Neighborhood of the supersingular elliptic curve isogeny graph at \(j = 0\) and 1728 (Q2291364)

In this paper, the authors focus on the supersingular elliptic curves \(E_0:y^2=x^3+1\) and \(E_{1728}:x^3+x\). In particular, they describe the neighborhood of the vertex \([E_0]\) and \([E_{1728}]\) in the \(\ell \)-isogeny graph \(G(\mathbb{F}_{p^2}, -2p)\) when \(p > 3\ell^2\) and \(p>4\ell^2\), respectively. The authors provide the number of loops, they count the number of adjacent vertices and, finally, they give a factorization of the modular polynomials \(\Phi_{\ell}(0,X)\) and \(\Phi_{\ell}(1728,X)\).