Suppose that we are given an elliptic curve over a number field $K$ and a finite morphism $\pi:E\longrightarrow \mathbf{P}^1_K$ of degree $d$. Assume $\pi$ is unramified outside a finite set $S\subset \mathbf{P}^1_K$ of points. Fix an embedding $K\longrightarrow \mathbf{C}$.

Consider $X=E\times_K \mathbf{C}$. We can write $X=\mathbf{C}/ (\mathbf{Z}+\tau \mathbf{Z})$, where $\tau$ is an element in the complex upper half plane. We define $\Vert \Delta\Vert(X) =\vert (\Im \tau)^6 \vert q\prod_{k=1}^\infty (1-q^k)^{24}$, where we write $q=\exp(2\pi i \tau)$ as usual. This is called the modular discriminant.

I would like to bound $\Vert\Delta\Vert(X)$ using constants depending on $d$, $K$ and the ramification of $\pi$ (and $\pi \times_K \mathbf{C}$).

For example, the bound could be something like
$$ \log \Vert \Delta \Vert(X) \leq [K:\mathbf{Q}]\cdot d^{10} \cdot \vert d_K\vert \cdot \vert S\vert^6. $$

This should be fairly standard. It comes down to choosing $\tau$ properly using $\pi$. How does one do this?

Note: This question is analogous to the same question for number fields, where one can use the ramification to bound the discriminant.