On primitive elements of algebraic function fields and models of $X_0(N)$

This paper is a continuation of our previous works where we study maps from $X_0(N)$, $N\ge 1$, into ${\mathbb{P}}^2$ constructed via modular forms of the same weight and criteria that such a map is birational (see [12]). In the present paper our approach is based on the theory of primitive elements in finite separable field extensions. We prove that in most of the cases the constructed maps are birational, and we consider those such that the resulting equation of the image in ${\mathbb{P}}^2$ is simplest possible.