On a simple model of $$X_0(N)$$ X 0 ( N )

Abstract

We find plane models for all $$X_0(N)$$ , $$N\ge 2$$ . We observe a map from the modular curve $$X_0(N)$$ to the projective plane constructed using modular forms of weight 12 for the group $$\Gamma _0(N)$$ ; the Ramanujan function $$\Delta $$ , $$\Delta (N\cdot )$$ and the third power of Eisestein series of weight 4, $$E_4^3$$ , and prove that this map is birational equivalence for every $$N\ge 2$$ . The equation of the model is the minimal polynomial of $$\Delta (N\cdot )/\Delta $$ over $${\mathbb {C}}(j)$$ .