@5fb5d869 - it does, thanks!   I was overlooking the difference between SL and PSL.

I'm going to call 2n "N".

Now that I think about it, there's an "obvious" element of order N in SL(2,F) where F is the rationals with a primitive Nth root of unity adjoined, say

ζ = exp(2πi/N) 

Namely, it's

ζ   0 
0 1/ζ

People who think about rotations a lot know this is conjugate to the real matrix

cos θ  -sin θ
sin θ   cos θ

where θ = 2π/N.  But  then the less obvious part is that's it's also conjugate to

0    1
-1 2cos θ

I think there's something I still don't understand well enough, some very classical stuff about the relation between the field F and ℚ[cos θ].