curve #3367

y² + xy = x³ − 1252656615138720x + 17064495157914929894400

a-invariants: [1, 0, 0, -1252656615138720, 17064495157914929894400]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 83973799470
discriminant (Δ): 1576189838975742673577701543174718016000000
Faltings height: 7.2745
naive height: 115.9057
primes of bad reduction: 2, 3, 5, 11, 71, 109, 131, 251
regulator: 265.29989440217662032409007268670884005413221
submitted by: David Renshaw
submitted at: 2026-09-26 21:13:32 UTC
last updated: 2026-09-26 21:13:32 UTC

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

(-22294560, 184159104480)
(19160427660/529, 1686132546866460/12167)
(34638558240/1681, 89367027636480/68921)

Commentary

Found in a rational-parameter search of the Kubert–Tate family for torsion ℤ/2ℤ × ℤ/6ℤ: y^2 + (1-c)xy - by = x^3 - bx^2 with c=(10-2t)/(t^2-9), b=c+c^2, at t = 19/30; converted to the global minimal model. The 3 witness points were found with PARI/GP ellrank. Conductor 83973799470, torsion ℤ/2ℤ × ℤ/6ℤ.

last edited by David Renshaw at 2026-09-26 21:13:32 UTC · history