are those for which the coefficient of qp in η(τ)2η(11τ)2 vanishes mod p, and are given by the list 2, 19, 29, 199, 569, 809, 1289, 1439, 2539, 3319, 3559, 3919, 5519, 9419, 9539, 9929,... If an elliptic curve over the rationals has complex multiplication then the set of primes for which it is supersingular has density 1/2. If it does not have complex multiplication then Serre showed that the set of primes for which it is supersingular has density zero. showed that any elliptic curve defined over the rationals is supersingular for an infinite number of primes. Classification For each positive characteristic there are only a finite number of possible j-invariants of supersingular elliptic curves. Over an algebraically closed field K an elliptic curve is determined by its j-invariant, so there are only a finite number of supersingular elliptic curves. If each such curve is weighted by 1/|Aut(E)| then the total weight of the supersingular curves is (p–1)/24. Elliptic curves have automorphism groups of order 2 unless their j-invariant is 0 or 1728, so the supersingular elliptic curves are classified as follows. There are exactly ⌊p/12⌋ supersingular elliptic curves with automorphism groups of order 2. In addition if p≡3 mod 4 there is a supersingular elliptic curve (with j-invariant 1728) whose automorphism group is cyclic or order 4 unless p=3 in which case it has order 12, and if p≡2 mod 3 there is a supersingular elliptic curve (with j-invariant 0) whose automorphism group is cyclic of order 6 unless p=2 in which case it has order 24. give a table of all j-invariants of supersingular curves for primes up to 307. For the first few primes the supersingular elliptic curves are given as follows. The number of supersingular values of j other than 0 or 1728 is the integer part of (p−1)/12. See also Supersingular prime Supersingular variety References Robin Hartshorne (1977), Algebraic Geometry, Springer. Joseph H. Silverman (2009), The Arithmetic of Elliptic Curves, Springer. Lawrence C. Washington (2003), Elliptic Curves, Chapman&Hall. Elliptic curves