on the [*$p$-extremal*]{} maximal orders ${\mathcal{O}}$ containing $\pi$ such that $\pi^2 = -p$. For a general order there exists a unique maximal $2$-sided ideal $\mathfrak{P}$ over $p$, and this ideal is principal if and only if there exists such an element $\pi$. The maximal ideal $\mathfrak{P}$ is a generator of the $2$-sided class group, and $p$-extremal orders are precisely those of trivial $2$-sided class number. In the context of supersingular elliptic curves, these are the maximal orders which are endomorphism rings of elliptic curves defined over ${\mathbb{F}}_p$ with Frobenius endomorphism $\pi$.

Secondly, we focus on orders with distinguished quadratic subring $R$. For a maximal order ${\mathcal{O}}$ we define $
d({\mathcal{O}}) = \min\{ {\mathrm{disc}}(R) : {\mathbb{Z}}\ne R \subsetneq {\mathcal{O}}\}.
$ Among all $p$-extremal maximal quaternion orders, we define a [*special*]{} $p$-extremal maximal order ${\mathcal{O}}$ to be a $p$-extremal maximal order such that $d({\mathcal{O}})$ is minimal.

The following lemma establishes the main properties we need for such an order, after which Lemmas \[lem:disc-4\], \[lem:disc-8\], and \[lem:disc-q\] provide for their existence by explicit construction.

\[lem:special-p-extremal-properties\] Let ${\mathcal{O}}$ be a maximal order in $B_{p,\infty}$ containing a subring ${\mathbb{Z}}\langle{i,j}\rangle$ with $i^2=-q$, $j^2=-p$, and $ij = -ji$, for $q$ coprime to $p$. Set $R = {\mathcal{O}}\cap 