curve #2679

y² = x³ − x² − 9x + 9

a-invariants: [0, -1, 0, -9, 9]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 192
discriminant (Δ): 36864
Faltings height: -0.4262
naive height: 18.3144
primes of bad reduction: 2, 3
regulator: 0.6758018672060416661469060359761463402574
submitted by: David Renshaw
submitted at: 2026-09-26 20:17:24 UTC
last updated: 2026-09-26 20:17:24 UTC

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

(5, 8)

Commentary

Cremona 192a2 = LMFDB 192.a2: https://www.lmfdb.org/EllipticCurve/Q/192/a/2 Mordell–Weil rank exactly 1 (Cremona's tables). Conductor 192, minimal discriminant 36864, torsion ℤ/2ℤ × ℤ/2ℤ. The witness is the generator from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:17:24 UTC · history