curve #1174

y² + xy + y = x³ − 33x + 68

a-invariants: [1, 0, 1, -33, 68]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/6ℤ
conductor (N): 130 ☆ record for ℤ/6ℤ torsion, rank ≥ 1
discriminant (Δ): 26000 ☆ record for ℤ/6ℤ torsion, rank ≥ 1
Faltings height: -0.3189 ☆ record for ℤ/6ℤ torsion, rank ≥ 1
naive height: 22.0592 ☆ record for ℤ/6ℤ torsion, rank ≥ 1
primes of bad reduction: 2, 5, 13
regulator: 1.170464153594722231116300800325189639761
submitted by: David Renshaw
submitted at: 2026-09-23 21:38:19 UTC
last updated: 2026-09-23 21:38:19 UTC

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

(2, 2)

Commentary

Cremona 130a1, 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 21:38:20 UTC · history