Let $T\subset {\rm GL}(N,{\mathbb Q})$ be an $n$-dimensional ${\mathbb Q}$-torus given by its Lie algebra $\mathfrak{t}\subset \mathfrak{gl}(N,{\mathbb Q})$. I want to compute, in some sense explicitly, the cocharacter group $$M=X_*(T):=X_*(T_{\overline{\mathbb Q}})={\rm Hom}({\mathbb G}_{{\rm m},\overline{\mathbb Q}}, T_{\overline {\mathbb Q}}); $$ of course, it is isomorphic to ${\mathbb Z}^n$. The absolute Galois group ${\rm Gal}(\overline {\mathbb Q}/{\mathbb Q})$ naturally acts on $M$.
Consider the effective Galois group $$\Theta:={\rm im}\big[{\rm Gal}(\overline {\mathbb Q}/{\mathbb Q})\to {\rm GL}(M)\big].$$ We obtain a finite group $\Theta\subseteq {\rm GL}(n,{\mathbb Z})$.
Question 1. How can I compute $\Theta\subseteq {\rm GL}(n,{\mathbb Z})$ using computer?
Let $L/{\mathbb Q}$ be the finite Galois extension corresponding to the kernel $$\ker\big[{\rm Gal}(\overline{\mathbb Q}/{\mathbb Q})\to \Theta\big].$$ Then ${\rm Gal}(L/{\mathbb Q})=\Theta$.
Let $V_{\mathbb Q}=\{\infty,2,3,5,7,11,\dots\}$ denote the set of places of ${\mathbb Q}$, and let $V_L$ denote the set of places of $L$. We have a natural surjective map $$V_L\to V_{\mathbb Q}\,,$$ and $\Theta$ acts on the fibers of this map via the action on $L$. Let $p\in V_{\mathbb Q}$ be a prime number, and let $v\in V_L$ be a place of $L$ over $p$. Consider the decomposition group $\Theta_v={\rm Stab}_\Theta(v)$. It is determined by $p$ uniquely up to conjugation.
Question 2. How can I compute the subgroup $\Theta_v\subseteq\Theta\subseteq {\rm GL}(M)= {\rm GL}(n,{\mathbb Z})$ using computer?
Assuming that I have computed $M$ and $\Theta_v\subseteq{\rm GL}(M)$, I can compute using computer the first Galois cohomology group $$ H^1({\mathbb Q}_p,T)\cong(M_{\Theta_v})_{\rm Tors}\,,$$ the torsion subgroup of the group of co-invariants of $\Theta_v$ in $M$.
