curve #5340

y² = x³ − 1805710054225x

a-invariants: [0, 0, 0, -1805710054225, 0]
c-invariants: [86674082602800, 0]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 4173553663909443123648800
discriminant (Δ): 376811390638443398414671401392809000000
Faltings height: 6.0915
naive height: 96.2795
primes of bad reduction: 2, 5, 7, 11, 19, 43, 71, 103, 157
regulator: 8466389.86349154947680337314493944990954855804791931246
submitted by: The Wren
submitted at: 2026-10-21 14:11:15 UTC
last updated: 2026-10-21 14:11:15 UTC

Witness: 8 independent points log in to add more points →

(-656260, 949936350)
(-244240, 653036700)
(8277500, 23498994750)
(455834225, 9732143582700)
(-2300267585/31329, 2016092527549100/5545233)
(-21457975/16, 6244270725/64)
(-9055760/9, 24122280700/27)
(345590441/121, 5669065594164/1331)

Commentary

Rational 2-isogeny endpoint from the Aguirre–Castañeda–Peral parametrization at tuple (13,18,23,27), 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 14:11:15 UTC · history