curve #5706

y² + y = x³ + 2783952957611498888

a-invariants: [0, 0, 1, 0, 2783952957611498888]
c-invariants: [0, -2405335355376335039448]
rank (lower bound): ≥ 12
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 31243715072587803983534516506247361 ☆ record for Mordell curves, rank ≥ 12
discriminant (Δ): -3348170238323726838287509392358985946843
Faltings height: 6.2630
naive height: 98.4640
primes of bad reduction: 3, 7, 8417091330647011
regulator: 555441473551185.47602791862879655450619649414070857068033669834
submitted by: jesper-petersen
submitted at: 2026-10-24 14:07:20 UTC
last updated: 2026-10-24 14:07:20 UTC

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

(462448, 1697896119)
(6087232, 15111010632)
(-53591927896152527/202656630276, 151713729615309471179166131/91230745877868024)
(-187576741352/564001, 702040616773780280/423564751)
(-71110277166/52441, 6473695730880019/12008989)
(-81834455087/553536, 686748263377897241/411830784)
(-1065143807/1156, 55607767453011/39304)
(56289048/289, 8208301041671/4913)
(161977794657/5776, 65194450960699057/438976)
(740016814818138/185204881, 20565436991614478947120/2520453225529)
(8702121860447316/505665169, 812000545371516986511233/11370892655303)
(390629260512520078/795951249921, 1209734893312833487383609920/710116663080769281)

Commentary

Constructed from the Burkhardt quartic using the rational Bruin–Nasserden parametrization, with parameters (u,v,w)=(2/3,-4/3,2/3) and specialization t=419/164. Candidates were selected by a three-stage Mestre–Nagao sieve over 328 primes, followed by PARI/GP 2.17.4 rank computations. Twelve rational points were obtained, and PARI returned matching rank bounds [12,12].

last edited by jesper-petersen at 2026-10-24 14:07:20 UTC · history