: 2026-10-23 18:40:08 UTC

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

(-131628, 35281669)
(-117530, 43610738)
(-108300, 47488261)
(-15960, 59340613)
(3892, 59375354)
(22974, 59476882)
(214242, 115581154)
(2516640, 3992819386)
(525777/16, 3819067591/64)
(1267204/25, 7557701233/125)

Commentary

Specialization at t = -79491/16285 of the generic rank-4 Mordell family associated with the Burkhardt quartic, with parameters (a,b,c,d) = (-43/36,-5/12,1/4,7/9). Found using a Mestre–Nagao sieve combined with p-adic coefficient reduction. The specialization satisfies m = 103n (mod 637), guaranteeing a sixth-power factor 637^6 in the integral Mordell coefficient. An additional factor 17^6 produces an especially compact model. PARI/GP 2.17.4 verifies exact rank 10, with bounds [10,10] and 10 independent rational points.

last edited by jesper-petersen at 2026-10-23 18:40:08 UTC · history