Concrete example of Weil Pairing I am trying to find a concrete example of the Weil Pairing.
What I have done until now is that I took $E=(x-1)(x-2)(x-3)$ over $F_5$.
I took $E[2]=\{\infty,(1,0),(2,0),(3,0)\}$.
I know that there exist a rational function f such that
\begin{equation*}
div(f)=2[T]-2[\infty]
\end{equation*}
How can I find this function f?
How can I find the points of order 4?
How should I continue?
The automatic way to find f is by using the Miller algorithm.

But you can also see this directly here:
Since the line y=0 is not tangent to E , y is a uniformizing function.
You can write  $(x-1) = y^2*((x-2)(x-3))^{-1}$, where $(x-2)*(x-3)$ has neither a zero nor a pole at (1, 0). This says exactely that (1,0) is a double zero of the function x-1. Since it has a double zero it must also have a double pole at $\infty$. This gives you 
\begin{equation*}
div((x-1))=2[(1,0)]-2[\infty]
\end{equation*}

In a similar way you can show this for the points (2,0) and (3,0)

Now you have all 2-torsion points, lets find the other points:
x=0 gives $y^2 = -1$, which solves to y=2 and y=3
x=4 gives $y^2 = 1$, which solves to y=1 and y=4
Since the group has 8 elements, at least two of those points must have order 4. 
That all 4 points have order 4 can be shown by a direct calculation(2 squarings are sufficient).
One can also see this by a direct argument:
If the group would be cyclic, it would have 4 elements of order 8. This would let no place for an element of order 4. Therefore the group cannot be cyclic and all 4 elements, which are not 2-torsion elements, must have order 4.