curve #3246

y² = x³ − 9x + 9

a-invariants: [0, 0, 0, -9, 9]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/3ℤ
conductor (N): 324
discriminant (Δ): 11664
Faltings height: -0.4944
naive height: 18.2053
primes of bad reduction: 2, 3
regulator: 0.3876245949042761905184396586176523590703
submitted by: David Renshaw
submitted at: 2026-09-26 20:45:15 UTC
last updated: 2026-09-26 20:45:15 UTC

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

(1, 1)

Commentary

Cremona 324c1 = LMFDB 324.a2: https://www.lmfdb.org/EllipticCurve/Q/324/a/2 Mordell–Weil rank exactly 1 (Cremona's tables). Conductor 324, minimal discriminant 11664, torsion ℤ/3ℤ. The witness is the generator from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:45:15 UTC · history