# On a simple model of X_0(N)

## TL;DR

This paper constructs explicit plane models for all modular curves $X_0(N)$ using modular forms, proving a birational map and providing minimal polynomial equations for these models.

## Contribution

It introduces a uniform method to obtain plane models for all $X_0(N)$ using modular forms of weight 12, establishing birational equivalence and explicit equations.

## Key findings

- Constructed plane models for all $X_0(N)$ with $N \\geq 2$.
- Proved the map from $X_0(N)$ to the plane is birational.
- Derived minimal polynomial equations for the models.

## Abstract

We find plane models for all $X_0(N)$, $N\geq 2$. 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 $N\geq 2$. The equation of the model is the minimal polynomial of $\Delta(N\cdot)/\Delta$ over $\mathbb{C}(j)$.