Let $E$ be a supersingular elliptic curve which is defined over field $\mathbb{F}_{p^2}$ and $l$ be a prime such that $gcd(l,p)=1$. Is there an endomorphism $\phi\in End(E)$ such that $deg(\phi)=l$? Suppose that $H$ is a cyclic subgroup of $E$ and $|H|=l$. I also want to know if I construct an isogeny using velo formula, is it possible to have $j(E/H)=j(E)$?