defined by an equation of the form with a3 nonzero is a supersingular elliptic curve, and conversely every supersingular curve is isomorphic to one of this form (see Washington2003, p. 122). Over the field with 2 elements any supersingular elliptic curve is isomorphic to exactly one of the supersingular elliptic curves with 1, 3, and 5 points. This gives examples of supersingular elliptic curves over a prime field with different numbers of points. Over an algebraically closed field of characteristic 2 there is (up to isomorphism) exactly one supersingular elliptic curve, given by , with j-invariant 0. Its ring of endomorphisms is the ring of Hurwitz quaternions, generated by the two automorphisms and where is a primitive cube root of unity. Its group of automorphisms is the group of units of the Hurwitz quaternions, which has order 24, contains a normal subgroup of order 8 isomorphic to the quaternion group, and is the binary tetrahedral group If K is a field of characteristic 3, every curve defined by an equation of the form with a4 nonzero is a supersingular elliptic curve, and conversely every supersingular curve is isomorphic to one of this form (see Washington2003, p. 122). Over the field with 3 elements any supersingular elliptic curve is isomorphic to exactly one of the supersingular elliptic curves Over an algebraically closed field of characteristic 3 there is (up to isomorphism) exactly one supersingular elliptic curve, given by , with j-invariant 0. Its ring of endomorphisms is the ring of quaternions of the form a+bj with a and b Eisenstein integers. , generated by the two automorphisms and where i is a primitive fourth root of unity. Its group of automorphisms is the group of units of these quaternions, which has order 12 and contains a normal subgroup of order 3 with quotient a cyclic group of order 4. For with p>3 the elliptic curve defined by with j-invariant 0 is supersingular if and only if and the elliptic curve defined by with j-invariant 1728 is supersingular if and only if (see Washington2003, 4.35). The elliptic curve given by is nonsingular over for . It is supersingular for p = 23 and ordinary for every other (see Hartshorne1977, 4.23.6). The modular curve X0(11) has j-invariant −21211−5313, and is isomorphic to the curve y2 + y = x3 − x2 − 10x − 20. The primes p for which it is supersingular