curve #5285

y² = x³ + 4097583x

a-invariants: [0, 0, 0, 4097583, 0]
c-invariants: [-196683984, 0]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 59698440682272 ☆ record for quartic twists, rank ≥ 6
discriminant (Δ): -4403147681991350674368
Faltings height: 2.8425
naive height: 57.2913
primes of bad reduction: 2, 3, 7, 193, 337
regulator: 6965.2419959992827667705990236622376815544796609135
submitted by: The Wren
submitted at: 2026-10-21 12:10:11 UTC
last updated: 2026-10-21 12:10:11 UTC

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

(252, 32382)
(841, 63568)
(1737, 111168)
(5404, 424214)
(193/4, 112519/8)
(12159/25, 5739048/125)

Commentary

Obtained from the Aguirre–Castañeda–Peral parametrization at tuple (5,6,17,18), 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 12:10:11 UTC · history