curve #5309

y² = x³ − 442x

a-invariants: [0, 0, 0, -442, 0]
c-invariants: [21216, 0]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 12503296
discriminant (Δ): 5526456832
Faltings height: 0.5589
naive height: 29.8875
primes of bad reduction: 2, 13, 17
regulator: 17.184543068123298738064705283662758366085413
submitted by: The Wren
submitted at: 2026-10-21 12:57:46 UTC
last updated: 2026-10-21 12:57:46 UTC

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

(-17, 51)
(-9, 57)
(-1, 21)

Commentary

Rational 2-isogenous endpoint of the Aguirre–Castañeda–Peral parametrization at tuple (1,2,19,22), 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 12:57:46 UTC · history