curve #1160

y² = x³ − 1889771x + 961189146

a-invariants: [0, 0, 0, -1889771, 961189146]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/4ℤ
conductor (N): 217328 ☆ record for ℤ/4ℤ torsion, rank ≥ 3
discriminant (Δ): 32806040800515080192
Faltings height: 2.5079
naive height: 54.9695
primes of bad reduction: 2, 17, 47
regulator: 18.249249326808464621019381621512796836236098
submitted by: David Renshaw
submitted at: 2026-09-23 21:37:57 UTC
last updated: 2026-09-23 21:37:57 UTC

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

(629, 4624)
(965, 6016)
(-774, 44274)

Commentary

Cremona 217328c3, rank exactly 3 according to Cremona's tables. Witness generators from https://github.com/JohnCremona/ecdata/blob/master/allgens/allgens.210000-219999 . 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 21:37:57 UTC · history