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.
Read more about this topic: Supersingular Elliptic Curve
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