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

Abstract

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