We develop methods to study 2-dimensional 2-adic Galois representations
ρ 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 detρ, whether or not ρ is residually reducible, and
further information about the size of the isogeny graph of ρ whose
vertices are homothety classes of stable lattices. The methods are illustrated
with examples for K=Q, and for K imaginary quadratic, ρ being
the representation attached to a Bianchi modular form.
These results form part of the first author's thesis.Comment: 40 pages, 3 figures. Numerous minor revisions following two referees'
report