Suppose that we are given an elliptic curve $E$ 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 $S=\{s_1,\ldots,s_n\}\subset\mathbf{P}^1_K$ 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 $\Vert\Delta\Vert(X)$ from below and above using constants depending on $d$ and the ramification of $\pi$. (Note that Hurwitz gives that $2d = \deg R$ with $R$ the ramification divisor.) 

For example, the bound could be something like  
$$ \vert \log \Vert \Delta \Vert(X) \vert  \leq 1020323 \cdot d^{10} \cdot  s_1\cdot \ldots \cdot s_n\cdot \vert S\vert^6. $$ 

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

**Example.**  If $\pi$ is the Weierstrass function then it is etale above $\{0,1,\infty,s\}$ and of degree $2$. The modular discriminant is given by $\Vert\Delta\Vert(X) = s^2(s-1)^2$. This is precisely what I'm looking for. Note that here we get an equality whereas in the general case I expect only an inequality involving $d$ and the branch points of $\pi$. 

So just to be clear. I want to bound $\Vert\Delta\Vert(X)$ from below and above in $d$ and constants depending on the branch locus $S=\{s_1,\ldots,s_n\}$.

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

Note: I've edited the question a couple of times in the past hour.