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There are several formulas for the number of orbits of the projective lineunder the action of subgroups of $GL_2$. We give an interpretation of two suchformulas in terms of the geometry of elliptic curves, and prove a more generalformula for a large class of congruence subgroups of Bianchi groups. Ourformula involves the number of walks on a certain graph called an isogenyvolcano. Underlying our results is a complete description of the group ofextensions of a pair of CM elliptic curves, and of a pair of lattices in aquadratic field.

Extensions of CM elliptic curves and orbit counting on the projective line