Let $U$ be the affine curve defined by $x^2=3+10t^4+3t^8$ and $V$ the curve defined by $w^2=z^6-1$. Both are smooth.
Let $F:U\to V$ be defined by $(x,y)\mapsto (z,w)= (\frac{1+t^2}{1-t^2},\frac{2tx}{(1-t^2)^3})$.
The exercise is to check that $F$ is unramified when $t\neq\pm 1$.
Now, however I try to do it, I get very complicated computations. For example in generic points (where the right hand side of each equation is non zero), respectively $t,z$ are coordinates for $U,V$. So writing $F$ in these local coordinates we have $z=\frac{2tx}{(1-t^2)^3}=\frac{2t\sqrt{3+10t^4+3t^8}}{(1-t^2)^3}$, but I just can't write down the roots of the derivative of this function, its too much computation.
Does anyone have an idea how to approach this problem in a computationally possible way? I realize that $U$ is of genus 3 and $V$ is of genus 2 by furthur materiel or the preceeding exercise in the book. Thus Huruwitz's formula says that degree of $F$ is at most 2 and in that case F is unramified. Also the preimage of $(0,i)$ is of size 2. So this basically proves F is unramified even in the projective closure. However, I am looking for a more elementary solution with not (too) much computation, if its possible.
Thank you!