curve #3371

y² + xy = x³ − 101871594x + 378135545508

a-invariants: [1, 0, 0, -101871594, 378135545508]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/8ℤ
conductor (N): 6981702
discriminant (Δ): 5888197514236558569414912
Faltings height: 3.5114
naive height: 66.9313
primes of bad reduction: 2, 3, 7, 13, 19, 673
regulator: 10.1813611658367064121620662777149449377111
submitted by: David Renshaw
submitted at: 2026-09-26 21:16:09 UTC
last updated: 2026-09-26 21:16:09 UTC

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

(-108, 623862)
(1602, 467226)

Commentary

Found in a rational-parameter search of the Kubert–Tate family for torsion ℤ/8ℤ: y^2 + (1-c)xy - by = x^3 - bx^2 with b=(2t-1)(t-1), c=b/t, at t = -6/7; converted to the global minimal model. The 2 witness points were found with PARI/GP ellrank. Conductor 6981702, torsion ℤ/8ℤ.

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