Let $K = \mathbb{Q}(\theta)$ be an (algebraic) number field, $f(X)$ be the minimal polynomial of $\theta$ and $O_K$ be its ring of integers. It's known that, if $K$ is monogenic, i.e. $O_K = \mathbb{Z}[\alpha]$ (ring extension) with some $\alpha \in K$, then the codifferent (fractional) ideal $O^\vee_K = [f'(\alpha)]^{-1} O_K$. In the other words, consider all of its respective integral ideals, which are achieved by suitably scaling $O^\vee_K$, $O_K$ is one of them and is principal, hence the rest of them is principal. Now we call that $O^\vee_K$ is principal. In general, it's equivalent to that the different (integral) ideal $(O^\vee_K)^{-1}$ is principal.
On the other hand, it's also known that, there is infinitely many $K$ that is not monogenic. $K = \mathbb{Q}(\beta)$ ($\beta$ is the real root of $X^3-X^2-2X-8$) is Dedekind's example. Despite that, it's not clear whether $O^\vee_K$ and $(O^\vee_K)^{-1}$ are principal or not. For this example, we have $O_K = \mathbb{Z} \left [ \beta, \frac{\beta+\beta^2}{2} \right ]$ and $O^\vee_K = \frac{207-18\beta-20\beta^2}{503}O_K$ is principal, leading to that $(O^\vee_K)^{-1} = (25+14\beta+8\beta^2)O_K$ is principal.
My curiosity makes me target the exhaustion of cases. Particularly, I want to find a $K$ that is not monogenic and has $O^\vee_K$, $(O^\vee_K)^{-1}$ be not principal. That is my question.