curve #3394

y² + xy + y = x³ − x² − 1315955739625988x + 18364818030478476137831

a-invariants: [1, -1, 1, -1315955739625988, 18364818030478476137831]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 4397377590
discriminant (Δ): 149982322605660868944713975913599161472001600
Faltings height: 7.4208
naive height: 116.0536
primes of bad reduction: 2, 3, 5, 17, 23, 29, 31, 139
regulator: 404.71006182935076537852967277095169335707894
submitted by: David Renshaw
submitted at: 2026-09-26 21:20:05 UTC
last updated: 2026-09-26 21:20:05 UTC

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

(77759467/4, 90891363693/8)
(590482719/25, 2677719415487/125)
(63685413811/2916, 1015219901991223/157464)

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 = 6/29; converted to the global minimal model. The 3 witness points were found with PARI/GP ellrank. Conductor 4397377590, torsion ℤ/2ℤ × ℤ/6ℤ.

last edited by David Renshaw at 2026-09-26 21:20:05 UTC · history