curve #3374

y² + xy + y = x³ − x² − 34040275269152x + 77498916550843571651

a-invariants: [1, -1, 1, -34040275269152, 77498916550843571651]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/9ℤ
conductor (N): 3054566970
discriminant (Δ): -70221113154273618228597016011264000000000
Faltings height: 6.6383
naive height: 105.1167
primes of bad reduction: 2, 3, 5, 7, 13, 17, 71, 103
regulator: 27.7831692355216756542207964105614157421384
submitted by: David Renshaw
submitted at: 2026-09-26 21:16:41 UTC
last updated: 2026-09-26 21:16:41 UTC

Witness: 2 independent points log in to add more points →

(-544928819/81, 1038948512921/729)
(36791732421/11881, 1726954254558901/1295029)

Commentary

Found in a rational-parameter search of the Kubert–Tate family for torsion ℤ/9ℤ: y^2 + (1-c)xy - by = x^3 - bx^2 with c=t^2(t-1), b=c(t^2-t+1), at t = 7/20; converted to the global minimal model. The 2 witness points were found with PARI/GP ellrank. Conductor 3054566970, torsion ℤ/9ℤ.

last edited by David Renshaw at 2026-09-26 21:16:41 UTC · history