example of computing ramification index I am trying to understand example 2.2.9 of Silverman's "Arithmetic of Elliptic Curves". In this example, Silverman considers a map 
$$
\phi:\mathbb{P}^1\to \mathbb{P}^1; [X,Y]\mapsto [X^3(X-Y)^2, Y^5]
$$
and he claims that the map $\phi$ ramifies at the points $[0,1]$ and $[1,1]$ and that the ramification indices are 
$$
e_\phi([0,1])=3
$$
$$
e_\phi([1,1]) = 2
$$
I am struggling to actually compute these ramification indices by hand. Here is my attempt so far...
We define the ramification index by 
$$
e_\phi([0,1]) = ord_{[0,1]}(\phi^*t_{[1,1]})
$$
where $t_{[1,1]}$ is a uniformizer for $\mathbb{P}^1$ at $[1,1] = \phi([0,1])$. I think of the function field $K(\mathbb{P}^1)$ as the subfield of $K(X, Y)$ generated by the rational functions whose numerator and denominator have the same degree. Under this identification, I can regard the local ring $K[\mathbb{P}^1]_{[1,1]}$ as the subring of the function field consisting of the rational functions whose denominator does not vanish at $[1,1]$. So I think that a uniformizer at $[1,1]$ is given by $(X-Y)/Y$. I similarly think that a uniformizer at $[0,1]$ is $X/Y$, but I am not certain these are correct. I am also uncertain of how to compute
$$
\phi^*((X-Y)/Y)
$$
and to compute the order of this. Any help is greatly appreciated! :)
 A: Mohan essentially gives the answer in his comment above. But I figured I would write it up as an answer to my question. 
Note that the fibre of $\phi$ at the point $[1,0]$ is precisely the point $[1,0]$. Thus restricting $\phi$ to the affine chart given by $Y=1$ gives a map
$$
\phi: \mathbb{A}^1\to \mathbb{A}^1
$$
and in this case $\phi$ is precisely the polynomial $x^3(x-1)^2$. Now in the field $K(x)$, a uniformizer for $0\in \mathbb{A}^1$ is $x$ and a uniformizer for $1\in \mathbb{A}^1$ is $x-1$. Thus, 
$$
ord_{[0,1]}(\phi) = ord_0(x^3(x-1)^2) = 3
$$
and similarly for the order of $\phi$ at $[1,1]$.
A: We have $\phi[0,1]=\phi[1,1]=[0,1]$, so by definition
$$e_{\phi}[0,1] = ord_{[0:1]}(\phi^{*}t_{\phi[0,1]}) =  ord_{[0:1]}(\phi^{*}t_{[0,1]}) = ord_{[0:1]}(t_{[0,1]}\circ \phi)$$
Analogously 
$$e_{\phi}[1,1] = ord_{[1:1]}(t_{[0,1]}\circ \phi)$$
A uniformizer in [0,1] is $t_{[0,1]}:= (x/y)$ because $ord_{[0,1]}(x/y)=1$, indeed, we will see $ord_{[0,1]}(x)=1$ and $ord_{[0,1]}(y)=0$:
First $y\in\mathcal{O}_{\mathbb{P}^1,[0,1]}$, so $ord_{[0,1]}(y)\geq 0$. 
Moreover $y[0,1]=1\neq 0 \Rightarrow ord_{[0,1]}(y)\leq 0$, thus $ord_{[0,1]}(y)=0$.
Second, $x[0,1]=0 \Rightarrow  x\in\mathfrak{m}_{[0,1]} \Rightarrow (x)\subseteq \mathfrak{m}_{[0,1]}$ We have to check $(x)=\mathfrak{m}_{[0,1]}$. We will suppose the opposite and find a contradiction:
If $(x)\subsetneq \mathfrak{m}_{[0,1]}$ then we have a chain of prime ideals of length equal to 2 of the ring $\mathcal{O}_{\mathbb{P}^1,[0,1]}$, this is a contradiction because $dim(\mathcal{O}_{\mathbb{P}^1,[0,1]})=1$.
So we have
$$ord_{[0,1]}(x/y) =  ord_{[0,1]}(x) - ord_{[0,1]}(y)=1-0=1 \Rightarrow  t_{[0,1]}=x/y$$
Making a traslation we wil have
$$t_{[1,1]} = \frac{x}{y}-1$$
Now we can compute the ramification index:
$$e_{\phi}[0,1] = ord_{[0:1]}(t_{[0,1]}\circ \phi) = ord_{[0:1]}(\frac{x}{y}\circ [x^3(x-y)^2,y^5])= ord_{[0:1]}(\frac{x^3(x-y)^2}{y^5}) = ord_{[0:1]}((\frac{x}{y})^3\frac{(x-y)^2}{y^2}) = ord_{[0:1]}((\frac{x}{y})^3)+ord_{[0:1]}((\frac{x}{y}-1)^2)) = 3+0=3$$
Analogously, remembering that $t_{[1,1]} = x/y -1$:
$$e_{\phi}[0,1]) =  ord_{[1:1]}(t_{[0,1]}\circ \phi) = ord_{[0:1]}((\frac{x}{y})^3)+ord_{[0:1]}((\frac{x}{y}-1)^2) = 0+2=2$$
