curve #1167

y² = x³ − x² − 180x + 900

a-invariants: [0, -1, 0, -180, 900]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 840
discriminant (Δ): 70560000
Faltings height: 0.2398
naive height: 27.1980 ☆ record for ℤ/2ℤ × ℤ/4ℤ torsion, rank ≥ 1
primes of bad reduction: 2, 3, 5, 7
regulator: 1.098298981876534643228948977213290779042
submitted by: David Renshaw
submitted at: 2026-09-23 21:38:10 UTC
last updated: 2026-09-23 21:38:10 UTC

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

(5, 10)

Commentary

Cremona 840f2, 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 21:38:10 UTC · history