curve #5574

y² + y = x³ − 14119816595652189

a-invariants: [0, 0, 1, 0, -14119816595652189]
c-invariants: [0, 12199521538643491080]
rank (lower bound): ≥ 11
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 1757704149799537651459229058603075
discriminant (Δ): -86127503340177344921502223871550675
Faltings height: 5.3823
naive height: 87.8959
primes of bad reduction: 3, 5, 7, 44267, 5207662997
regulator: 182088979359.938976738665502766879649052591375318439342561767
submitted by: jesper-petersen
submitted at: 2026-10-23 18:07:02 UTC
last updated: 2026-10-23 18:07:02 UTC

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

(262311, 62681993)
(455799, 283855014)
(9629001, 29879162028)
(2284639/9, 1277301491/27)
(32403009/64, 174127182751/512)
(15127809/25, 56933172252/125)
(17367231/25, 70835686254/125)
(40253899/81, 240255317746/729)
(562668891/289, 13334093900802/4913)
(2185717419/9025, 7911869246803/857375)
(4148506341/17161, 5991864933016/2248091)

Commentary

Specialization at t = 211/374 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 11 (ellrank bounds [11,11]), with 11 independent rational points.

last edited by jesper-petersen at 2026-10-23 18:07:02 UTC · history