curve #1168

y² = x³ − x² − 784x + 8704

a-invariants: [0, -1, 0, -784, 8704]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 336 ☆ record for ℤ/2ℤ × ℤ/4ℤ torsion, rank ≥ 1
discriminant (Δ): 88510464
Faltings height: 0.4208
naive height: 31.6081
primes of bad reduction: 2, 3, 7
regulator: 1.492598417389027417818875207263106353784
submitted by: David Renshaw
submitted at: 2026-09-23 21:38:11 UTC
last updated: 2026-09-23 21:38:11 UTC

Witness: 1 independent point log in to add more points →

(18, 10)

Commentary

Cremona 336e4, rank exactly 1 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 21:38:11 UTC · history