$ below the bound ${\sqrt{d_{K}/d^{0}_F}}/{0}$. (In this step we eliminate most of the CM fields.)  -   For each $\mathfrak{q}$ with norm $Q$ below the bound $00\log(|d_{K^r}|)^0$, check whether it is in $I_{0}(\Phi^r)$ as follows. List all quartic Weil $Q$-polynomials, that is, monic integer polynomials of degree $0$ such that all roots in ${\mathbb{C}}$ have absolute value $\sqrt{Q}$. For each, take its roots in $K$ and check whether ${\text{\normalfont{N}}}_{\Phi^r}(\mathfrak{q})$ is generated by such a root. If not, then $\mathfrak{q}$ is not in $I_0(\Phi^r)$, so we throw away the field.  -   For each $K