  We give a method to describe all congruence images of a finitely generated
Zariski dense group $H \leq \mathrm{SL}(n, \mathbb{Z})$. The method is applied
to obtain efficient algorithms for solving this problem in odd prime degree
$n$; if $n=2$ then we compute all congruence images only modulo primes. We
propose a separate method that works for all $n$ as long as $H$ contains a
known transvection. The algorithms have been implemented in GAP, enabling
computer experiments with important classes of linear groups that have recently
emerged.
