Constructing the group SL$_2$ of $\mathbb{Z}[i]$ mod $p$ in GAP

Question

I'm trying to construct, in GAP, the group $$\mbox{SL}_2(\mathbb{Z}[i]/p),$$ where $p$ is a prime in $\mathbb{Z}[i]$, the Gaussian integers. $p$ could be real or complex (for $p$ real would be enough).

First, I do not know how to define the ring $\mathbb{Z}[i]$ mod $p$. I think it could be described as a group with some presentation, but maybe there is a more simple way to define it including the complete ring structure.

I'm thankful for any help.

Answer 1

If $p$ is a real prime in $\mathbb{Z}[i]$, then $p\equiv 3\pmod{4}$, and the quotient ring is isomorphic to the field with $p^2$ elements (the isomorphism being to reduce real numbers mod $p$ and $i$ being mapped to a root of $x^2+1$ in the field with $p^2$ elements. (If $p\equiv 1\pmod 4$, then $p$ is not prime in $\mathbb{Z}[i]$, but splits into two conjugate primes, $p=2$ is (associated to) a square of a prime).

You then can construct SL over this field. E.g. in GAP:

gap> G:=SL(2,7^2);;
gap> Size(G);
117600

Note that there also is the ring $\mathbb{Z}/(p^2)$ which is different, but also allows for an SL as SL(2,Integers mod 7^2);.