Is there an algorithm for generating (some or all) subfields of a certain genus of a given function field (even a random one,I mean for example generating a random elliptic subfield of a certain given function field). I did a quick search and it seems to me that the problem is heavily treated in the case of cyclic and Hermitian function fields, but I was wondering what do we know in general case. Is there something that I can do in Magma?
On the other hand, do we have an algorithm to check if $F$ is a subfield of $E$, When $F, E$ are function fields (of one variable)? Florian Hess told me that somebody developed such an algorithm using his automorphism algorithm but I don't have much luck finding it.
In order to stick to the tradition, I give a motivation also: Subfields of function fields with a rich automorphism group are subject to cover attack in cryptography when they are not one of those few which are fixed by an automorphism of the cover.
Thank you very much indeed!