curve #5315

y² = x³ + 3211754495x

a-invariants: [0, 0, 0, 3211754495, 0]
c-invariants: [-154164215760, 0]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 330091741956886560800
discriminant (Δ): -2120347271984821015708981592000
Faltings height: 4.5086
naive height: 77.2839
primes of bad reduction: 2, 5, 17, 29, 43, 157, 193
regulator: 3129438.333288427805625328752664174422827147886939354
submitted by: The Wren
submitted at: 2026-10-21 13:05:59 UTC
last updated: 2026-10-21 13:05:59 UTC

Witness: 7 independent points log in to add more points →

(3625, 3419100)
(5041/9, 36215396/27)
(21199/25, 206308668/125)
(935680/9, 1030826960/27)
(978895/49, 2913191520/343)
(8073895/961, 156443225220/29791)
(56278705/9, 422215311140/27)

Commentary

Rational 2-isogenous endpoint of the Aguirre–Castañeda–Peral parametrization at tuple (2,16,19,22), 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 13:05:59 UTC · history