Galois proved that $\text{PSL}(2,\mathbb{F}_p)$ has a permutation action on $p$ elements for and only for $$\text{PSL}(2,\mathbb{F}_5) \ \ \ \text{PSL}(2,\mathbb{F}_7) \ \ \ \text{ and } \ \ \ \text{PSL}(2,\mathbb{F}_{11}) $$ The Klein quartic $X(7)=\mathbb{H}/\text{PSL}(2,\mathbb{F}_7)$ is an extremely interesting Riemann surface, which I know about. Thus I suspect that the three Riemann surfaces $$X(5) \ \ \ X(7) \ \ \ \text{ and } \ \ \ X(11)$$ should likely also satisfy remarkable properties.
Does anyone know what these properties are, and where I might read about them?
(I mean properties beyond just a simple ``taking the quotient of the Galois proof'', for instance for $X(7)$ its relation to modular forms, quaternions, representations of $\text{PSL}(2,\mathbb{F}_7)$ etc.)
$$\text{}$$ (The title is a reference to Arnold's Trinities)