Elliptic Curves


Let \(K\) be a field. An elliptic curve \(E\) is a plane curve defined over the field \(K\) as follows: $$E(K)=\{y^2=x^3+ax+b : (x,y) \in K^2\} \cup \{O\}$$ 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.