## Which elements in SL2(Q) are conjugated to an element in SL2(Z)

Dear all,

once again my question is all about $SL_2(\mathbb{Z})$ and $SL_2(\mathbb{Q})$ ! Which elements in $M \in SL_2(\mathbb{Q})$ can you write in the following form:

$M= NBN^{-1}$

with $N \in GL_2(\mathbb{Q})$ and $B \in SL_2(\mathbb{Z})$?

It is surely nor all of $SL_2(\mathbb{Q})$ (look at traces), but I do not have any guess which matrices I get!

Thank you very much again! Karl

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 Thank you for your comment! I will edit my post at once! – Karl Oct 26 2010 at 9:58

You can do this if and only if the trace of $M$ is an integer. By the theory of the rational canonical form if matrices $A$ and $B$ over $\mathbb{Q}$ have the same characteristic polynomial and neither has a repeated eigenvalue they are conjugate by a matrix over $\mathbb{Q}$. This almost does it, save for some fiddling about when the eignvalue of $M$ is repeated.
What if you work over an arbitrary number field instead of $\mathbb{Q}$? Does this argument still hold? – Karl Oct 26 2010 at 10:10