curve #3353

y² = x³ − 84556x + 4408560

a-invariants: [0, 0, 0, -84556, 4408560]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 43513792
discriminant (Δ): 30295201507508224
Faltings height: 1.8576
naive height: 45.6491
primes of bad reduction: 2, 7, 23, 41, 103
regulator: 63.10616281289593059100285295072107653336707633
submitted by: David Renshaw
submitted at: 2026-09-26 21:10:45 UTC
last updated: 2026-09-26 21:10:45 UTC

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

(-152, 3708)
(-90, 3360)
(1702, 69216)
(-2810/9, 16480/27)

Commentary

Found in a Mestre–Nagao-sieved search of the family y^2 = x(x+a)(x+b), at a = 206, b = 574; converted to the global minimal model. The 4 witness points were found with PARI/GP ellrank. Conductor 43513792, torsion ℤ/2ℤ × ℤ/2ℤ.

last edited by David Renshaw at 2026-09-26 21:10:45 UTC · history