Let $k$ be a finite field, $X/k$ a smooth curve, $f$ a polynomial of 2 variables which gives an affine model of $X$ and $J$ its Jacobian.

Then how can I compute $J(k)$?

If $X$ is a hyperelliptic curve, then there is an algorithm computing it. (Although I don’t know its theory...)

But what about for non-hyperelliptic curves?

In 12.3.1 of Poonen, Schaefer, Stoll’s “Twists of X(7) and primitive solutions to x^2+y^3=z^7”, the authors give $J(k)$ (and the orders of some particular points) for some non hyperelliptic curves with no arguments. So I think that we can compute them even for non hyperelliptic curves.

It seems that we can compute its order using the Zeta function (since we can compute the explcit action of the Frobenius on the curve). But I can’t find how to compute the groups structures.

And can we compute their torsion part if $k$ is a number field, in particular the rationals?