[Next][Prev] [Right] [Left] [Up] [Index] [Root]
The quaternion algebra in which S is an order.
The base ring of the quaternion order or ideal S.
The integral basis of the order or ideal S.
Given a quaternion order or ideal, returns the matrix defining the
embedding in its quaternion algebra.
Given an order S over Z in a quaternion algebra A over Q,
returns the reduced discriminant of S.
Given an order S over Z in a quaternion algebra A over Q,
returns the index 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. For an ideal, this
returns the level of the left or right order.
We note that the printing of order or order elements is in terms of
the algebra which contains them, while the coordinates of the basis
elements are with respect to the ideal or order elements.
> S := QuaternionOrder(11);
> i := S.1;
> P<x> := PolynomialRing(Integers());
> MinimalPolynomial(i);
x^2 + 1
> // Create an ideal generated by 2, 1 + i (of norm 2)
> I := LeftIdeal(S, [2,1+i]);
> I;
Quaternion Ideal of level (11, 1) with base ring Integer Ring
> Basis(I);
[ 1 + j + k, i + j + k, 2*j, 2*k ]
> [ Eltseq(x) : x in Basis(I) ];
[
[ 1, 0, 0, 0 ],
[ 0, 1, 0, 0 ],
[ 0, 0, 1, 0 ],
[ 0, 0, 0, 1 ]
]
To get the rational coordinates with respect to the quaternion
algebra, it is necessary to explicitly coerce into the algebra.
This data is equivalent to the function embedding matrix (which
is only applied at the time of printing).
> A := QuaternionAlgebra(S);
> [ Eltseq(A!x) : x in Basis(I) ];
[
[ 1, 0, 1, 1 ],
[ 0, 1, 0, 1 ],
[ 0, 0, 2, 0 ],
[ 0, 0, 0, 2 ]
]
> EmbeddingMatrix(I);
[1 0 1 1]
[0 1 0 1]
[0 0 2 0]
[0 0 0 2]
[Next][Prev] [Right] [Left] [Up] [Index] [Root]