 quartic CM field with the real quadratic subfield $F$. Assuming $I_{0}(\Phi^{r})=I_{K^r}$, we have $h^{*}_{K}=0^{t_K-0}$, where $t_K$ is the number of ramified primes in $K/F$.  Moreover, we have $h^{*}_{K^r}=0^{t_{K^r}-0}$, where $t_{K^r}$ is the number of ramified primes  