curve #5722

y² + y = x³ + 7344109857102388481

a-invariants: [0, 0, 1, 0, 7344109857102388481]
c-invariants: [0, -6345310916536463647800]
rank (lower bound): ≥ 12
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 1278481768134837687888432658099974675
discriminant (Δ): -23300330224257416861766685193872038451875
Faltings height: 6.4247
naive height: 100.4040
primes of bad reduction: 3, 5, 19181, 756315294737
regulator: 148542158347.77049875037519649278631823836639296944983655693820
submitted by: jesper-petersen
submitted at: 2026-10-24 16:53:37 UTC
last updated: 2026-10-24 16:53:37 UTC

Witness: 12 independent points log in to add more points →

(-2091879/4, 21467861015/8)
(-451799/4, 21677887545/8)
(252009, 2712953110)
(701569, 2772980609)
(1196811, 3009712603)
(11091609/4, 42831703319/8)
(4285899, 9277472155)
(8153541, 23439142426)
(18839259, 81815219860)
(-24134949/25, 317321935799/125)
(-17463221/9, 5310444916/27)
(59862609/169, 5971861947037/2197)

Commentary

Constructed from the rational Bruin–Nasserden parametrization of the Burkhardt quartic with parameters (u,v,w)=(3/5,16/5,-14/5) and specialization t=1102/751. The specialization was selected using a three-stage Mestre–Nagao sieve involving 328 primes, followed by rational-point reconstruction. All twelve points were verified by exact arithmetic on the minimal model. PARI/GP 2.17.4 returned matching computational rank bounds [12,12].

last edited by jesper-petersen at 2026-10-24 16:53:37 UTC · history