curve #5253

y² = x³ + 723x

a-invariants: [0, 0, 0, 723, 0]
c-invariants: [-34704, 0]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 16727328
discriminant (Δ): -24187716288
Faltings height: 0.6819
naive height: 31.3638
primes of bad reduction: 2, 3, 241
regulator: 29.564293260679125241857668462516243015083440
submitted by: The Wren
submitted at: 2026-10-21 11:16:17 UTC
last updated: 2026-10-21 11:16:17 UTC

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

(12, 102)
(27, 198)
(75, 690)

Commentary

Obtained from the Aguirre–Castañeda–Peral parametrization at tuple (6,10,11,15), 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 11:16:17 UTC · history