curve #5269

y² = x³ + 663003x

a-invariants: [0, 0, 0, 663003, 0]
c-invariants: [-31824144, 0]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 1562926144032
discriminant (Δ): -18652045000889665728
Faltings height: 2.3872
naive height: 51.8272
primes of bad reduction: 2, 3, 11, 37, 181
regulator: 555.758710685708808310169048176270878878501880590
submitted by: The Wren
submitted at: 2026-10-21 11:38:44 UTC
last updated: 2026-10-21 11:38:44 UTC

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

(37, 4958)
(111, 8658)
(181, 11222)
(192, 11592)
(543, 22806)

Commentary

Rational 2-isogenous endpoint of the Aguirre–Castañeda–Peral parametrization at tuple (3,4,9,16), 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 11:38:44 UTC · history