curve #5747

y² = x³ + 1089044458375800897601029

a-invariants: [0, 0, 0, 0, 1089044458375800897601029]
c-invariants: [0, -940934412036691975527289056]
rank (lower bound): ≥ 13
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 24393495144904469512782813122768868609422244
discriminant (Δ): -512359703561825942614075542124354932291137123019312
Faltings height: 8.4091
naive height: 124.2178
primes of bad reduction: 2, 3, 7, 140983, 1714441, 486517463
regulator: 15024928066151.89551090038826074910934845742144064426394172417120
submitted by: jesper-petersen
submitted at: 2026-10-24 18:40:56 UTC
last updated: 2026-10-24 18:40:56 UTC

Witness: 13 independent points log in to add more points →

(-102854465, 30706011602)
(-2419809, 1043566140330)
(26063935, 1052022070898)
(87607891, 1327195355500)
(125282623, 1747983806414)
(146143263, 2051913943926)
(211731555, 3252848489352)
(227789023, 3592843527014)
(-4488843975/64, 441630524574099/512)
(-685502489/25, 129206008671866/125)
(-326530743/4, 5906215198107/8)
(-204680175/4, 7818182614809/8)
(16011650487/169, 3059667250520958/2197)

Commentary

Constructed using the rational Bruin–Nasserden parametrization of the Burkhardt quartic with parameters (u,v,w)=(13/9,4/9,16/9) and specialization t=119/4. The specialization was identified by a three-stage Mestre–Nagao sieve using 328 primes. PARI/GP 2.17.4 returned matching computational rank bounds [13,13].

last edited by jesper-petersen at 2026-10-24 18:40:56 UTC · history