Wrong quaternion order membership testing
Let Q = QuaternionAlgebra(-2167, -7)
be a rational quaternion algebra
with standard basis {1, i , j, k}
as follows.
sage: Q.<i, j, k> = QuaternionAlgebra(-2161,-7)
sage: B = [1/2 + 2/7*j + 1/14*k, 1/32*i + 13/32*j + 19/8*k, 4/7*j + 1/7*k, 4*k]
sage: O = Q.quaternion_order(B)
sage: O
Order of Quaternion Algebra (-2161, -7) with base ring Rational Field
with basis (1/2 + 2/7*j + 1/14*k, 1/32*i + 13/32*j + 19/8*k, 4/7*j + 1/7*k, 4*k)
I think this (maximal Z-)order doesn't contain k
but I get
sage: k in O
True
The is inconsistent with the result on Magma:
> K := Rationals();
> Q<i, j, k> := QuaternionAlgebra<K|-2161, -7>;
> B := [ 1/2 + 2/7*j + 1/14*k, 1/32*i + 13/32*j + 19/8*k, 4/7*j + 1/7*k, 4*k ];
> O := QuaternionOrder(B);
> O;
Order of Quaternion Algebra with base ring Rational Field, defined by i^2 = -2161, j^2 = -7
with coefficient ring Integer Ring
> k in O;
false
> 2*k in O;
false
> 4*k in O;
true
What is the issue here?