curve #5259

y² = x³ − 777x

a-invariants: [0, 0, 0, -777, 0]
c-invariants: [37296, 0]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 19319328
discriminant (Δ): 30022235712
Faltings height: 0.6999
naive height: 31.5799
primes of bad reduction: 2, 3, 7, 37
regulator: 10.031929271311089512659115714673236156500951
submitted by: The Wren
submitted at: 2026-10-21 11:24:47 UTC
last updated: 2026-10-21 11:24:47 UTC

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

(-27, 36)
(-21, 84)
(-3, 48)

Commentary

Rational 2-isogenous endpoint of the Aguirre–Castañeda–Peral parametrization at tuple (2,3,5,6), 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 11:24:48 UTC · history