curve #5337

y² = x³ − 10605x

a-invariants: [0, 0, 0, -10605, 0]
c-invariants: [509040, 0]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 3598912800
discriminant (Δ): 76332940488000
Faltings height: 1.3533
naive height: 39.4208
primes of bad reduction: 2, 3, 5, 7, 101
regulator: 71.29286774053252313852017820637084825742284939
submitted by: The Wren
submitted at: 2026-10-21 13:54:45 UTC
last updated: 2026-10-21 13:54:45 UTC

Witness: 4 independent points log in to add more points →

(-101, 202)
(-84, 546)
(169, 1742)
(505, 11110)

Commentary

Rational 2-isogeny endpoint from the Aguirre–Castañeda–Peral parametrization at tuple (4,7,16,27), 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 13:54:46 UTC · history