TITLE: Which elements in SL2(Q) are conjugated to an element in SL2(Z)
QUESTION [11 upvotes]: 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

REPLY [17 votes]: 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.