I tried writing the full motivation for this question, but it turned out to be too long a detour for a question that is really quite specific. So I will only sketch the motivation this time.

### Sketch of motivation

I'm taking a $G$-Galois cover of $\mathbb{P}^1_{\overline{\mathbb{C}(t)}}$ which is defined (not including the group action) over $\mathbb{C}(t)$. Then I descend to $\mathbb{C}[t]$ and observe the branching behavior. The following specific example came up from a very explicit computation of a $D_8$-cover, $X$ over $\mathbb{P}^1_{\overline{\mathbb{C}(t)}}$, where the interesting part (the part in the question) is the $C_4$-cover $X \rightarrow X/C_4$. In the question, I'm actually looking formally about $t=0$.

While this is the motivation, the question will require none of this.

### Question

Let $K=\mathbb{C}((t))(y)$, and $L=Quot(\mathbb{C}((t))[y,z]/z^4-(y-\sqrt{1-t})^2(y+\sqrt{1-t})^2(\sqrt{2(2-t)}-y)(\sqrt{2(2-t)}+y)^3)$. Let's look at the surface $R:=\mathbb{P}^1_{\mathbb{C}[[t]]}$ (with parameter $y$, so its function field is $K$), and the branched covering of it: $S:=$ the normalization of $R$ in $L$ (obviously with function field $L$). My question is: what is the ramification index of the vertical divisor, $t$, of $R$, in the branched covering $S \rightarrow R$?

It seems to me, through roundabout ways, that the ramification index should equal $2$, but surely there should be a systematic way of doing this that isn't completely terrible.

Do you have any tricks up your sleeves for computing vertical ramification in situations like this?

### Remark

Fix a choice for $\sqrt{1-t}$ and $\sqrt{2(2-t)}$ in $\mathbb{C}[[t]]$ throughout.