# Checking conjugacy of two matrices

I want to check whether or not two matrices are conjugate. I need to check conjugacy for two matrices in $SL(2,\mathbb{Z})$. How to do it ? How to do it for any general ring or field ? (Notice that I just want to verify whether two given matrices are conjugate or not, I don't need the actual matrices; meaning that say $X$ and $Y$ are conjugates and $AXA^{-1}=Y$. I just want to know if or not $X$ and $Y$ are conjugates, I don't need the matrix $A$)

Extending the previous question, say $AXA^{-1}=Y$. Then we can solve the system of linear equations $AX=YA$ to find out $A$. But is there any command which can directly find $A$ ?