curve #1142

y² + xy + y = x³ − x² − 54788x + 369542

a-invariants: [1, -1, 1, -54788, 369542]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 174915 ☆ record for ℤ/2ℤ × ℤ/2ℤ torsion, rank ≥ 3
discriminant (Δ): 10470461903825625
Faltings height: 1.7635
naive height: 44.3473
primes of bad reduction: 3, 5, 13, 23
regulator: 7.3623119496216410158977551849184508975454207
submitted by: David Renshaw
submitted at: 2026-09-23 21:37:26 UTC
last updated: 2026-09-23 21:37:26 UTC

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

(-68, 1977)
(270, 2146)
(387, 5890)

Commentary

Cremona 174915e2, rank exactly 3 according to Cremona's tables. Witness generators from https://github.com/JohnCremona/ecdata/blob/master/allgens/allgens.170000-179999 . 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 21:37:26 UTC · history