curve #5310

y² = x³ + 2020343x

a-invariants: [0, 0, 0, 2020343, 0]
c-invariants: [-96976464, 0]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 130617146804768
discriminant (Δ): -527782876453970790848 ☆ record for quartic twists, rank ≥ 6
Faltings height: 2.6657 ☆ record for quartic twists, rank ≥ 6
naive height: 55.1699 ☆ record for quartic twists, rank ≥ 6
primes of bad reduction: 2, 13, 23, 29, 233
regulator: 7065.3336253517796691711269526374116711846161054094
submitted by: The Wren
submitted at: 2026-10-21 12:59:38 UTC
last updated: 2026-10-21 12:59:38 UTC

Witness: 6 independent points log in to add more points →

(29, 7656)
(299, 25116)
(637, 39312)
(2691, 157872)
(667/9, 330832/27)
(3393/4, 385671/8)

Commentary

Rational 2-isogenous endpoint of the Aguirre–Castañeda–Peral parametrization at tuple (8,16,19,22), 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 12:59:39 UTC · history