Let $l$ be an odd prime and let $L=\mathbb{Q}(\zeta_{l^n})$. Suppose $x,y\in \mathbb{Z}[\zeta_{l^n}]$. Let $\mathfrak{p}\mid p$ be a split prime (so in particular $p\equiv 1 \ (\text{mod} \ l^n)$) so that $\mathcal{O}_L/\mathfrak{p} \cong \mathbb{Z}/p\mathbb{Z}$. Let $G=Gal(L/\mathbb{Q})$. Let $(a,b)$ denote the greatest common divisor of $a$ and $b$. For $g\in G$ define $\iota(g)=v_{\mathfrak{p}}(x^g,y^g)$ and $(x(g),y(g))=(x^g,y^g)p^{-\iota(g)}$. Then $x(g),y(g)$ are $\mathbb{p}-$integral and at most one of them is divisible by $\mathfrak{p}$. What I would like to know is if there is an algorithm that allows you to compute a list of possible residues of the $(x(g),y(g))$ from the residues of $(x(1),y(1))$ and a choice of $\zeta_{l^n} \ (\text{mod} \ \mathfrak{p})$.
Even if we begin with the hypothesis that $\mathfrak{p} \nmid x^g$ for all $g \in G$, i.e. that $(N_{L/\mathbb{Q}}(x),p)=1$ so that $(x(g),y(g))=(x^g,y^g)$, the knowledge of the residue of $x(1)$ is probably not enough to find the residues of the conjugates. But maybe there is a way to shorten the list of possible residues. Any insight or reference recs would be appreciated. Thanks!