curve #3354

y² = x³ − x² − 875284x + 140750404

a-invariants: [0, -1, 0, -875284, 140750404]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 5245656
discriminant (Δ): 34394042302997854464
Faltings height: 2.4433 ☆ record for ℤ/2ℤ × ℤ/4ℤ torsion, rank ≥ 3
naive height: 52.6605 ☆ record for ℤ/2ℤ × ℤ/4ℤ torsion, rank ≥ 3
primes of bad reduction: 2, 3, 13, 17, 23, 43
regulator: 22.460458169304061878432358751804900994264210
submitted by: David Renshaw
submitted at: 2026-09-26 21:10:55 UTC
last updated: 2026-09-26 21:10:55 UTC

Witness: 3 independent points log in to add more points →

(-731, 19734)
(868, 5850)
(1492, 46410)

Commentary

Found in a rational-parameter search of the Kubert–Tate family for torsion ℤ/2ℤ × ℤ/4ℤ: y^2 + (1-c)xy - by = x^3 - bx^2 with b=t^2-1/16, c=0, at t = -17/276; converted to the global minimal model. The 3 witness points were found with PARI/GP ellrank. Conductor 5245656, torsion ℤ/2ℤ × ℤ/4ℤ.

last edited by David Renshaw at 2026-09-26 21:10:55 UTC · history