curve #1002

y² + y = x³ − 24x + 45

a-invariants: [0, 0, 1, -24, 45]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/3ℤ
conductor (N): 189
discriminant (Δ): 189
Faltings height: -0.5400
naive height: 21.1478
primes of bad reduction: 3, 7
regulator: 1.863243552212362977779362145064929045038
submitted by: David Renshaw
submitted at: 2026-09-23 15:13:17 UTC
last updated: 2026-09-23 15:13:17 UTC

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

(-3, 9)

Commentary

Cremona 189b1 = LMFDB 189.b3: https://www.lmfdb.org/EllipticCurve/Q/189/b/3 Mordell–Weil rank exactly 1 (Cremona's tables). Conductor 189, minimal discriminant 189, torsion ℤ/3ℤ. The witness is the generator from Cremona's tables.

last edited by David Renshaw at 2026-09-23 15:13:17 UTC · history