curve #3383

y² + xy + y = x³ − 12006994666815083x + 506406870636197175470306

a-invariants: [1, 0, 1, -12006994666815083, 506406870636197175470306]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 9692581470
discriminant (Δ): 9153080591662607617971811845581585062500
Faltings height: 7.5933
naive height: 122.6864
primes of bad reduction: 2, 3, 5, 11, 13, 83, 163, 167
regulator: 235.97059532833888803600363768958787067572706
submitted by: David Renshaw
submitted at: 2026-09-26 21:18:16 UTC
last updated: 2026-09-26 21:18:16 UTC

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

(62632515, 8653617142)
(63263325, -23805218)
(198398184105/3136, -3944085519193/175616)

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 = 157/165; converted to the global minimal model. The 3 witness points were found with PARI/GP ellrank. Conductor 9692581470, torsion ℤ/2ℤ × ℤ/6ℤ.

last edited by David Renshaw at 2026-09-26 21:18:16 UTC · history