curve #1176

y² + xy + y = x³ − 164041x + 25558926

a-invariants: [1, 0, 1, -164041, 25558926]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/6ℤ
conductor (N): 18746
discriminant (Δ): 326311622 ☆ record for ℤ/6ℤ torsion, rank ≥ 2
Faltings height: 1.3841
naive height: 47.6372
primes of bad reduction: 2, 7, 13, 103
regulator: 22.5355209728951259200478688207043034138741
submitted by: David Renshaw
submitted at: 2026-09-23 21:38:24 UTC
last updated: 2026-09-23 21:38:24 UTC

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

(-410, 5097)
(3909/16, 10181/64)

Commentary

Cremona 18746i2, rank exactly 2 according to Cremona's tables. Witness generators from https://github.com/JohnCremona/ecdata/blob/master/allgens/allgens.10000-19999 . Found by screening the local allgens tables for missing torsion size records; torsion and independence independently verified by the site.

last edited by David Renshaw at 2026-09-23 21:38:25 UTC · history