On a simple model of X_0(N)

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 ${\mathrm{\Gamma }}_0(N)$; the Ramanujan function $\mathrm{\Delta }$, $\mathrm{\Delta }(N\cdot )$ 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 $\mathrm{\Delta }(N\cdot )\mathrm{/}\mathrm{\Delta }$ over $\mathbb{C}(j)$.