  Let $p>3$ be a prime and $E$ be a supersingular elliptic curve defined over
$\mathbb{F}_{p^2}$. Let $c$ be a prime with $c < 3p/16$ and $G$ be a subgroup
of $E[c]$ of order $c$. The pair $(E,G)$ is called a supersingular elliptic
curve with level-$c$ structure, and the endomorphism ring $\text{End}(E,G)$ is
isomorphic to an Eichler order with level $c$. We construct two kinds of
Eichler orders $\mathcal{O}_c(q,r)$ and $\mathcal{O}'_c(q,r')$ with level $c$.
Interestingly, we can reduce each $\mathcal{O}_c(q,r)$ or
$\mathcal{O}'_c(q,r')$ to a primitive reduced binary quadratic form with
discriminant $-16cp$ or $-cp$ respectively. If the curve $E$ is
$\mathbb{Z}[\sqrt{-cp}]$-oriented, then we prove that $\text{End}(E,G)$ is
isomorphic to $\mathcal{O}_c(q,r)$ or $\mathcal{O}'_c(q,r')$. Due to the fact
that $\mathbb{Z}[\sqrt{-cp}]$-oriented isogenies between
$\mathbb{Z}[\sqrt{-cp}]$-oriented elliptic curves could be represented by
quadratic forms, we show that these isogenies are reflected in the
corresponding Eichler orders via the composition law for their corresponding
quadratic forms.
