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 \(\mathbb{F}_p\).


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 \(c_\ell\).


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.