Wrong quaternion order membership testing

Anonymous

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?

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This bug seems to have been reported already.

A fix needs review at:

more

I'm using older version but does it work properly on Sage 9.6? It seems there is no ticket saying it has been fixed yet.

( 2022-03-22 15:46:09 +0200 )edit

Someone needs to review the ticket. Then it can be merged in the next version of Sage.

See Reviewing tickets in the developer guide.

( 2022-03-22 16:51:12 +0200 )edit