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For a definite quaternion order or ideal defined over Z, one can
compute a Minkowski-reduced Gram matrix.
NormSpace(S) : AlgQuatOrd -> ModTupRng, Map
NormModule(S) : AlgQuatOrd -> ModTupRng
Given a quaternion algebra A over Q or an order or ideal S
over Z, returns the underlying
space or module over its base ring, with inner product respect to the
norm, followed by the map into the structure.
GramMatrix(I) : AlgQuatOrdIdl -> AlgMatElt
The Gram matrix of the quaternion order S or ideal I over Z
with respect to the norm on the basis for S.
ReducedBasis(S) : AlgQuatOrdIdl -> SeqEnum
ReducedBasis(S) : AlgAssVOrdIdl -> SeqEnum
Given an order or ideal S over Z in a definite quaternion algebra,
returns a Minkowski-reduced basis for S.
ReducedGramMatrix(S) : AlgQuatOrdIdl -> AlgMatElt
The unique Minkowski-reduced Gram matrix of a reduced basis for the
definite quaternion order or ideal S over Z.
The quaternion ideal machinery makes use of a Minkowski basis reduction
algorithm which returns a uniquely normalized reduced Gram matrix for
any definite quaternion ideal. This forms the core of the isomorphism
testing for quaternion ideals.
> 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]
]
OptimisedRepresentation(O) : AlgAssVOrd -> AlgQuat, Map
Given an order O contained in a quaternion algebra A, returns a new
quaternion algebra A' such that A' = ((a, b)/F) where a and b
are small (with respect to O), and an isomorphism A |-> A'.
OptimisedRepresentation(A) : AlgQuat -> AlgQuat, Map
Given a quaternion algebra A, returns a new
quaternion algebra A' such that A' = ((a, b)/F) where a and b
are small, and an isomorphism A - > A'.
Enumerate(O, B) : AlgAssVOrd[RngOrd], RngElt -> [AlgAssVOrdElt]
Enumerate(O, B) : AlgAssVOrd[RngOrd], [RngElt] -> [AlgAssVOrdElt]
Enumerate(I, B) : AlgAssVOrdIdl[RngOrd], [RngElt] -> [AlgAssVOrdElt]
The sequence of elements x in the definite quaternion order O
up to sign such that the trace of the norm of x lies in the
interval [A, ... B] or [0, ... B], respectively.
ReducedBasis(I) : AlgAssVOrdIdl[RngOrd] -> [AlgAssVOrdElt]
Returns a "reduced" basis of the order O or the ideal I. 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.
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