Conventions

By an absolute number field we mean finite extensions of $\mathbf{Q}$, which is of type AbsSimpleNumField and whose elements are of type AbsSimpleNumFieldElem. Such an absolute number field $K$ is always given in the form $K=\mathbf{Q}(\alpha )=\mathbf{Q}[X]/(f)$, where $f\in \mathbf{Q}[X]$ is an irreducible polynomial. See here for more information on the different types of fields supported.

We call $(1,\alpha ,{\alpha }^2,\dots ,{\alpha }^{d-1})$, where $d$ is the degree $[K:\mathbf{Q}]$ the power basis of $K$. If $\beta$ is any element of $K$, then the representation matrix of $\beta$ is the matrix representing $K\to K,\gamma \mapsto \beta \gamma$ with respect to the power basis, that is,

$$\beta \cdot (1,\alpha ,\dots ,{\alpha }^{d-1})=M_{\alpha }(1,\alpha ,\dots ,{\alpha }^{d-1}).$$

Let $(r,s)$ be the signature of $K$, that is, $K$ has $r$ real embeddings ${\sigma }_i:K\to \mathbf{R}$, $1\le i\le r$, and $2s$ complex embeddings ${\sigma }_i:K\to \mathbf{C}$, $1\le i\le 2s$. In Hecke the complex embeddings are always ordered such that ${\sigma }_i=\overline{{\sigma }_{i+s}}$ for $r+1\le i\le r+s$. The $\mathbf{Q}$-linear function

$$\begin{array}{c}K⟶{\mathbf{R}}^d\\ \alpha ⟼({\sigma }_1(\alpha ),\dots ,{\sigma }_r(\alpha ),\sqrt{2}\mathrm{Re}({\sigma }_{r+1}(\alpha )),\sqrt{2}\mathrm{Im}({\sigma }_{r+1}(\alpha )),\dots ,\sqrt{2}\mathrm{Re}({\sigma }_{r+s}(\alpha )),\sqrt{2}\mathrm{Im}({\sigma }_{r+s}(\alpha )))\end{array}$$

is called the Minkowski map (or Minkowski embedding).

If $K=\mathbf{Q}(\alpha )$ is an absolute number field, then an order $\mathcal{O}$ of $K$ is a subring of the ring of integers ${\mathcal{O}}_K$, which is free of rank $[K:\mathbf{Q}]$ as a $\mathbf{Z}$-module. The natural order $\mathbf{Z}[\alpha ]$ is called the equation order of $K$. In Hecke orders of absolute number fields are constructed (implicitly) by specifying a $\mathbf{Z}$-basis, which is referred to as the basis of $\mathcal{O}$. If $({\omega }_1,\dots ,{\omega }_d)$ is the basis of $\mathcal{O}$, then the matrix $B\in {\mathrm{Mat}}_{d\times d}(\mathbf{Q})$ with

is called the basis matrix of $\mathcal{O}$. We call $det(B)$ the generalized index of $\mathcal{O}$. In case $\mathbf{Z}[\alpha ]\subseteq \mathcal{O}$, the determinant $det(B{)}^{-1}$ is in fact equal to $[\mathcal{O}:\mathbf{Z}[\alpha ]]$ and is called the index of $\mathcal{O}$. The matrix

$$\left(\begin{array}{ccccccc}{\sigma }_1({\omega }_1)& \dots & {\sigma }_r({\omega }_1)& \sqrt{2}\mathrm{Re}({\sigma }_{r+1}({\omega }_1))& \sqrt{2}\mathrm{Im}({\sigma }_{r+1}({\omega }_1))& \dots & \sqrt{2}\mathrm{Im}({\sigma }_{r+s}({\omega }_1))\\ \\ {\sigma }_1({\omega }_2)& \dots & {\sigma }_r({\omega }_2)& \sqrt{2}\mathrm{Re}({\sigma }_{r+1}({\omega }_2))& \sqrt{2}\mathrm{Im}({\sigma }_{r+1}({\omega }_2))& \dots & \sqrt{2}\mathrm{Im}({\sigma }_{r+s}({\omega }_2))\\ \\ ⋮& \ddots & ⋮& ⋮& ⋮& \ddots & ⋮\\ \\ {\sigma }_1({\omega }_d)& \dots & {\sigma }_r({\omega }_d)& \sqrt{2}\mathrm{Re}({\sigma }_{r+1}({\omega }_d))& \sqrt{2}\mathrm{Im}({\sigma }_{r+2}({\omega }_d))& \dots & \sqrt{2}\mathrm{Im}({\sigma }_{r+s}({\omega }_d))\end{array}\right)\in {\mathrm{Mat}}_{d\times d}(\mathbf{R}).$$

is called the Minkowski matrix of $\mathcal{O}$.