Klein's modular function $J(z)$ is defined and studied in e.g. Apostol's book Modular functions and Dirichlet series in number theory.

Certain specific evaluations are available, for example,

$$J(i) = 1$$

Additionally, it is known that $J(z)$ takes on all complex values.

Question: Can one solve for that $z$ in the fundamental domain

(either explicitly or numerically) which satisfies $J(z) = \mathrm{i}$?