On degrees and birationality of the maps X0(N) → P2 constructed via modular forms

In this paper we prove a formula which relates the degree of a curve which is the image of a mapping constructed out of three linearly independent modular forms of the same integral or half-integral weight into and the degree of that map. Based on the formula, we present a test for birationality of the map. As an example, we compute the formula for the total degree i.e., the degree considered as a polynomial of two (independent) variables of the classical modular polynomial (or the degree of the canonical model of ). We give an interesting example to the test for birationality which leads us to make a question on existence of specific explicit and simple model of . We prove our claim when is a prime.