maximal order in the quaternion algebra $B_{p,\infty}$ ramified only at $p$ and $\infty$. Such an algebra is isomorphic to ${\mathrm{End}}(E) \otimes {\mathbb{Q}}$ for any supersingular elliptic curve $E/{\mathbb{F}}_{p^2}$. Here we denote ${\mathrm{End}}(E) = {\mathrm{End}}_{\bar{{\mathbb{F}}}_p}(E)$ and if we assume $\#E({\mathbb{F}}_{p^2}) = (p+1)^2$, then the full endomorphism ring ${\mathrm{End}}(E)$ is defined over ${\mathbb{F}}_{p^2}$. Any definite quaternion algebra over ${\mathbb{Q}}$ has a presentation of the form ${\mathbb{Q}}\langle{i,j}\rangle$, where $i^2 = a$, $j^2 = b$, $k = ij = -ji$ for negative integers $a,b$. The canonical involution on $B_{p,\infty}$ is given by $$\alpha = x_0 + x_1 i + x_2 j + x_3 k \longmapsto
\bar{\alpha} = x_0 - x_1 i - x_2 j - x_3 k.$$ from which the reduced trace and norm take the form $${\mathrm{Trd}}(\alpha) = \alpha + \bar{\alpha} = 2x_0
\mbox{ and }
{\mathrm{Nrd}}(\alpha) = \alpha \bar{\alpha} = x_0^2 - a x_1^2 - b x_2^2 + ab x_3^2.$$ The integral basis $\{1,i,j,k\}$ has the nice property of being an orthogonal basis with respect to the bilinear form $\langle{x,y}\rangle = {\mathrm{Nrd}}(x + y) - {\mathrm{Nrd}}(x) - {\mathrm{Nrd}}(y)$ associated to the reduced norm. Nevertheless, the order ${\mathcal{O}}= {\mathbb{Z}}\langle{i,j}\rangle$ is never maximal.

Extremal orders\[sec:prob:extremal\]
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In this work we first place the focus 