  The coupling constants of fixed points in the $\epsilon$ expansion at one
loop are known to satisfy a quadratic bound due to Rychkov and Stergiou. We
refer to fixed points that saturate this bound as extremal fixed points. The
theories which contain such fixed points are those which undergo a saddle-node
bifurcation, entailing the presence of a marginal operator. Among bifundamental
theories, a few examples of infinite families of such theories are known. A
necessary condition for extremality is that the sizes of the factors of the
symmetry group of a given theory satisfy a specific Diophantine equation, given
in terms of what we call the extremality polynomial. In this work we study such
Diophantine equations and employ a combination of rigorous and probabilistic
estimates to argue that these infinite families constitute rare exceptions. The
Pell equation, Falting's theorem, Siegel's theorem, and elliptic curves figure
prominently in our analysis. In the cases we study here, more generic classes
of multi-fundamental theories saturate the Rychkov-Stergiou bound only in
sporadic cases or in limits where they degenerate into simpler known examples.
