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In this article, we study the combinatorics of congruence subgroups of themodular group. More precisely, we consider the notion of minimal monomialsolutions. These are the solutions of a matrix equation (also appearing in thestudy of Coxeter friezes), modulo an integer $N$, whose components areidentical and minimal for this property. The objective here is to characterizethe solutions of this type verifying a property of irreducibility modulo aprime power.

Solutions monomiales minimales irr\'eductibles dans $SL_{2}(\mathbb{Z}/p^{n}\mathbb{Z})$