curve #3364

y² = x³ − x² − 272209x − 34702511

a-invariants: [0, -1, 0, -272209, -34702511]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 438156096
discriminant (Δ): 767923057847844864
Faltings height: 2.1337
naive height: 49.1566
primes of bad reduction: 2, 3, 7, 17, 127, 151
regulator: 86.22304763333035504130517826597479716413056583
submitted by: David Renshaw
submitted at: 2026-09-26 21:12:33 UTC
last updated: 2026-09-26 21:12:33 UTC

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

(-375, 3808)
(-341, 4284)
(-312, 4445)
(6289, 496944)

Commentary

Found in a Mestre–Nagao-sieved search of the family y^2 = x(x+a)(x+b), at a = -302, b = -1016; converted to the global minimal model. The 4 witness points were found with PARI/GP ellrank. Conductor 438156096, torsion ℤ/2ℤ × ℤ/2ℤ.

last edited by David Renshaw at 2026-09-26 21:12:33 UTC · history