[[Was08], Chapter XII.1] Let \(E_1:y^2=x^3+a_1x+b_1\) and \(E_2:y^2=x^3+a_2x+b_2\) be elliptic curves over a field \(K\). An isogeny from \(E_1\) to \(E_2\) is a nonconstant homorphism \(\alpha:E_1 \rightarrow E_2\) that is given by rational functions such that \(\alpha(O)=O\).

This means \(\alpha(P+Q)=\alpha(P)+\alpha(Q)\) for all \(P,Q \in E_1\) and there exists rational functions \(P, Q\) such that if \(\alpha(x_1, y_1)=(P(x_1, y_1),Q(x_1, y_1))\).

In fact, it can be proved that we can write \(\alpha\) in the form \(\alpha(x_1, y_1)=(p(x_1), y_1q(x_1))\).

If \(p(x)=\dfrac{r(x)}{s(x)}\) for polynomials \(r\) and \(s\) without common roots, define the degree of \(\alpha\) to be \(Max(deg(r(x)),deg(s(x)))\).

We say an isogeny is seperable if \(s(x)\) have no repeated roots.


Consider two curves \(E_1:y^2=x^3+x\) and \(E_2:y^2=x^3-4x\) over \(\mathbb{F}_{11}\). Then the map $$\alpha: E_1 \rightarrow E_2$$ $$(x,y) \mapsto \left(\dfrac{x^2+1}{x},\dfrac{y(x^2-1)}{x}\right)$$ is an isogeny from \(E_1\) to \(E_2\).


We mention several important properties of isogenies.

  1. Isogenies are uniquely determined by their kernel: Given an elliptic curve \(E\) and a subgroup \(L\), there is an unique elliptic curve \(E'\) and an isogeny \(\alpha: E \rightarrow E'\) such that the kernel of \(\alpha\) is \(L\).

  2. [[Sil09], Chapter III.6, Theorem 6.2] For every isogeny \(\alpha: E \rightarrow E'\) of degree \(l\), there exists an unique dual isogeny \(\hat{\alpha}: E' \rightarrow E\) such that \(\alpha \hat{\alpha}= \hat{\alpha} \alpha=l\)

  3. [[Unknown bib ref: Gha21], Proposition 2.2] (Decomposition of isogenies) Let \(\alpha: E \rightarrow E'\) be a seperable isogeny. Then there exists an integer \(k\) elliptic curves \(E=E_0, E_1,...,E_n=E'\) and isogenies \(\beta_i: E_i \rightarrow E_{i+1}\) of prime degree such that \(\alpha=\beta_{n-1} \beta_{n-2} ... \beta_{0} [k]\)

The edges of Supersingular isogeny graphs are determined by isogenies between the curves, we will talk about it later in the definition of the graph.