Black Box Galois Representations

We develop methods to study $2$-dimensional $2$-adic Galois representations $\rho$ of the absolute Galois group of a number field $K$, unramified outside a known finite set of primes $S$ of $K$, which are presented as Black Box representations, where we only have access to the characteristic polynomials of Frobenius automorphisms at a finite set of primes. Using suitable finite test sets of primes, depending only on $K$ and $S$, we show how to determine the determinant $\det\rho$, whether or not $\rho$ is residually reducible, and further information about the size of the isogeny graph of $\rho$ whose vertices are homothety classes of stable lattices. The methods are illustrated with examples for $K=\mathbb{Q}$, and for $K$ imaginary quadratic, $\rho$ being the representation attached to a Bianchi modular form. These results form part of the first author's thesis.