curve #1127

y² + xy + y = x³ + x² − 9x + 6

a-invariants: [1, 1, 1, -9, 6]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/2ℤ
conductor (N): 1443
discriminant (Δ): 4329
Faltings height: -0.5457 ☆ record for ℤ/2ℤ torsion, rank ≥ 2
naive height: 18.2122
primes of bad reduction: 3, 13, 37
regulator: 0.790463101484975738716171228728583121090222
submitted by: David Renshaw
submitted at: 2026-09-23 21:36:52 UTC
last updated: 2026-09-23 21:36:52 UTC

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

(2, 0)
(4, 5)

Commentary

Cremona 1443c1, rank exactly 2 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 21:36:52 UTC · history