Vtyjkyuk

I would like to know the set $X=J^-1(mathbbR)$, where $J$ is Klein's $J$-invariant.
Since $J$ is a modular function, it suffices to know the intersection of $X$ and the fundamental domain $mathrmRe(tau)$.

The special values of $J$ suggest that $J(tau)>0$ for $tau=i y$ with $yge 1$, but I cannot prove that it holds generally.

The $j$-invariant is real on the fundamental domain, if and only if $tau$
lies on the boundary (both vertical lines and the circular arc) or if $tau$ has zero real part (lies on the imaginary axis).

To prove this, note that on the vertical lines with integer and half-integer real
values, the $h$-function is real, and the unit semicircle is equivalent uunder
$textSL_2(Bbb Z)$ to one of these vertical lines, so $j$ is real where I said.

Also $j$ tends to $infty$ and $-infty$ as $tautoinfty$ on these vertical
lines, so takes every real value, by the IVT, on the stated set. Now since
the $j$-function is bijective on the "strict" fundamental region, then elsewhere
the $j$-function is non-real.