curve #2795

y² = x³ − 13251x + 474370

a-invariants: [0, 0, 0, -13251, 474370]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 199728
discriminant (Δ): 51699091083264
Faltings height: 1.3474
naive height: 40.0891
primes of bad reduction: 2, 3, 19, 73
regulator: 6.2323966483756369199046582453994941097286232
submitted by: David Renshaw
submitted at: 2026-09-26 20:22:46 UTC
last updated: 2026-09-26 20:22:46 UTC

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

(113, 648)
(98, 342)
(17, 504)

Commentary

Cremona 199728a2 = LMFDB 199728.b2: https://www.lmfdb.org/EllipticCurve/Q/199728/b/2 Mordell–Weil rank exactly 3 (Cremona's tables). Conductor 199728, minimal discriminant 51699091083264, torsion ℤ/2ℤ × ℤ/2ℤ. The witnesses are the generators from Cremona's tables.

last edited by David Renshaw at 2026-09-26 20:22:46 UTC · history