Let $K$ be a number field with the ring of integers $\mathcal{O}$. Let $S=\{\mathfrak{p}_1,\dots ,\mathfrak{p}_k\}$ be a finite set of prime ideals in $\mathcal{O}$. Consider a fractional ideal defined as $\mathcal{L}=\alpha\cdot\mathcal{O}\cdot\mathfrak{p}_1^{-v_1}\cdots\mathfrak{p}_k^{-v_k}$ for some $\alpha\in K$ and $v_j\in\mathbb{Z}$. How can we express a basis matrix for $\mathcal{L}$ as a lattice in terms of a basis matrix $B$ for $\mathcal{O}$ and the norms $\mathcal{N}(\mathfrak{p}_j)$ of the prime ideals?
In this paper: https://fangsong.info/files/pubs/BS_SODA16.pdf , for two fractional ideals $\mathcal{L}=\alpha\cdot\mathcal{O}\cdot\mathfrak{p}_1^{-v_1}\cdots\mathfrak{p}_k^{-v_k}$ and $\mathcal{L}^\prime=\beta\cdot\mathcal{O}\cdot\mathfrak{p}_1^{-w_1}\cdots\mathfrak{p}_k^{-w_k}$, a basis for the fractional ideal $\mathcal{L}^\prime\mathcal{L}^{-1}$ is written as $e^{\text{diag}(a_j)}\cdot B\cdot\text{diag}(d_j/d)$, where $a=(a_j)\in\mathcal{O}$, $B$ is a basis for the ring of integers $\mathcal{O}$, and $d_j, d\in\mathbb{Z}$. (The definition for the inverse ideal: $\mathcal{L}^{-1}=\{x\in K\, |\,x\mathcal{L}\subset\mathcal{O}\}$.) My guess is that $a$ depends on $\beta\alpha^{-1}$, and that $d_j, d$ depend on $\mathcal{N}(\mathfrak{p}_j)$. Is it correct? And how exactly are they expressed?
Thanks in advance for any help.