curve #1140

y² + xy + y = x³ + x² − 813x + 3906

a-invariants: [1, 1, 1, -813, 3906]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 2175 ☆ record for ℤ/2ℤ × ℤ/2ℤ torsion, rank ≥ 2
discriminant (Δ): 26609765625
Faltings height: 0.6958
naive height: 31.7159
primes of bad reduction: 3, 5, 29
regulator: 1.92989989018885637930651116130325729428639
submitted by: David Renshaw
submitted at: 2026-09-23 21:37:22 UTC
last updated: 2026-09-23 21:37:22 UTC

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

(0, 62)
(30, 72)

Commentary

Cremona 2175c2, 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 21:37:22 UTC · history