Elliptic Curves


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.