curve #1155

y² + xy + y = x³ − x² − 22x + 44

a-invariants: [1, -1, 1, -22, 44]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/4ℤ
conductor (N): 205
discriminant (Δ): 1025 ☆ record for ℤ/4ℤ torsion, rank ≥ 1
Faltings height: -0.4983 ☆ record for ℤ/4ℤ torsion, rank ≥ 1
naive height: 20.8438
primes of bad reduction: 5, 41
regulator: 1.602601397324700955066904509537931569839
submitted by: David Renshaw
submitted at: 2026-09-23 21:37:49 UTC
last updated: 2026-09-23 21:37:49 UTC

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

(-1, 8)

Commentary

Cremona 205a1, 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 21:37:49 UTC · history