For further information about orders of associative algebras, see Section Orders.
For a quaternion order S over Z or Fq[X], Magma additionally defines the following functions.
The quaternion algebra for which S is an order.
Returns the basis matrix of the quaternion order S over Z or Fq[X]. The rows of the matrix give the basis elements of S with respect to the basis of the container algebra.
Given an order S over Z or Fq[X], this function returns the reduced discriminant of S as a positive integer or a normalized polynomial.
Given a quaternion order S, this function returns the factorisation of the reduced discriminant of S (that is, Factorization(Discriminant(S))).
Given an order S over Z or Fq[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.
Let S be an order in a definite quaternion algebra A over a field F where F is the rationals, Fq(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.