curve #1128

y² + xy = x³ − 51x + 136

a-invariants: [1, 0, 0, -51, 136]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/2ℤ
conductor (N): 2465
discriminant (Δ): 2465 ☆ record for ℤ/2ℤ torsion, rank ≥ 2
Faltings height: -0.3417
naive height: 23.4103
primes of bad reduction: 5, 17, 29
regulator: 1.71777543023359452600715033818409177601626
submitted by: David Renshaw
submitted at: 2026-09-23 21:36:54 UTC
last updated: 2026-09-23 21:36:54 UTC

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

(3, 2)
(8, 12)

Commentary

Cremona 2465a1, 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 21:36:54 UTC · history