November 28, 2020 Dataset Open Access

Isogeny graphs of supersingular elliptic curves

Enric Florit; Gerard Finol

This database contains a collection of data on graphs of supersingular elliptic curves. It contains all graphs for characteristic up to 30,000 and isogeny degrees 2 through 11, and a basic list of invariants for each graph. This will be expanded to include some examples for larger characteristic and also new invariants. A long-term goal is to list examples for higher genera, and to give substantial evidence to discover behavior in such cases.

Tables and visualizations

To consult the tables with the graph invariants, we recommend you to visit our website isogenies.enricflorit.com.

Description of the dataset

The dataset contains one folder for each prime p between 13 and 29989. Each folder contains:

• A JSON file {p}_metadata.json, with the following information:
  • The prime p, the number of supersingular elliptic curves modulo p, and the polynomial used to generate the quadratic extension of Z/pZ.
  • The size of the spine, i.e., the number of supersingular j-invariants defined over Z/pZ.
  • A flag "undirected" to indicate whether the adjacency matrices are symmetric or not.
  • For each isogeny degree ell (2, 3, 5, 7, 11), the diameter of the ell-isogeny graph, the number of Frobenius-conjugate isogenous pairs, and the relevant eigenvalues of the adjacency matrix (second and last, when sorted in decreasing order).
• A plaintext file {p}_nodes.txt, with all supersingular j-invariants modulo p. Note that the minimal polynomial is needed to parse this file, see our repo for a parsing example.
• Adjacency matrices {p}_{ell}.npz, one for each isogeny degree (2, 3, 5, 7, 11). Numpy's npz compressed format is used for convenience, there are packages to parse this format for languages other than python (in particular, for C++, Matlab and R).

Code

The data has been generated using Sagemath and Lithops. The code is available in our github repo github.com/gfinol/IsogenyGraph.