I follow the notation of Georges Gras: Class Field Theory, some of which I recall for convenience; feel free to skip the following lines if you are familiar with the notation. Let $K$ be a number field, $T$ a finite set of finite places of $K$, and $S=S_0\cup S_\infty$ a finite set of respectively finite and infinite places, with $S_0$ disjoint from $T$. Let $\mathfrak m=\prod_{v\in T}\mathfrak p_v^{m_v}$ ($m_v\ge0$) be a modulus built from $T$.
He defines the generalized class group $\mathcal C\ell^S_\mathfrak{m}$ of $K$ as $$\mathcal C\ell^S_\mathfrak{m}=I_T/P_{T,\mathfrak m, \Delta_\infty}\cdot\langle S_0\rangle$$ where $\Delta_\infty=\text{Pl}_\infty\setminus S_\infty$; $I_T$ are the fractional ideals prime to $T$; $P_{T,\mathfrak m, \Delta_\infty}$ is the group of principal ideals $(x)$ where $v(x)=0$ for all $v\in T$, $x\equiv1\pmod{\mathfrak m}$ and positive on $\Delta_\infty$; and where $\langle S_0\rangle$ is the free group on $S_0$.
Later Gras states a result saying that $$|\mathcal C\ell^S_\mathfrak{m}|=|\mathcal C\ell^S|\dfrac{\varphi(\mathfrak m)}{(E^S:E^S_\mathfrak{m})}$$ where $\varphi$ is the generalized Euler function; $E^S$ the $S$-units and $E^S_\mathfrak m$ the $S$-units congruent to 1 modulo $\mathfrak m$.
I guess I simply do not understand everything going on... If I let $K=\mathbb Q(\sqrt{-5})$, $S=\text{Pl}_\infty$ (or $S=\varnothing$, don't think it matters?) $\mathfrak m=(2,1+\sqrt{-5})$ and $T$ the place corresponding to $\mathfrak m$, I would suppose that $\mathcal C\ell^S_\mathfrak{m}$ was trivial? Since any fractional ideal prime to $\mathfrak m$ is already principal? But I get that $\varphi(\mathfrak m)=1$, that $E^S=\{\pm1\}=E^S_\mathfrak{m}$ and $|\mathcal C\ell^S|=2$ as being the ordinary class group of $K$. What's the stupid mistake(s)? Thanks in advance :)