curve #3250

y² = x³ + x² − 13x + 15

a-invariants: [0, 1, 0, -13, 15]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/3ℤ
conductor (N): 172
discriminant (Δ): -11008
Faltings height: -0.4578
naive height: 19.4545
primes of bad reduction: 2, 43
regulator: 0.7601396630066703586209147466874986291317
submitted by: David Renshaw
submitted at: 2026-09-26 20:45:54 UTC
last updated: 2026-09-26 20:45:54 UTC

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

(2, 1)

Commentary

Cremona 172a1 = LMFDB 172.a1: https://www.lmfdb.org/EllipticCurve/Q/172/a/1 Mordell–Weil rank exactly 1 (Cremona's tables). Conductor 172, minimal discriminant -11008, torsion ℤ/3ℤ. The witness is the generator from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:45:54 UTC · history