curve #5583

y² = x³ + 44163212175736

a-invariants: [0, 0, 0, 0, 44163212175736]
c-invariants: [0, -38157015319835904]
rank (lower bound): ≥ 10
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 52660511361335064931572825792
discriminant (Δ): -842568181781361038905165212672
Faltings height: 4.4211
naive height: 76.3610
primes of bad reduction: 2, 3, 406661, 13574947
regulator: 520425652.2295826188312557341780255706037170749230466750495
submitted by: jesper-petersen
submitted at: 2026-10-23 20:01:05 UTC
last updated: 2026-10-23 20:01:05 UTC

Witness: 10 independent points log in to add more points →

(-33930, 2258644)
(-20103, 6003247)
(-16110, 6323144)
(-7278, 6616472)
(62337, 16923317)
(126354, 45403160)
(-796206/25, 430465072/125)
(-244814/9, 132371920/27)
(-122462/9, 174236696/27)
(20625/4, 53246777/8)

Commentary

Specialization t = 59/19 of a Mordell family constructed from the Burkhardt quartic, using the affine quadruple (a,b,c,d) = (-11/2,12,6,3/2). Found by varying the Burkhardt quartic parameters using the rational parametrization of Bruin–Nasserden, followed by a sixth-power-residue Mestre–Nagao sieve. PARI/GP 2.17.4 verifies exact rank 10, with bounds [10,10] and 10 rational points.

last edited by jesper-petersen at 2026-10-23 20:01:05 UTC · history