Elliptic Curves

Definition

Let $K$ be a field. An elliptic curve $E$ is a plane curve defined over the field $K$ as follows: $$E(K)=\{y^2=x^3+ax+b : (x,y) \in K^2\} \cup \{O\}$$ where $(a,b) \in K^2$. The point $O$ is called the infinity point of the curve. The set $E(K)$ forms an abelian group with identity element $O$.

In addition, we need the curve to have no cusps, self-intersections, or isolated points. Algebraically, this can be defined by the condition $4a^3+27b^2 \neq 0$ in the field $K$.

The $j-$ invariant of an elliptic curve is defined to be $-1728\frac{4a^3}{4a^3+27b^2}$. Two elliptic curves are isomorphic to each other if and only if they have the same $j-$ invariant value.

The endomorphism ring of $E$ is denoted $End(E)$. The structure of $End(E)$ can be found in Chapter 3.9 of Silverman's book.

For an integer $n$, we define $E[n]=\{(x,y) \in E(K) | n*(x,y)=O\}$

Over a field $\mathbb{F}_p$, there are two types of curves: Ordinary and Supersingular, based on the set $E[p]$. We are interested in studying Supersingular curves, since the isogeny graph on these curves has nice structure and properties.