curve #5742

y² + y = x³ + 2130477241169640

a-invariants: [0, 0, 1, 0, 2130477241169640]
c-invariants: [0, -1840732336370569176]
rank (lower bound): ≥ 11
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 653606391620419410901518477276489
discriminant (Δ): -1960819174861258232704555431829467
Faltings height: 5.0671
naive height: 84.1134
primes of bad reduction: 3, 8521908964678561
regulator: 545730934020.995325053588803135471869483480415478392624536084
submitted by: jesper-petersen
submitted at: 2026-10-24 18:11:55 UTC
last updated: 2026-10-24 18:11:55 UTC

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

(-302263/4, 329749769/8)
(1185849/4, 1343106951/8)
(319100, 186071959)
(3700785/4, 7128926797/8)
(7983425/4, 22560153797/8)
(-63119506/3025, 7662995559097/166375)
(-33578335/484, 451324244557/10648)
(-790688/9, 1028975032/27)
(1254726641/12100, 75826349677811/1331000)
(9723474882/9409, 959734302315503/912673)
(10135616694/3025, 1020440214751872/166375)

Commentary

Constructed using the rational Bruin–Nasserden parametrization of the Burkhardt quartic with parameters (u,v,w)=(-9/6,-3/6,-1/6) and specialization t=-233/690. The specialization was identified through a three-stage Mestre–Nagao sieve over 328 primes. PARI/GP 2.17.4 returned matching computational rank bounds [11,11] for this model and its rationally 3-isogenous partner.

last edited by jesper-petersen at 2026-10-24 18:11:55 UTC · history