$I$ to $J$ is a map given by $\alpha \mapsto \alpha\gamma$ for $\gamma$ in ${B_{p,\infty}}^*$, which is an isomorphism if $J = I\gamma$. By the multiplicativity of the reduced norm, isomorphisms are similitudes of quadratic modules (with respect to the reduced norm). In particular, an isomorphism sends a reduced basis to a reduced basis. In fact the normalized norm map $$q_I = \frac{{\mathrm{Nrd}}}{{\mathrm{Nrd}}(I)} : I \longrightarrow {\mathbb{Z}}$$ remains invariant under this isomorphism, in the sense that $q_I(\alpha)
= q_J(\beta)$ for $\alpha$ in $I$ and $\beta = \alpha\gamma$ in $J$. The normalized norm $q_I$ is a positive-definite integral quadratic map, whose bilinear module given by $
\langle{x,y}\rangle = q_I(x+y) - q_I(x) - q_I(y)
$ has determinant $p^2$. This follows from the same property for any maximal order (see Pizer [@Pizer1980 Proposition 1.1]), since $|{\mathcal{O}}/I| = {\mathrm{Nrd}}(I)^2$, and the fact that any submodule of index $m$ in a quadratic module $L$ has determinant $m^2\det(L)$.

The following lemma serves to replace an ideal $I$ with an isomorphic one of different reduced norm.

\[lem:ideal\_norm\_rep\] Let $I$ be a left ${\mathcal{O}}$-ideal of reduced norm $N$ and $\alpha$ an element of $I$. Then $I\gamma$, where $\gamma = \bar{\alpha}/N$, is a left ${\mathcal{O}}$-ideal of norm $q_I(\alpha)$.

By 