curve #3399

y² + xy + y = x³ + x² − 150244560x + 59718809265

a-invariants: [1, 1, 1, -150244560, 59718809265]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/8ℤ
conductor (N): 9409530
discriminant (Δ): 215514333746490320100000000
Faltings height: 3.7426
naive height: 68.0969
primes of bad reduction: 2, 3, 5, 13, 23, 1049
regulator: 13.1564070133689794141150680723178581411636
submitted by: David Renshaw
submitted at: 2026-09-26 21:20:53 UTC
last updated: 2026-09-26 21:20:53 UTC

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

(-1637, 549713)
(15199, 1127105)

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 = 13/3; converted to the global minimal model. The 2 witness points were found with PARI/GP ellrank. Conductor 9409530, torsion ℤ/8ℤ.

last edited by David Renshaw at 2026-09-26 21:20:53 UTC · history