Attributes of Orders

For further information about orders of associative algebras, see Section Orders.

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.