Ray class group (${\mathrm{Cl}}_{\mathfrak{m}}(K)$)

The ray class group modulo $\mathfrak{m}$ is $I_K(\mathfrak{m})/P_K(\mathfrak{m})$, where the numerator consists of fractional ideals prime to ${\mathfrak{m}}_0$ and the denominator consists of $(\alpha )$ with $\alpha \equiv 1$ at the finite modulus and positive at every real place in ${\mathfrak{m}}_{\infty }$.

If $U_{\mathfrak{m},1}$ consists of units congruent to one modulo ${\mathfrak{m}}_0$ and positive at the selected real places, then