Asvin's profile - activity

I want to investigate the tower of modular curves $X(\ell^n) \to X(1)$ over a finite field $\mathbb F_q$ with $\ell \neq 0 0 \in \mathbb F_q$ and pullbacks of this tower by maps $C \to X(1)$.

In particular, I want to investigate the characteristic polynomial of the Frobenius on the $\ell$-adic cohomology. If I knew defining equations for $X(n)$ in terms of the parameter $t$ on $X(1)$, I guess I could do it but I am not sure...

I would like some way to iterate over all finite groups in increasing size till I manually stop the program (preferably as permutation groups). Is there an easy command to do this?

I am in the situation where I have groups $A,G$ with $G$ normal in $A$ such that $A/G$ is cyclic. I would like to find the coset of $G$ corresponding to a generator of $A/G$.

I feel fairly stupid for asking this but where is the list of commands for group theory in sage? For instance, I would like to know how to find the number of fixed points of a given permutation.

I tried to google it but all I come across are various introductory tutorials and none of them seem to be complete documentations. I am sure I am missing something simple...