It is known that $$j(\tau)=\frac{256(1-x)^3}{x^2}$$ where $x=\lambda (\tau)(1-\lambda (\tau))$ and $\lambda (\tau)$ is the modular lambda function.
But I came across the following statement (https://en.wikipedia.org/wiki/J-invariant):
Rational functions of $j$ are modular, and in fact give all modular functions.
So $\lambda (\tau)$ is a rational function of $j(\tau)$. I tried to solve
$$j(\tau)=\frac{256(1-x)^3}{x^2}$$
for $x$ but ended up with long rooty expressions, not hinting at anything rational. That doesn't look good...
How can I express $\lambda (\tau)$ as a rational function of $j(\tau)$?