Does K and its Hilbert class field have same conductor?

Question

Let $K$ be an abelian number field and $H(K)$ be the Hilbert class field of $K.$

Definition: (conductor of a abelian number field) Let $K$ be a number field with the abelian Galois group over $\Bbb{Q}.$ The conductor $n$ is the smallest even number such that $K\subset Q(\zeta_n).$

Will $K$ and $H(K)$ have the same conductor? Assuming that $H(K)$ is abelian over $\Bbb Q$.

Answer 1

Yes. Under the conditions you describe, $H(K)$ agrees with the genus class field $G(K)$, which has the same conductor as $K$. For an explicit description of $G(K)$, see Xianke Zhang's paper from 1985.