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

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