# Attributes of Orders

For further information about orders of associative algebras, see Section [Orders](../AssociativeAlgebras/alg_ass_ord.md#secalgassvord).

For a quaternion order $S$ over ${\mathbb{Z}}$ or ${\mathbb{F}}_q[X]$, Magma additionally defines the following functions.

## `Algebra(S): AlgQuatOrd -> AlgQuat`

## `QuaternionAlgebra(S): AlgQuatOrd -> AlgQuat`

The quaternion algebra for which $S$ is an order.

## `BasisMatrix(S): AlgQuatOrd -> AlgMatElt`

## `EmbeddingMatrix(S): AlgQuatOrd -> AlgMatElt`

Returns the basis matrix of the quaternion order $S$ over ${\mathbb{Z}}$ or ${\mathbb{F}}_q[X]$. The rows of the matrix give the basis elements of $S$ with respect to the basis of the container algebra.

## `Discriminant(S): AlgQuatOrd -> RngElt`

Given an order $S$ over ${\mathbb{Z}}$ or ${\mathbb{F}}_q[X]$, this function returns the reduced discriminant of $S$ as a positive integer or a normalized polynomial.

## `FactoredDiscriminant(S): AlgQuatOrd -> SeqEnum`

Given a quaternion order $S$, this function returns the factorisation of the reduced discriminant of $S$ (that is, `Factorization(Discriminant(S))`).

## `Conductor(S): AlgQuatOrd -> RngElt`

## `Level(S): AlgQuatOrd -> RngElt`

Given an order $S$ over ${\mathbb{Z}}$ or ${\mathbb{F}}_q[X]$ in a quaternion algebra $A$, this function returns the reduced index of $S$ in a maximal order of $A$ containing it. Together with the reduced discriminant of the order, this serves to classify the local isomorphism class of an Eichler order.

## `Normalizer(S): AlgAssVOrd -> Grp, Map`

Let $S$ be an order in a definite quaternion algebra $A$ over a field $F$ where $F$ is the rationals, ${\mathbb{F}}_q(t)$ or a number field. This function returns a matrix group $G$ isomorphic to the normalizer of $S$ in $A^*$ modulo $F^*$. A homomorphism from $G$ to $A^*$ is also returned.