curve #1148

y² = x³ + x² − 2x + 1

a-invariants: [0, 1, 0, -2, 1]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/3ℤ
conductor (N): 124
discriminant (Δ): -496
Faltings height: -0.7718 ☆ record for ℤ/3ℤ torsion, rank ≥ 1
naive height: 14.6318
primes of bad reduction: 2, 31
regulator: 0.5205306938826742774502387930373645426292
submitted by: David Renshaw
submitted at: 2026-09-23 21:37:36 UTC
last updated: 2026-09-23 21:37:36 UTC

Witness: 1 independent point log in to add more points →

(2, 3)

Commentary

Cremona 124a1, rank exactly 1 according to Cremona's tables. Witness generators from https://github.com/JohnCremona/ecdata/blob/master/allgens/allgens.00000-09999 . 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 21:37:36 UTC · history