curve #1152

y² + y = x³ + 306x + 2135

a-invariants: [0, 0, 1, 306, 2135]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/3ℤ
conductor (N): 308691
discriminant (Δ): -3803381811
Faltings height: 0.5245 ☆ record for ℤ/3ℤ torsion, rank ≥ 3
naive height: 28.8558 ☆ record for ℤ/3ℤ torsion, rank ≥ 3
primes of bad reduction: 3, 37, 103
regulator: 15.380671430305728503508639138525967336646338
submitted by: David Renshaw
submitted at: 2026-09-23 21:37:44 UTC
last updated: 2026-09-23 21:37:44 UTC

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

(-3, 34)
(15, 100)
(21, 133)

Commentary

Cremona 308691b1, rank exactly 3 according to Cremona's tables. Witness generators from https://github.com/JohnCremona/ecdata/blob/master/allgens/allgens.300000-309999 . 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 21:37:44 UTC · history