curve #3249

y² + y = x³ − 84x + 315

a-invariants: [0, 0, 1, -84, 315]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/3ℤ
conductor (N): 171
discriminant (Δ): -5000211
Faltings height: 0.0334
naive height: 25.0299
primes of bad reduction: 3, 19
regulator: 0.6779006064770846362536787946564408518636
submitted by: David Renshaw
submitted at: 2026-09-26 20:45:44 UTC
last updated: 2026-09-26 20:45:44 UTC

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

(5, 4)

Commentary

Cremona 171b2 = LMFDB 171.b2: https://www.lmfdb.org/EllipticCurve/Q/171/b/2 Mordell–Weil rank exactly 1 (Cremona's tables). Conductor 171, minimal discriminant -5000211, torsion ℤ/3ℤ. The witness is the generator from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:45:44 UTC · history