Introduction

In Hecke, maximal orders (aka ring of integers), due to their special properties normal orders don't share, come with their own type NfMaximalOrder. While the elements have type NfOrderElem, the ideals and fractional ideals have types NfMaximalOrderIdeal and NfMaximalOrderFracIdeal respectively.

While theoretically a number field contains a unique maximal order (the set of all integral elements), for technical reasons in Hecke a number field admits multiple maximal orders, which are uniquely determined by the number field and a chosen integral basis.

Let $K$ be a number field of degree $d$ with primitive element $\alpha$ and $\mathcal O$ a maximal order $K$ with $\mathbf{Z}$-basis $(\omega_1,\dotsc,\omega_d)$. The basis matrix of $\mathcal O$ is the unique matrix $M_{\mathcal O} \in \operatorname{Mat}{d \times d}(\mathbf{Q})$ such that \begin{align} \begin{pmatrix} \omega_1 \\ \omega_2 \\ \vdots \\ \omega_d \end{pmatrix} = M{\mathcal O} \begin{pmatrix} 1 \\ \alpha \\ \vdots \\ \alpha^{d - 1} \end{pmatrix}. \end{align} If $\beta$ is an element of $\mathcal O$, we call the unique integers $(x_1,\dotsc,x_d) \in \mathbf Z^d$ with \begin{align} \beta = \sum_{i=1}^d x_i \omega_i \end{align} the coefficients of $\beta$ with respect to $\mathcal O$ or just the coefficient vector.

For an ideal $I$ of $\mathcal O$, a basis matrix of $I$ is a matrix $M \in \operatorname{Mat}{d \times d}(\mathbf{Z})$, such that the elements $(\alpha_1,\dotsc,\alpha_d)$ definied by \begin{align} \begin{pmatrix} \alpha_1 \\ \alpha_2 \\ \vdots \\ \alpha_d \end{pmatrix} = M{\mathcal O} \begin{pmatrix} \omega_1 \\ \omega_2 \\ \vdots \\ \omega_d \end{pmatrix} \end{align} form a $\mathbf{Z}$-basis of $I$.

Let $(r,s)$ be the signature of $K$, that is, $K$ has $r$ real embeddings $\sigma_i \colon K \to \mathbf{R}$, $1 \leq i \leq r$, and $2s$ complex embeddings $\sigma_i \colon K \to \mathbf{C}$, $1 \leq i \leq 2s$. We order the complex embeddings such that $\sigma_i = \overline{\sigma_{i+s}}$ for $r + 1 \leq i \leq r + s$. The $\mathbf{Q}$-linear function \[ K \longrightarrow \mathbf R^{d}, \ \alpha \longmapsto (\sigma_1(\alpha),\dotsc,\sigma_r(\alpha),\sqrt{2}\operatorname{Re}(\sigma_{r+1}(\alpha)),\sqrt{2}\operatorname{Im}(\sigma_{r+1}(\alpha)),\dotsc,\sqrt{2}\operatorname{Re}(\sigma_{r+s}(\alpha)),\sqrt{2}\operatorname{Im}(\sigma_{r+s}(\alpha)) \] is called the Minkowski map (or Minkowski embedding). For our maximal order $\mathcal O$ with basis $\omega_1,\dotsc,\omega_d$, the matrix \[ \begin{pmatrix} \sigma_1(\omega_1) & \dotsc & \sigma_r(\omega_1) & \sqrt{2}\operatorname{Re}(\sigma_{r+1}(\omega_1)) & \sqrt{2}\operatorname{Im}(\sigma_{r+1}(\omega_1)) & \dotsc & \sqrt{2}\operatorname{Im}(\sigma_{r+s}(\omega_1)) \\ \sigma_1(\omega_2) & \dotsc & \sigma_r(\omega_2) & \sqrt{2}\operatorname{Re}(\sigma_{r+1}(\omega_2)) & \sqrt{2}\operatorname{Im}(\sigma_{r+1}(\omega_2)) & \dotsc & \sqrt{2}\operatorname{Im}(\sigma_{r+s}(\omega_2)) \\ \vdots & \dotsc & \vdots & \vdots & \dotsc & \vdots & \vdots\\ \sigma_1(\omega_d) & \dotsc & \sigma_r(\omega_d) & \sqrt{2}\operatorname{Re}(\sigma_{r+1}(\omega_d)) & \sqrt{2}\operatorname{Im}(\sigma_{r+2}(\omega_d)) & \dotsc & \sqrt{2}\operatorname{Im}(\sigma_{r+s}(\omega_d)) \end{pmatrix} \in \operatorname{Mat}_{d\times d}(\mathbf R). \] is called the Minkowski matrix of $\mathcal O$.