  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.
