Let $X$ be a curve of genus two over a field $k$ with a $k$-rational point. Let $J$ be the Jacobian of $X$.

Can we write down an explicit equation for the abelian surface $J$?

I know $X$ can be given by the equation $y^2 =f(x)$ with $f(x)\in k[x]$ of degree $5$ or $6$.

(Note that I'm actually asking for an equation for a surface birational to $J$.)