  We study the Coulomb branches of the rank-one 4d $\mathcal{N} = 2$ quantum
field theories, including the KK theories obtained from the circle
compactification of the 5d $\mathcal{N}= 1$ $E_n$ Seiberg theories. The focus
is set on the relation between their Seiberg-Witten geometries and rational
elliptic surfaces, with more attention being given to the modular surfaces,
which we completely classify using the classification of subgroups of the
modular group ${\rm PSL}(2,\mathbb{Z})$. We derive closed-form expressions for
the modular functions for all congruence and some of the non-congruence
subgroups associated with these geometries. Moreover, we show how the BPS
quivers of these theories can be determined directly from the fundamental
domains of the monodromy groups and study how changes of these domains can be
interpreted as quiver mutations. This approach can also be generalized to
theories whose Coulomb branches contain `undeformable' singularities, leading
to known quivers of such theories.
