curve #3365

y² + xy + y = x³ − 213900147005623x − 158026545479889387994

a-invariants: [1, 0, 1, -213900147005623, -158026545479889387994]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 23094404970
discriminant (Δ): 615556365357650147808428076523365852365062500
Faltings height: 7.2841
naive height: 110.6032
primes of bad reduction: 2, 3, 5, 7, 13, 17, 41, 53, 229
regulator: 112.02231211828037997631402219144849096163569
submitted by: David Renshaw
submitted at: 2026-09-26 21:12:43 UTC
last updated: 2026-09-26 21:12:43 UTC

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

(-51170372870/4489, 8503560453346376/300763)
(-1890438050/841, 430429347854308/24389)
(-1413837695/169, 71061873398294/2197)

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 = -89/41; converted to the global minimal model. The 3 witness points were found with PARI/GP ellrank. Conductor 23094404970, torsion ℤ/2ℤ × ℤ/6ℤ.

last edited by David Renshaw at 2026-09-26 21:12:43 UTC · history