  In this paper we analyze the intersection between the norm-trace curve over
$\mathbb{F}_{q^3}$ and the curves of the form $y=ax^3+bx^2+cx+d$, giving a
complete characterization of the intersection between the curve and the
parabolas, as well as sharp bounds for the other cases. This information is
used for the determination of the weight distribution of some one-point AG
codes constructed on the curve.
