Smoothness of Shortest Isogenies to Frobenius Conjugates

Let $E$ be chosen uniformly from the geometric isomorphism classes of supersingular elliptic curves over ${\bar{\mathbb{F}}}_p$ and ${\delta }_E$ be the minimum degree of isogenies from $E$ to its Frobenius conjugate $E^{(p)}$. We prove the smoothness of ${\delta }_E$ and hence affirm Wesolowski’s $p^{1\mathrm{/}3+o(1)}$ attack on the supersingular endomorphism ring problem is unconditional.