Isomorphism Classes of Modules over Iwasawa Algebra with $\lambda =4$

We classify the isomorphism classes of finitely generated torsion $\mathcal{O}_E[[T]]$-modules which are free over $\mathcal{O}_E$ of rank $4$, where $\mathcal{O}_E$ is the ring of the integers of a local field $E$. We apply this classification to the Iwasawa module associated to the cyclotomic $\mathbb{Z}_p$-extension of an imaginary quadratic field.