### Elliptic Curves

#### Definition

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

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 $$E[n]=\{(x,y) \in E(K) | n*(x,y)=\mathcal{O}\}$$

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.