curve #3395

y² + xy = x³ − x² − 4790128743364914x + 127164280714358001627348

a-invariants: [1, -1, 0, -4790128743364914, 127164280714358001627348]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 17490168990
discriminant (Δ): 48544738631941129429564892966832301053924000000
Faltings height: 7.8150
naive height: 119.9296
primes of bad reduction: 2, 3, 5, 7, 17, 19, 23, 37, 101
regulator: 97.707129290817376298092396520146374019840969
submitted by: David Renshaw
submitted at: 2026-09-26 21:20:14 UTC
last updated: 2026-09-26 21:20:14 UTC

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

(-63280988, 420603499494)
(2857296927/121, 219570109599039/1331)
(73548364548/1681, 2520469429880406/68921)

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 = -15/133; converted to the global minimal model. The 3 witness points were found with PARI/GP ellrank. Conductor 17490168990, torsion ℤ/2ℤ × ℤ/6ℤ.

last edited by David Renshaw at 2026-09-26 21:20:14 UTC · history