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Let Q 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 + 2/7j + 1/14k, 1/32i + 13/32j + 19/8k, 4/7j + 1/7k, 4k]; O
Order of Quaternion Algebra (-2161, -7) with base ring Rational Field with basis (1/2 + 2/7j + 1/14k, 1/32i + 13/32j + 19/8k, 4/7j + 1/7k, 4k)

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 + 2/7j + 1/14k, 1/32i + 13/32j + 19/8k, 4/7j + 1/7k, 4k ];
O:= QuaternionOrder(B);
O;
k in O;
2k in O;
4k in O;

Order of Quaternion Algebra with base ring Rational Field, defined by i^2 = -2161, j^2 = -7 with coefficient ring Integer Ring
false
false
true

What is the issue here?

Let Q = (-2167, -7) be a rational quaternion algebra with standard basis {1, i , j, k} as follows.

I think this (maximal Z-)order doesn't contain k but I get

The is inconsistent with the result on Magma:

K:=Rationals();
Q<i, j,="" k="">:=QuaternionAlgebra<k|-2161, -7="">;
B:=[ 1/2 + 2/7j + 1/14k, 1/32i + 13/32j + 19/8k, 4/7j + 1/7k, 4k ];
O:= QuaternionOrder(B);
O;
k in O;
2k in O;
4k in O;

Order of Quaternion Algebra with base ring Rational Field, defined by i^2 = -2161, j^2 = -7 with coefficient ring Integer Ring
false
false
true

What is the issue here?

Let Q = (-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 + 2/7j + 1/14k, 1/32i + 13/32j + 19/8k, 4/7j + 1/7k, 4k]; O
Order of Quaternion Algebra (-2161, -7) with base ring Rational Field with basis (1/2 + 2/7j + 1/14k, 1/32i + 13/32j + 19/8k, 4/7j + 1/7k, 4k)

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 + 2/7j + 1/14k, 1/32i + 13/32j + 19/8k, 4/7j + 1/7k, 4k ];
O:= QuaternionOrder(B);
O;
k in O;
2k in O;
4k in O;

Order of Quaternion Algebra with base ring Rational Field, defined by i^2 = -2161, j^2 = -7 with coefficient ring Integer Ring
false
false
true

What is the issue here?

Let Q = (-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 + 2/7j + 1/14k, 1/32i + 13/32j + 19/8k, 4/7j + 1/7k, 4k]; O
Order of Quaternion Algebra (-2161, -7) with base ring Rational Field with basis (1/2 + 2/7j + 1/14k, 1/32i + 13/32j + 19/8k, 4/7j + 1/7k, 4k)

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 + 2/7j + 1/14k, 1/32i + 13/32j + 19/8k, 4/7j + 1/7k, 4k ];
O:= QuaternionOrder(B);
O;
k in O;
2k in O;
4k in O;

O;
Order of Quaternion Algebra with base ring Rational Field, defined by i^2 = -2161, j^2 = -7 with coefficient ring Integer Ring
false
false
true

What is the issue here?

Let Q = (-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 + 2/7j + 1/14k, 1/32i + 13/32j + 19/8k, 4/7j + 1/7k, 4k]; O
Order of Quaternion Algebra (-2161, -7) with base ring Rational Field with basis (1/2 + 2/7j + 1/14k, 1/32i + 13/32j + 19/8k, 4/7j + 1/7k, 4k)

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 + 2/7j + 1/14k, 1/32i + 13/32j + 19/8k, 4/7j + 1/7k, 4k ];
O:= QuaternionOrder(B);
O;
k in O;
2k in O;
4k in O;
Order of Quaternion Algebra with base ring Rational Field, defined by i^2 = -2161, j^2 = -7 with coefficient ring Integer Ring
false
false
true

What is the issue here?

Let Q = (-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 + 2/7j + 1/14k, 1/32i + 13/32j + 19/8k, 4/7j + 1/7k, 4k]; O
Order of Quaternion Algebra (-2161, -7) with base ring Rational Field with basis (1/2 + 2/7j + 1/14k, 1/32i + 13/32j + 19/8k, 4/7j + 1/7k, 4k)

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 + 2/7j + 1/14k, 1/32i + 13/32j + 19/8k, 4/7j + 1/7k, 4k ];
O:= QuaternionOrder(B);
O;
k in O;
2k in O;
4k in O;
Order of Quaternion Algebra with base ring Rational Field, defined by i^2 = -2161, j^2 = -7 with coefficient ring Integer Ring
false
false
true

What is the issue here?

Let Q = (-2167, -7) -7) be a rational quaternion algebra with standard basis {1, i , j, k} k} as follows.

I think this (maximal Z-)order doesn't contain k k but I get

The is inconsistent with the result on Magma:

K:=Rationals();
Magma:

> K := Rationals();
> Q<i, j,="" k="">:=QuaternionAlgebra<k|-2161, -7="">; 

  B:=[ j, k> := QuaternionAlgebra<K|-2161, -7>;
> B := [ 1/2 + 2/7j + 1/14k, 1/32i + 13/32j + 19/8k, 4/7j + 1/7k, 4k ]; 

  O:= QuaternionOrder(B); 

  O; 

  k in O; 

  2k in O; 

  4k in O; 
 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 

  false 

  false 

  true  
 

What is the issue here?

Let Q = (-2167, QuaternionAlgebra(-2167, -7) be a rational quaternion algebra 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?

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?