Attributes of Orders

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

For a quaternion order S over Z or 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 Z or 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 Z or 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 Z or 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, 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.