Given an extension $L/K$ of number fields we define the Takagi group as the subgroup $$T_{L/K} = N_{L/K} (D_L) \cdot H_K \subseteq D_K$$ where $N_{L/K}$ is the relative norm, $D_\bullet$ is the multiplicative group of nonzero fractional ideals and $H_\bullet$ is the multiplicative group of principal fractional ideals. For example, if $K$ has class number $1$ then trivially $T_{L/K} \cong D_K$.
I'm interested in trying to compute some less trivial examples, for example the case $K = \mathbb Q(\sqrt{-5})$, $L = \mathbb Q(i, \sqrt{-5})$. How would one do this manually?
In this case, $T_{L/K}$ has index $2$ in $D_K$ by class field theory, hence we must have $T_{L/K} = H_K$. In fact, the norms of principal ideals are principal, but so are the norms of non-principal ideals in the case at hand: since the class group of $K$ is cyclic of order $2$ and generated by $(2,1+\sqrt{-5})$, every nonprincipal ideal has the form $(\alpha) \cdot (2,1+\sqrt{-5})$, and its norm is a principal ideal times the norm of $(2,1+\sqrt{-5})$, which is $(2)$ and thus also principal. This proves that $T_{L/K} = H_K$.
For all other quadratic extensions of $K$, the Takagi group is $D_K$ by class field theory.