An elliptic curve $E$ over a field of characteristic $p$ defined by the cubic equation $f(w,x,y) = 0$ is called supersingular if the coefficient of $(wxy)^{p-1}$ in $f(w,x,y)^{p-1}$ is zero.

A supersingular elliptic curve is said to have Hasse invariant $0$ an ordinary (i.e. non-supersingular) elliptic curve is said to have Hasse invariant $1$

This is equivalent to many other conditions. $E$ is supersingular iff the invariant differential is exact. Also, $E$ is supersingular iff $F^* : H^1(E,\mathcal{O}_E)\to H^1(E,\mathcal{O}_E)$ is nonzero where $F^*$ is induced from the Frobenius morphism $F : E\to E$