For definite quaternion orders or ideals one can compute reduced bases and Gram matrices.
If the base ring of the order or ideal is Z or Fq[X] with q odd, the Gram matrices can be made unique up to isomorphism. In fact, in these cases, the isomorphism testing of ideals and orders is based on this reduction.
Given a quaternion algebra A over any field F not of characteristic 2, this function returns the underlying F-space with inner product the norm form. A map from A into the structure is returned as second value.
Given a quaternion order S over Z or Fq[X] (with q odd), this function returns the underlying module over its base ring, with inner product respect to the norm. A map from O into the structure is returned as second value.
The Gram matrix of the quaternion order S or ideal I over Z or Fq[X] with respect to the norm on the basis for S.
Given an order or ideal S over Z in a definite quaternion algebra, this function returns the Gram matrix G of the corresponding lattice.The quaternion ideal machinery makes use of a Minkowski basis reduction algorithm which returns a unique normalized reduced Gram matrix G for any definite quaternion ideal. This forms the core of the isomorphism testing for quaternion ideals.
Given an order or ideal S over Z in a definite quaternion algebra, this function returns some basis B of S the Gram matrix G of the corresponding lattice associated with S. Note that while there exists a unique Minkowski-reduced Gram matrix G, the basis B is not unique.
> A := QuaternionOrder(19,2);
> ideals := LeftIdealClasses(A);
> #ideals;
5
> [ (1/Norm(I))*ReducedGramMatrix(I) : I in ideals ];
[
[ 2 0 1 1]
[ 0 2 1 1]
[ 1 1 20 1]
[ 1 1 1 20],
[6 0 1 3]
[0 6 3 1]
[1 3 8 1]
[3 1 1 8],
[6 0 1 3]
[0 6 3 1]
[1 3 8 1]
[3 1 1 8],
[ 4 0 1 -1]
[ 0 4 1 1]
[ 1 1 10 0]
[-1 1 0 10],
[ 4 0 1 -1]
[ 0 4 1 1]
[ 1 1 10 0]
[-1 1 0 10]
]
Canonical: BoolElt Default: false
Given an order or ideal S over Fq[X] in a quaternion algebra A, this function returns a Gram matrix and/or a basis of S whose Gram matrix is in dominant diagonal form (see the function DominantDiagonalForm in Section Automorphism Group and Isometry Testing over Fq[t]). The Gram matrix will not be unique unless A is definite and Canonical is set to true.
Returns a "reduced" basis for the order O or the ideal I over some number ring. If O or I arise from a definite quaternion algebra, then this basis is LLL-reduced with respect to the norm form; otherwise, the basis is reduced with respect to a Minkowski-like embedding (see [KV10, Section 4]).
Given an order O contained in a quaternion algebra A over Q or a number field F, this function returns a new quaternion algebra A' such that A' = ((a, b)/F) where a and b are small (with respect to O), and, as second return value, an isomorphism A to A'.
Given a quaternion algebra A over Q or a number field F, this function returns a new quaternion algebra A' such that A' = ((a, b)/F) where a and b are small. An isomorphism A to A' is returned as second value.
The sequence of all elements x (up to sign) in the definite quaternion order O or ideal I over Z such that the reduced norm of x lies in the interval [A, ... B] or [0, ... B], respectively.
The sequence of elements x (up to sign) in the definite quaternion order O or ideal I over a number ring such that the absolute trace of the norm of x lies in the interval [A, ... B] or [0, ... B], respectively.