curve #2944

y² + xy + y = x³ − 173x + 852

a-invariants: [1, 0, 1, -173, 852]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ
conductor (N): 189926
discriminant (Δ): 4178372
Faltings height: 0.1014
naive height: 27.0652
primes of bad reduction: 2, 11, 89, 97
regulator: 2.8089144858378304944019278715238272391242215
submitted by: David Renshaw
submitted at: 2026-09-26 20:29:27 UTC
last updated: 2026-09-26 20:29:27 UTC

Witness: 3 independent points log in to add more points →

(9, 2)
(8, -5)
(5, 8)

Commentary

Cremona 189926b1 = LMFDB 189926.a1: https://www.lmfdb.org/EllipticCurve/Q/189926/a/1 Mordell–Weil rank exactly 3 (Cremona's tables). Conductor 189926, minimal discriminant 4178372, torsion ℤ/2ℤ. The witnesses are the generators from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:29:27 UTC · history