Abelian varieties over $\mathbf{Q}$ associated with an imaginary quadratic field

For an imaginary quadratic field $K$ with class number $h$, we shall characterize $h$-dimensional CM abelian varieties over $K$ which descend to abelian varieties over $\mathbf{Q}$. These CM abelian varieties have minimal dimension $h$ both over $K$ and over $\mathbf{Q}$.