Neighborhood of the supersingular elliptic curve isogeny graph at $j=0$ and $1728$

We describe the neighborhood of the vertex $[E_0]$ (resp. $[E_{1728}]$) in the $\mathrm{\ell }$-isogeny graph ${\mathcal{G}}_{\mathrm{\ell }}({\mathbb{F}}_{p^2},-2p)$ of supersingular elliptic curves over the finite field ${\mathbb{F}}_{p^2}$ when $p>3{\mathrm{\ell }}^2$ (resp. $p>4{\mathrm{\ell }}^2$) with $E_0:y^2=x^3+1$ (resp. $E_{1728}: y^2=x^3+x$) supersingular.