curve #1150

y² + xy + y = x³ − 75x + 242

a-invariants: [1, 0, 1, -75, 242]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/3ℤ
conductor (N): 1862 ☆ record for ℤ/3ℤ torsion, rank ≥ 2
discriminant (Δ): -182476
Faltings height: -0.1334
naive height: 24.5537
primes of bad reduction: 2, 7, 19
regulator: 0.852269403355797170597902634418901820270701
submitted by: David Renshaw
submitted at: 2026-09-23 21:37:40 UTC
last updated: 2026-09-23 21:37:40 UTC

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

(5, -4)
(6, 1)

Commentary

Cremona 1862a1, 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 21:37:40 UTC · history