curve #5713

y² + y = x³ − 14023117714644044

a-invariants: [0, 0, 1, 0, -14023117714644044]
c-invariants: [0, 12115973705452453800]
rank (lower bound): ≥ 11
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 1132691503327278269384717379947025
discriminant (Δ): -84951862749545870203853803496026875
Faltings height: 5.3812
naive height: 87.8821
primes of bad reduction: 3, 5, 895889, 2504438423
regulator: 17379000714.6776085300486224962075806406150599986193029149943
submitted by: jesper-petersen
submitted at: 2026-10-24 14:44:45 UTC
last updated: 2026-10-24 14:44:45 UTC

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

(245536, 27923803)
(1006361/4, 348891255/8)
(294926, 107842211)
(554816, 395930276)
(779876, 678456036)
(950116, 918512923)
(1042456, 1057747108)
(5316049/4, 12220313789/8)
(1393834, 1641304435)
(1469704, 1777802404)
(3176354, 5659763795)

Commentary

Constructed from the Burkhardt quartic with the rational Bruin–Nasserden parameters (u,v,w)=(-2,-1,-2/5), specialized at t=79/671. The specialization was selected through a three-stage Mestre–Nagao sieve and verified using PARI/GP 2.17.4. Eleven rational-point witnesses satisfy the minimal Weierstrass equation by exact arithmetic. PARI returned matching computational rank bounds [11,11].

last edited by jesper-petersen at 2026-10-24 14:44:45 UTC · history