curve can have rank less than 4, and it may be necessary to take a finite extension of the base field K to make the rank of the endomorphism ring 4. In particular the endomorphism ring of an elliptic curve over a field of prime order is never of rank 4, even if the elliptic curve is supersingular. Let G be the formal group associated to E. Since K is of positive characteristic, we can define its height ht(G), which is 2 if and only if E is supersingular and else is 1. We have a Frobenius morphism , which induces a map in cohomology . The elliptic curve E is supersingular if and only if equals 0. We have a Verschiebung operator , which induces a map on the global 1-forms . The elliptic curve E is supersingular if and only if equals 0. An elliptic curve is supersingular if and only if its Hasse invariant is 0. An elliptic curve is supersingular if and only if the group scheme of points of order p is connected. An elliptic curve is supersingular if and only if the dual of the Frobenius map is purely inseparable. An elliptic curve is supersingular if and only if the "multiplication by p" map is purely inseparable and the j-invariant of the curve lies in a quadratic extension of the prime field of K, a finite field of order p2. Suppose E is in Legendre form, defined by the equation , and p is odd. Then for , E is supersingular if and only if the sum vanishes, where . Using this formula, one can show that there are only finitely many supersingular elliptic curves over K (up to isomorphism). Suppose E is given as a cubic curve in the projective plane given by a homogeneous cubic polynomial f(x,y,z). Then E is supersingular if and only if the coefficient of (xyz)p–1 in fp–1 is zero. If the field K is a finite field of order q, then an elliptic curve over K is supersingular if and only if the trace of the q-power Frobenius endomorphism is congruent to zero modulo p. When q=p is a prime greater than 3 this is equivalent to having the trace of Frobenius equal to zero (by the Hasse bound); this does not hold for p=2 or 3. Examples If K is a field of characteristic 2, every curve