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### Determine Two quadratic form is integer congruence (rational equivalent)?

Fixed $n \geq 2$ and consider $A,B \in GL(n,\mathbb{Z}).$ We know that we have the Smith normal form. One can find $U, V \in SL(n,\mathbb{Z})$ such that $A=UDV.$ So as $B$. The Smith normal form is easy to compute by using Mathematica.

We also define Two matrices $A,B$ are congruent if there exists $X \in PSL(n,\mathbb{Z})$ such that $X^TAX=B$.

Now we have the following, if two matrices are congruent, then they have the same $D$. But now I want to ask is that I have two 4 by 4 matrices. I have checked they have the same $D$ in the smith normal form. But how do I know how to verify they are congruent. If it is, I have to find such $X.$

To be more specific, suppose $A=\left( \begin{array}{cccc} 2 & -1 & -1 & -1 \ -1 & 2 & 0 & 0 \ -1 & 0 & 2 & 0 \ -1 & 0 & 0 & 2 \ \end{array} \right), B=\left( \begin{array}{cccc} 4 & -6 & 4 & -2 \ -6 & 12 & -8 & 4 \ 4 & -8 & 6 & -3 \ -2 & 4 & -3 & 2 \ \end{array} \right).$ I want to ask whether $A,B$ are congruent. I listed what I already tried.

1. $A$, $B$ both have the same $D=\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 2 & 0 \ 0 & 0 & 0 & 2 \ \end{array} \right)$ in the smith normal form. So they define the same integer lattices. But I do not know whether they are congruent.

I suspect that they are congruent but I do have an idea on how to solve them. Any ideas or comments are really appreciated. Btw, $A$ is the Cartan matrix of $SO(8).$ I think it will not help.

I used sage and type them in as the quadratic form. But I found that "is_rationally_isometric" function only works for number field but not integers ring (if I use integers, there will be an error like no real_embedding). The "is_rationally_isometric" is True. But I do not know whether they are isometric in Integer. If the answer is true. Can I find such X?

A=QuadraticForm(QQ, 4,[4,-12,8,-4,12,-16,8,6,-6,2])
print(A)
print(A1)
A1.is_rationally_isometric(A)
# the output is True. 2 None

### Determine Two quadratic form is integer congruence (rational equivalent)?

Fixed Fix $n \geq 2$ and consider $A,B \in GL(n,\mathbb{Z}).$ \operatorname{GL}(n,\mathbb{Z}).$We know that we have the Smith normal form. form. One can find$U, V \in SL(n,\mathbb{Z})$\operatorname{SL}(n,\mathbb{Z})$ such that $A=UDV.$ So as $A=UDV.$ Likewise for $B$. The Smith normal form is easy to compute by using Mathematica.

We also define Two call two matrices $A,B$ are congruent if there exists exists $X \in PSL(n,\mathbb{Z})$ \operatorname{PSL}(n,\mathbb{Z})$such that$X^TAX=B$. Now we have the following, following: if two matrices are congruent, then they have the same$D$.$D$. But now I want to ask is that I have now, given two 4 by 4 matrices. I have checked they have the matrices with same$D$in the smith Smith normal form. But how do I know form$D$, how to verify they are congruent. If it is, I have congruent? An if they are, how to find a matrix$X$such$X.$that$X^T A X = B$. To be more specific, suppose$A=\left( \begin{array}{cccc} 2 & -1 & -1 & -1 \ -1 & 2 & 0 & 0 \ -1 & 0 & 2 & 0 \ -1 & 0 & 0 & 2 \ \end{array} \right), B=\left( \begin{array}{cccc} suppose $A = \begin{bmatrix} 2 & -1 & -1 & -1 \\ -1 & 2 & 0 & 0 \\ -1 & 0 & 2 & 0 \\ -1 & 0 & 0 & 2 \\ \end{bmatrix}$ and $B = \begin{bmatrix} 4 & -6 & 4 & -2 \ \\ -6 & 12 & -8 & 4 \ \\ 4 & -8 & 6 & -3 \ \\ -2 & 4 & -3 & 2 \ \end{array} \right).$ \\ \end{bmatrix}.$I want to ask whether$A,B$are congruent. I listed what I already tried. 1. Here is how far I got.$A$,$B$both have the same$D=\left( \begin{array}{cccc} Smith normal form $D = \begin{bmatrix} 1 & 0 & 0 & 0 \ \\ 0 & 1 & 0 & 0 \ 0 & 0 & 2 & 0 \ 0 & 0 & 0 & 2 \ \end{array} \right)$ in the smith normal form. \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \\ \end{bmatrix}.$So they define the same integer lattices. lattices. But I do not know whether they are congruent. I suspect that they are congruent but I do have an no idea on how to solve them. prove it. Any ideas or comments are really appreciated. Btw, appreciated. By the way,$A$is the Cartan matrix of$SO(8).SO(8).\$ I think it will not help.

I used sage Sage and type typed them in as the quadratic form. forms. But I found that "is_rationally_isometric" the is_rationally_isometric function only works works for number field but not integers ring (if I use integers, integers, there will be an error like no real_embedding). real_embedding). The "is_rationally_isometric" is True. is_rationally_isometric is True. But I do not know whether they are isometric in Integer. over the integers. If the answer is true. Can true, can I find such X?

A=QuadraticForm(QQ, 4,[4,-12,8,-4,12,-16,8,6,-6,2])
sage: A = QuadraticForm(QQ, 4, [4, -12, 8, -4, 12, -16, 8, 6, -6, 2])
sage: print(A)