curve #2173

y² + xy + y = x³ − 244941793x + 318865017056

a-invariants: [1, 0, 1, -244941793, 318865017056]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 6507930
discriminant (Δ): 896592149773469856698062500
Faltings height: 3.8623
naive height: 69.5632 ☆ record for ℤ/2ℤ × ℤ/6ℤ torsion, rank ≥ 2
primes of bad reduction: 2, 3, 5, 11, 13, 37, 41
regulator: 6.72047630415166334083307881045590206836068
submitted by: Steps Unbounded
submitted at: 2026-09-26 14:55:46 UTC
last updated: 2026-09-26 14:55:46 UTC

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

(-1685/4, 5198431/8)
(-20, 569012)

Commentary

Height-ordered search of the universal family for torsion Z/2Z x Z/6Z (Tate/Kubert normal form, parameter t=35/13), smallest curves first; rank lower bound from PARI ellrank: 2 independent points.

last edited by Steps Unbounded at 2026-09-26 14:55:46 UTC · history