  The theory of elliptic pairs, as investigated in a paper by Castravet,
Laface, Tevelev, and Ugaglia, provides useful conditions to determine
polyhedrality of the pseudo-effective cone, which give rise to interesting
arithmetic questions when reducing the variety modulo $p$. In this paper, we
examine one such case, namely the blow-up $X$ of 9 points in $\mathbb{P}^2$
lying on the nodal cubic, and study the density of primes $p$ for which the
pseudo-effective cone of the reduction of $X$ modulo $p$ is polyhedral. This
problem reduces to an analogue of Artin's Conjecture on primitive roots like
that investigated by Stephens and then Moree and Stevenhagen. As a result, we
find that the density of such "polyhedral primes" hover around a higher
analogue of the Stephens' Constant under the assumption of the Generalized
Riemann Hypothesis. Finally, in order to determine a precise value for the
density of polyhedral primes, we look at the containment of rank 8 root
sublattices of $\mathbb{E}_8$.
