By an order of rank $n$ we mean a unital ring $R$ which is isomorphic to $\mathbb{Z}^n$. An order $R$ is irreducible if it is isomorphic to a subring of a degree $n$ number field $K$ and the fraction field of $R$ is equal to $K$. An order $R$ is monogenic if it is of the form $\mathbb{Z}[\alpha]$ for some $\alpha \in R$.
Let $R$ be an irreducible cubic order. Then the famous Delone-Faddeev correspondence states that $R$ is canonically associated with the $\text{GL}_2(\mathbb{Z})$-equivalence class an integral binary cubic form $F_R$.
We then have the following theorem: Let $p$ be a prime. Then an irreducible cubic order $R$ contains a monogenic suborder of index $p$ if and only if $p$ is representable by the form $F_R$.
What is the corresponding statement for composite $m$? Is it still true that representation by $F_R$ characterized the existence of monogenic suborders of index $m$?
