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 the same weight and criteria that such map birational (see Muic in Monatsh Math 180(3):607–629, 2016). In present paper, approach based on theory primitive elements finite separable field extensions. We prove most cases are birational. consider particular their equations .