Comparing Galois representations in the residually reducible case

Let $n\ge 2$ and $p$ be a prime. Let $K$ be a number field and consider two Galois representations ${\rho }_1,{\rho }_2:\mathrm{Gal}(\bar{K}\mathrm{/}K)\to {\mathrm{GL}}_n({\mathbb{Z}}_p)$ having residual image a $p$-group. We explain and implement an algorithm that makes effective a result of Loïc Grenié to decide wether the semisimplifications of ${\rho }_1$ and ${\rho }_2$ are isomorphic. As an application, we show that an irreducible representation $\rho :G_{\mathbb{Q}(\sqrt{-3})}\to {\mathrm{GL}}_2({\mathbb{Z}}_3)$ unramified outside 3 is determined by the characteristic polynomials of Frobenius elements at five primes of small norm. As an additional check, we apply it to a 2-adic example studied by Grenié, recovering Grenié’s result in a fully automated way.