In the answers to this question it was shown that for closed geodesics on $\mathbb{H}^2/\Gamma(2)$, the projection under the modular function $\lambda$ is an immersed topological component of a real algebraic subset of the plane with equation defined over $\mathbb{Q}$. I’m wondering if there is more that one can say about this algebraic curve? In particular, I’m wondering about the following:
If one takes other geodesics on $\mathbb{H}^2/\Gamma(2)$ with the same length, then they are conjugate over $PGL_2(\mathbb{Q})$, and hence their projections under $\lambda$ will give curves that are related by lifting to branched covers over the plane (sometimes called “hidden symmetries”). Can one conclude any relation between the algebraic curves containing these components using this commensurability property?
Will the smallest field of definition of the algebraic curve be over a finite extension of $\mathbb{Q}$, maybe somehow related to the real quadratic field $\mathbb{Q}(\sqrt{tr(A)^2-4})$ (using the notation in this answer)?
Do the other components of the algebraic closure over $\mathbb{Q}$ correspond to projections of geodesics under $\lambda$ as well?
