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.
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Jyrki Lahtonen
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1A closely related older question. I haven't looked at it closely enough, but I hope/suspect that you can rewrite the numerator of the generator I found in a form that fits this. – Jyrki Lahtonen Nov 02 '19 at 08:50