In the context of my work, I am trying to develop an algorithm to factorize some operators on algebraic function fields of positive characteristic $p$. To this end I need to be able to compute information on the divisor class group of those function field.
Suppose that I am working on a field $K$ which is an algebraic extension of $\mathbb{F}_p(x)$ and let $\mathfrak{G}$ be its divisor class group. Note that since $K$ can be seen as the regular function field of a curve $\mathcal{C}$ over a finite field, $\mathfrak{G}$ is a finite commutative group.
Ideally I would need to be able to compute a lift of a generating family of its cokernel by the multiplication by $p$, but it is my understanding that there is no easier way to do this than to compute the whole divisor class group which is hard.
A more realistic approach I think would be to compute the $p$-rank of $\mathfrak{G}$ (the dimension of the $p$-torsion of $\mathfrak{G}$ as a $\mathbb{F}_p$-vector space).
I know that this should be doable more easily since Magma has an algorithm to do this task faster than computing $\mathfrak{G}$ as a whole.
Would you happen to know some references that tackle this question (and its algorithmic aspects) ?