Computing the torsion of the $p$-ramified module of a number field

We fix a prime number $p$ and a number field $K$, and denote by $M$ the maximal abelian $p$-extension of $K$ unramified outside $p$. Our aim is to study the $\mathbb {Z}_p$-module $\mathfrak {X}=\mathrm {Gal}(M/K)$ and to give a method to effectively compute its structure as a $\mathbb {Z}_p$-module. We also give numerical results, for real quadratic fields, cubic fields and quintic fields, together with their interpretations via Cohen-Lenstra heuristics.