  In this paper we give a complete characterization of the intersections
between the Norm-Trace curve over $\mathbb{F}_{q^3}$ and the curves of the form
$y=ax^3+bx^2+cx+d$, generalizing a previous result by Bonini and Sala,
providing more detailed information about the weight spectrum of one-point AG
codes arising from such curve. We also derive, with explicit computations, some
general bounds for the number of rational points on a cubic surface defined
over $\mathbb{F}_{q}$.
