TITLE: Modular curves of genus zero and normal forms for elliptic curves
QUESTION [31 upvotes]: 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 $(\omega, 0), (\omega^3, 0), (\omega^5, 0)$ (where $\omega$ 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?

REPLY [11 votes]: I think the answer to your question is the content of Velu's thesis: Courbes elliptiques munies d'un sous-groupe $Z/NZ\times \mu_N$. In there, he explicitly writes down the universal elliptic curve over $X(p)$ for $p>3$.