curve #5908

y² = x³ − 557468344237467

a-invariants: [0, 0, 0, 0, -557468344237467]
c-invariants: [0, 481652649421171488]
rank (lower bound): ≥ 6
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 180619743532939308
discriminant (Δ): -134253052485204818827736600870448
Faltings height: 4.8437
naive height: 81.4320
primes of bad reduction: 2, 3, 13, 19, 229, 241
regulator: 42026.811741193263894837351171441179438208579855938
submitted by: jesper-petersen
submitted at: 2026-10-25 21:06:56 UTC
last updated: 2026-10-25 21:06:56 UTC

Witness: 6 independent points log in to add more points →

(92287, 15117206)
(435487, 286412594)
(416689/4, 191498359/8)
(13631683/49, 49673852852/343)
(18722353/64, 80103332777/512)
(259001977/36, 4168255132691/216)

Commentary

Burkhardt quartic specialization b = -1, c = -a - d with (a,d) = (3,12), using the auxiliary quartic rational point t = 16179/2645, w = 25201350720/279841. The resulting cubesum curve is u^3 + v^3 = 245370294 after rational cube scaling. PARI/GP 2.17.4 ellrank bounds [6,6].

last edited by jesper-petersen at 2026-10-25 21:06:56 UTC · history