curve #5256

y² = x³ + 323x

a-invariants: [0, 0, 0, 323, 0]
c-invariants: [-15504, 0]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 3338528
discriminant (Δ): -2156689088
Faltings height: 0.4805
naive height: 28.9466
primes of bad reduction: 2, 17, 19
regulator: 16.860433458433346303259776317680119272494596
submitted by: The Wren
submitted at: 2026-10-21 11:20:34 UTC
last updated: 2026-10-21 11:20:34 UTC

Witness: 3 independent points log in to add more points →

(1, 18)
(17, 102)
(1292/49, 56202/343)

Commentary

Rational 2-isogenous endpoint of the Aguirre–Castañeda–Peral parametrization at tuple (8,10,11,14), 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 11:20:34 UTC · history