# Glossary

## Supersingular elliptic curves

An elliptic curve over a finite field of characteristic p is supersingular if it has no p-torsion points. This is equivalent to the fact that the endomorphism ring of the curve is an order in an imaginary quadratic field. The j-invariant of a supersingular curve is defined over $\mathbb{F}_{p^2}$. There are $[p/12] + \varepsilon$ supersingular j-invariants, where $\varepsilon = 0,1,1,2$ whenever $p \equiv 1, 5, 7, 11 \mod 12$ (respectively).

## Isogenies and degree

Given two elliptic curves $E$ and $E'$, an isogeny between them is a non-constant (and thus surjective) morphism of curves $E\to E'$ fixing the point at infinity. The isogeny is called separable or inseparable according to the extension of fields $K(E)/K(E')$. The degree of the isogeny is $[K(E):K(E)]$, and it equals the size of the kernel whenever it is separable. If the degree $\ell$ is different from the characteristic $p$, the isogeny is always separable.

## The supersingular isogeny graph $\Gamma_1(\ell;p)$

For $\ell \neq p$, the graph $\Gamma_1(\ell;p)$ has as vertices the set of supersingular j invariants (i.e., the isomorphism classes of supersingular elliptic curves), and as edges the set of $\ell$-degree isogenies between them.

## The spine

The spine of $\Gamma_1(\ell;p)$ is the induced subgraph of vertices that are defined over $\mathbb{F}_p$.

## Diameter

The diameter of a graph is the largest minimal distance between any two nodes. The diameter of a graph is dominated by the eigenvalues of its adjacency matrix (see below). The fact that $\Gamma_1(\ell;p)$ is an expander family (for fixed $\ell$) implies that the diameters are proportional to $\log p$.

## Isogenous conjugate pairs

Each j-invariant outside of the spine, $j\in\mathbb{F}_{p^2}$, has a Frobenius conjugate, $j^p$. This corresponds to the Frobenius (inseparable) isogeny that any curve has: $E \to E^{(p)}$, given by $(x,y)\mapsto (x^p,y^p)$. An $\ell$-isogenous conjugate pair is a pair of vertices $(j,j^p)$ that are connected by an $\ell$-degree isogeny. The number of such pairs $(j,j^p)$ with $j\neq j^p$ is denoted by $c_\ell$.

## Eigenvalues

The supersingular isogeny graph $\Gamma_1(\ell;p)$ is an expander Ramanujan graph. This means that if we order the eigenvalues of its adjacency matrix, $\lambda_1 > \lambda_2 \geq \cdots \geq \lambda_n$, we will have $\lambda_1 = \ell + 1$ (the regular degree of the graph), and $\max(|\lambda_2|,|\lambda_n|) \leq 2\sqrt{\ell}$. We list here these two eigenvalues, except in the cases where the graph has fewer than 4 vertices.

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The Isogeny Database.