curve #3278

y² = x³ − x² − 129x + 609

a-invariants: [0, -1, 0, -129, 609]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/4ℤ
conductor (N): 192
discriminant (Δ): 98304
Faltings height: -0.0796
naive height: 26.2008
primes of bad reduction: 2, 3
regulator: 1.351603734412083332293812071952292680515
submitted by: David Renshaw
submitted at: 2026-09-26 20:50:28 UTC
last updated: 2026-09-26 20:50:28 UTC

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

(8, 5)

Commentary

Cremona 192a3 = LMFDB 192.a1: https://www.lmfdb.org/EllipticCurve/Q/192/a/1 Mordell–Weil rank exactly 1 (Cremona's tables). Conductor 192, minimal discriminant 98304, torsion ℤ/4ℤ. The witness is the generator from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:50:28 UTC · history