Let $p$ be an odd prime, and denote by $Cl_p(H)$ the $p$-part of the ideal class group of a number field $H$. Let $\Delta:=Gal(\mathbb{Q}(\mu_p)/\mathbb{Q})$ and $\omega : \Delta \longrightarrow \mathbb{Z}_p^\times$ be the Teichmuller character. For a $\mathbb{Z}_p[\Delta]$-module $C$ and an integer $i$, consider $C^{(\omega^i)}$ the $(\omega^i)$-isotypic component of $C$.

The Iwasawa main conjecture states that for odd $i$, the characteristic ideal of the Iwasawa module $\varprojlim_n Cl_p(\mathbb{Q}(\mu_{p^n})^{(\omega^i)}$ (where transition maps are norm maps) is generated by the Kubota-Leopoldt $p$-adic $L$-function (see 6.1 for more details). It was proven by Mazur and Wiles, who also obtained the following formula :

$$|\#Cl_p(\mathbb{Q}(\mu_p))^{(\omega^i)}|_p = |B_{1,\omega^{p-i}}|_p$$

where $B_{1,\omega^{p-i}}$ is a generalized Bernouilli number.

Thus, we know the order of $Cl_p(\mathbb{Q}(\mu_p))^{(\omega^i)}$. My question is the following : can we (at least conjecturally) compute the full $\mathbb{Z}_p$-structure of this finite module in terms of an analytic invariant such as the Kubota-Leopoldt $p$-adic $L$-function ?