curve #5709

y² + y = x³ + 5204694100501747

a-invariants: [0, 0, 1, 0, 5204694100501747]
c-invariants: [0, -4496855702833509624]
rank (lower bound): ≥ 11
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 1624653501828766194979144041489 ☆ record for Mordell curves, rank ≥ 11
discriminant (Δ): -11702379173672602902434774530845267
Faltings height: 5.2160
naive height: 85.8998
primes of bad reduction: 3, 7, 60696141113723
regulator: 12436118604.2532182712774679345435574947075014943710989690744
submitted by: jesper-petersen
submitted at: 2026-10-24 14:41:56 UTC
last updated: 2026-10-24 14:41:56 UTC

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

(77875, 75345658)
(93319, 77571622)
(1140433/4, 1347714077/8)
(440419, 301051957)
(8932567, 26697218089)
(-1288931/25, 8898426128/125)
(-2753303/16, 668225365/64)
(288085/9, 1954003832/27)
(3071425/16, 7091768449/64)
(8058505/64, 43447607717/512)
(13602505/36, 52532622503/216)

Commentary

Constructed from the Burkhardt quartic using the rational Bruin–Nasserden parametrization with parameters (u,v,w)=(-1/5,0,6/5), specialized at t=-340/57. Candidates were selected using a three-stage Mestre–Nagao sieve, followed by PARI/GP 2.17.4 rank computations. Eleven rational-point witnesses were verified exactly on the minimal Weierstrass model. PARI returned matching computational rank bounds [11,11].

last edited by jesper-petersen at 2026-10-24 14:41:56 UTC · history