Supersingular Elliptic Curve - Equivalent Conditions

Equivalent Conditions

There is a number of equivalent conditions to supersingularity:

• Supersingular elliptic curves have many endomorphisms in the sense that an elliptic curve is supersingular if and only if its endomorphism algebra (over ) is an order in a quaternion algebra. Thus, their endomorphism group has rank 4, while the endomorphism group of every other elliptic curve has only rank 1 or 2.

• 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.

• Suppose E is in Legendre form, defined by the equation . Then 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 for every K.