curve #1154

y² + xy = x³ − x² − 3516x + 81116

a-invariants: [1, -1, 0, -3516, 81116]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/3ℤ
conductor (N): 201042 ☆ record for ℤ/3ℤ torsion, rank ≥ 3
discriminant (Δ): 1045820484
Faltings height: 0.7249
naive height: 36.1090
primes of bad reduction: 2, 3, 17, 73
regulator: 3.5650006521571686183568398472462834100617117
submitted by: David Renshaw
submitted at: 2026-09-23 21:37:47 UTC
last updated: 2026-09-23 21:37:47 UTC

Witness: 3 independent points log in to add more points →

(34, -14)
(31, 19)
(35, -21)

Commentary

Cremona 201042o1, rank exactly 3 according to Cremona's tables. Witness generators from https://github.com/JohnCremona/ecdata/blob/master/allgens/allgens.200000-209999 . 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 21:37:47 UTC · history