The curve $$(X-16)^3=XY\tag{1}\label{1}$$ is essential to Heegner's approach to the class number one problem for imaginary quadratic fields. We have the following “modular” parametrization \begin{equation}\tag{2}\label{2}(X,Y)=\left(2^{12}\Phi(\tau),j(\tau)\right),\end{equation} where $\Phi(\tau)=\frac{\Delta(2\tau)}{\Delta(\tau)}$. Note that the function $\Phi(\tau)$ is a Hauptmodul for the group $\Gamma_0(2)$.

My questions are:

- The parametrization \eqref{2} can be deduced, quite laboriously, using Weierstrass elliptic functions and the product expansion for $\Delta$. However, when we clear the denominators, equation \eqref{1} (possibly) becomes an identity between modular forms and such identities should be easy to prove using the fact, that the space of modular forms has a finite dimension. Can this be done?
- What can we say about $\Delta(2\tau)$? It is a modular form with respect to some group?