curve #3355

y² + xy + y = x³ + x² − 1076354x − 86661394

a-invariants: [1, 1, 1, -1076354, -86661394]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 5657379
discriminant (Δ): 76596896839955187969
Faltings height: 2.5058
naive height: 53.2809
primes of bad reduction: 3, 7, 13, 17, 23, 53
regulator: 16.035375261801572831898021866819509534952328
submitted by: David Renshaw
submitted at: 2026-09-26 21:11:06 UTC
last updated: 2026-09-26 21:11:06 UTC

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

(-982, 5446)
(-948, 9577)
(7263, 608980)

Commentary

Found in a rational-parameter search of the Kubert–Tate family for torsion ℤ/2ℤ × ℤ/4ℤ: y^2 + (1-c)xy - by = x^3 - bx^2 with b=t^2-1/16, c=0, at t = -159/92; converted to the global minimal model. The 3 witness points were found with PARI/GP ellrank. Conductor 5657379, torsion ℤ/2ℤ × ℤ/4ℤ.

last edited by David Renshaw at 2026-09-26 21:11:07 UTC · history