2 added 59 characters in body

This is maybe the first question I actually need to know the answer to!

Let $N$ be a positive integer such that $\mathbb{H}/\Gamma(N)$ has genus zero. Then the function field of $\mathbb{H}/\Gamma(N)$ is generated by a single function. When $N = 2$, the cross-ratio $\lambda$ is such a function. A point of $\mathbb{H}/\Gamma(2 )$ at which $\lambda = \lambda_0$ is precisely an elliptic curve in Legendre normal form

$$y^2 = x(x - 1)(x - \lambda_0)$$

where the points $(0, 0), (1, 0)$ constitute a choice of basis for the $2$-torsion. When $N = 3$, there is a modular function $\gamma$ such that a point of $\mathbb{H}/\Gamma(3)$ at which $\gamma = \gamma_0$ is precisely an elliptic curve in Hesse normal form

$$x^3 + y^3 + 1 + \gamma_0 xy = 0$$

where (I think) the points $(1, (\omega, 0), (\omega, \omega^3, 0), (\omega^2, \omega^5, 0)$ (I thinkwhere $\omega$ is a primitive sixth root of unity) constitute a choice of basis for the $3$-torsion.

Question: Does this picture generalize? That is, for every $N$ above does there exist a normal form for elliptic curves which can be written in terms of a generator of the function field of $\mathbb{H}/\Gamma(N)$ and which "automatically" equips the $N$-torsion points with a basis? (I don't even know if this is possible when $N = 1$, where the Hauptmodul is the $j$-invariant.) If not, what's special about the cases where it is possible?

1

# Modular curves of genus zero and normal forms for elliptic curves

This is maybe the first question I actually need to know the answer to!

Let $N$ be a positive integer such that $\mathbb{H}/\Gamma(N)$ has genus zero. Then the function field of $\mathbb{H}/\Gamma(N)$ is generated by a single function. When $N = 2$, the cross-ratio $\lambda$ is such a function. A point of $\mathbb{H}/\Gamma(2 )$ at which $\lambda = \lambda_0$ is precisely an elliptic curve in Legendre normal form

$$y^2 = x(x - 1)(x - \lambda_0)$$

where the points $(0, 0), (1, 0)$ constitute a choice of basis for the $2$-torsion. When $N = 3$, there is a modular function $\gamma$ such that a point of $\mathbb{H}/\Gamma(3)$ at which $\gamma = \gamma_0$ is precisely an elliptic curve in Hesse normal form

$$x^3 + y^3 + 1 + \gamma_0 xy = 0$$

where the points $(1, 0), (\omega, 0), (\omega^2, 0)$ (I think) constitute a choice of basis for the $3$-torsion.

Question: Does this picture generalize? That is, for every $N$ above does there exist a normal form for elliptic curves which can be written in terms of a generator of the function field of $\mathbb{H}/\Gamma(N)$ and which "automatically" equips the $N$-torsion points with a basis? (I don't even know if this is possible when $N = 1$, where the Hauptmodul is the $j$-invariant.) If not, what's special about the cases where it is possible?