curve #1151

y² + y = x³ + x² − 1373x + 19131

a-invariants: [0, 1, 1, -1373, 19131]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/3ℤ
conductor (N): 4123
discriminant (Δ): -4123 ☆ record for ℤ/3ℤ torsion, rank ≥ 2
Faltings height: 0.2464
naive height: 33.2886
primes of bad reduction: 7, 19, 31
regulator: 8.66205414346498453038665192864581676451785
submitted by: David Renshaw
submitted at: 2026-09-23 21:37:42 UTC
last updated: 2026-09-23 21:37:42 UTC

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

(17, 31)
(149/4, 1157/8)

Commentary

Cremona 4123c1, 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 21:37:42 UTC · history