of the norm form is $D^2 p^2$, from which we conclude that $R + Rj$ has index $|D|$ in any maximal order.

By convention, for our special $p$-extremal order ${\mathcal{O}}$, we fix ${\mathbb{Z}}[i] \subseteq R$ with $i^2 = -q$ and $D = {\mathrm{disc}}(R) 
= -d({\mathcal{O}})$, and $j^2 = -p$ (i.e. $j = \pi$ above). Being of smallest discriminant, $R$ is necessarily a maximal order whose discriminant is the first of the sequence $$-3, -4, -7, -8, -q \text{ for prime } q \equiv 3 \bmod 4,$$ such that $p$ is ramified or inert in $R$. The next three lemmas establish existence for $q = 1$, $q = 2$, and $q \equiv 3 \bmod 4$ prime. These lemmas incorporate and expand on Propositions 5.1 and 5.2 of Pizer [@Pizer1980]. We recall that an order in a quaternion algebra is [*Eichler*]{} if it is the intersection of two maximal orders.

\[lem:disc-4\] Let $p \equiv 3 \bmod 4$ be a prime, and let $B = {\mathbb{Q}}\langle{i,j}\rangle$ be the quaternion algebra given by the presentation $i^2 = -1$, $j^2 = -p$, and $k = ij = -ji$, and set $R = {\mathbb{Z}}[i]$. Then $B$ is ramified only at $p$ and $\infty$, and ${\mathbb{Z}}\langle{i,j}\rangle$ is 