Loops of isogeny graphs of supersingular elliptic curves at j = 0

Abstract We improve Adj et al.'s bound in [1, Theorem 12] from p > 4 l to p > 3 l for the loops of E 0 : y 2 = x 3 + 1 in the l -isogeny graph G l ( F p 2 , − 2 p ) of supersingular elliptic curves over F p 2 with trace − 2 p .