curve #3205

y² = x³ − x² − 4x − 2

a-invariants: [0, -1, 0, -4, -2]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ
conductor (N): 192
discriminant (Δ): 192
Faltings height: -0.7728
naive height: 16.0126
primes of bad reduction: 2, 3
regulator: 1.351603734412083332293812071952292680515
submitted by: David Renshaw
submitted at: 2026-09-26 20:42:36 UTC
last updated: 2026-09-26 20:42:36 UTC

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

(3, 2)

Commentary

Cremona 192a1 = LMFDB 192.a3: https://www.lmfdb.org/EllipticCurve/Q/192/a/3 Mordell–Weil rank exactly 1 (Cremona's tables). Conductor 192, minimal discriminant 192, torsion ℤ/2ℤ. The witness is the generator from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:42:36 UTC · history