How to find explicitly the Klein j-invariant

Question

Let $K$ be a field, and let $K(x)$ be the associated field of rational functions. I want to find the subfield $L$ of $K(x)$ of the rational functions that are invariant under this set of transformations: $$x\mapsto x$$ $$x\mapsto \frac{1}{x}$$ $$x\mapsto 1-x$$ $$x\mapsto \frac{x}{x-1}$$ $$x\mapsto \frac{1}{1-x}$$ $$x\mapsto 1-\frac{1}{x}$$ By Lüroth theorem I know that there exist a rational function $\ell(x)$ such that $L=K(\ell(x))$ and I know that I can take $$\ell(x)=\frac{(x^2-x+1)^3}{x^2(x-1)^2}$$ In each book or lecture notes I have found, this object is only defined, and later is proved to be invariant, but I was interested in how to find it without too many calculations. Are there some references or can you give a hint? Many thanks in advance.