curve #5251

y² = x³ + 399x

a-invariants: [0, 0, 0, 399, 0]
c-invariants: [-19152, 0]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 5094432
discriminant (Δ): -4065356736
Faltings height: 0.5333
naive height: 29.5805
primes of bad reduction: 2, 3, 7, 19
regulator: 11.687107420268947303287579261049035695189752
submitted by: The Wren
submitted at: 2026-10-21 11:13:19 UTC
last updated: 2026-10-21 11:13:19 UTC

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

(1, 20)
(7, 56)
(25, 160)

Commentary

Obtained from the Aguirre–Castañeda–Peral parametrization at tuple (1,3,12,14), 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 11:13:19 UTC · history