curve #5336

y² = x³ + 336399x

a-invariants: [0, 0, 0, 336399, 0]
c-invariants: [-16147152, 0]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 3621257190432
discriminant (Δ): -2436374595208268736
Faltings height: 2.2176
naive height: 49.7918
primes of bad reduction: 2, 3, 7, 83, 193
regulator: 1340.86522749026340161677902817073098193002595568
submitted by: The Wren
submitted at: 2026-10-21 13:45:14 UTC
last updated: 2026-10-21 13:45:14 UTC

Witness: 5 independent points log in to add more points →

(1, 580)
(193, 8492)
(249, 9960)
(6727, 553784)
(15687, 1966104)

Commentary

Rational 2-isogeny endpoint from the Aguirre–Castañeda–Peral parametrization at tuple (7,11,13,26), 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 13:45:14 UTC · history