How do you construct a supersingular elliptic curve in the field Fq?

Question

If anyone can give me any links to articles or any information about how to construct a supersingular elliptic curve in Fq then that would be extremely helpful - thanks!

Answer 1

I'm no expert on elliptic curves, but
I have the book "Elliptic Curves" by
Dale Husemöller and I found the following
theorem on p.249:
In characteristic q > 3 the curve
y² = x(x-1)(x-λ) is supersingular if and only if λ is a root
of the polynomial
H_q(λ) = (-1)^m* Σ(i = 0..m) (mCi)²λ^i,
where m = (q-1)/2.
He also proves that the only supersingular
elliptic curve in characteristic 3 is y² = x³ - x.
Hope that helps!