curve #3361

y² = x³ − x² − 76169x + 5644809

a-invariants: [0, -1, 0, -76169, 5644809]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 121000128
discriminant (Δ): 14641030976016384
Faltings height: 1.8072
naive height: 45.3357
primes of bad reduction: 2, 3, 73, 89, 97
regulator: 63.30217928691081122069388058741387885043208495
submitted by: David Renshaw
submitted at: 2026-09-26 21:12:04 UTC
last updated: 2026-09-26 21:12:04 UTC

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

(-211, 3504)
(-65, 3212)
(419, 6864)
(665, 15768)

Commentary

Found in a Mestre–Nagao-sieved search of the family y^2 = x(x+a)(x+b), at a = 146, b = 534; converted to the global minimal model. The 4 witness points were found with PARI/GP ellrank. Conductor 121000128, torsion ℤ/2ℤ × ℤ/2ℤ.

last edited by David Renshaw at 2026-09-26 21:12:05 UTC · history