Bielliptic curves of genus 2

Question

Given a hyperelliptic equation y^2 = f(x) for a curve of genus 2, is there a method to decide whether the curve is bielliptic?

Answer 1

The roots of $f$ (and the point at infinity, when $f$ has degree five) give you six points on $\mathbf P^1$. The curve is bi-elliptic if and only if there is an involution of $\mathbf P^1$ which switches these points pairwise with each other.

After a quadratic extension of your base field you can assume that the involution has the form $x \mapsto -x$, in which case your equation can be written in the form $y^2 = f(x^2)$ where $f$ is a squarefree cubic which does not vanish at the origin.