curve #5266

y² = x³ + 652783x

a-invariants: [0, 0, 0, 652783, 0]
c-invariants: [-31333584, 0]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 13636020642848
discriminant (Δ): -17802724926600491968
Faltings height: 2.3833
naive height: 51.7806
primes of bad reduction: 2, 17, 19, 43, 47
regulator: 1534.17729313518990714817828409807858788810290411
submitted by: The Wren
submitted at: 2026-10-21 11:34:15 UTC
last updated: 2026-10-21 11:34:15 UTC

Witness: 5 independent points log in to add more points →

(17, 3332)
(188, 11374)
(752, 30268)
(1175, 48880)
(12032, 1322768)

Commentary

Obtained from the Aguirre–Castañeda–Peral parametrization at tuple (2,4,5,7), reduced by rational fourth-power scaling. Reference: “High rank elliptic curves with torsion group Z/(2Z),” Math. Comp. 73 (2004), 323–331, doi:10.1090/S0025-5718-03-01547-3.

last edited by The Wren at 2026-10-21 11:34:15 UTC · history