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 $J$.)



Equation for simple Jacobian of a genus two curve
In Mumford's ``Tata Lectures on Theta II" (Progress in Math. 43, 1984) there are explicit equations for a certain open affine (dense) subset $Z$ of the jacobian $Jac(C)$ for any hyperelliptic curve $C$; the jacobian is covered by all the translations of $Z$ by points of order $2$. (Recall that every genus 2 curve is hyperelliptic.) 


