I have an algebraic function field $\mathbb{Q}(x,y)$, where $y$ satisfies $$ (y^2-1)^2 = x^2(1+x^2), $$ and I need to find a rational function that has a first order root at $x=0,y=1$ a first order pole at $x=0,y=-1$, and no other roots or poles.
Is this possible? How do I go about finding such a function or proving that it cannot exist? Is it an issue that $y+1$ has a branch point of order $\frac12$ at $z=\pm i$? I need to automate this process, so is there any standard algorithm for this that I need to look at?
This might be easy (I don't know), but I asked a related question on MSE and got no responses at all, so I want to ask here this time.