Numerical verification of the Cohen-Lenstra-Martinet heuristics \and of Greenberg's p-rationality conjecture

In this paper we make a series of numerical experiments to support Greenberg's p-rationality conjecture, we present a family of p-rational biquadratic fields and we find new examples of p-rational multiquadratic fields. In the case of multiquadratic and multicubic fields we show that the conjecture is a consequence of the Cohen-Lenstra-Martinet heuristic and of the conjecture of Hofmann and Zhang on the p-adic regulator, and we bring new numerical data to support the extensions of these conjectures. We compare the known algorithmic tools and propose some improvements.