Isogeny graphs of supersingular elliptic curves

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. It is available in our github repo github.com/gfinol/IsogenyGraph.本数据库收录了超奇异椭圆曲线（supersingular elliptic curves）图的相关数据集，涵盖特征数不超过30000、同源度（isogeny degree）为2至11的全部图结构，以及每张图的基础不变量列表。后续将扩充更大特征数的示例与新增不变量内容。本项目的长期目标为列出更高亏格的相关示例，并提供充分的实证依据以探索此类场景下的规律。 表格与可视化 如需查阅图不变量相关表格，可访问我们的官方网站：isogenies.enricflorit.com。 数据集说明 本数据集为13至29989之间的每个素数p对应一个独立文件夹，每个文件夹包含以下内容： 1. 名为{p}_metadata.json的JSON文件，包含以下信息： - 素数p、模p的超奇异椭圆曲线数量，以及用于生成有限域Z/pZ的二次扩域的多项式； - 骨干（spine）的大小，即定义在Z/pZ上的超奇异j不变量（j-invariants）的数量； - 一个名为"undirected"的标记，用于指示邻接矩阵（adjacency matrix）是否对称； - 针对每个同调度ℓ（2、3、5、7、11），ℓ-同源图的直径、弗罗贝尼乌斯共轭同源对（Frobenius-conjugate isogenous pairs）的数量，以及邻接矩阵的相关特征值（按降序排序后的第二和最后一个特征值）。 2. 名为{p}_nodes.txt的纯文本文件，收录了模p的全部超奇异j不变量。请注意，解析该文件需使用极小多项式，解析示例可参见我们的代码仓库。 3. 针对每个同调度（2、3、5、7、11）的邻接矩阵文件{p}_{ell}.npz。为便于使用，本数据集采用NumPy的npz压缩格式存储，除Python外，C++、Matlab与R等其他语言均有对应的格式解析工具包。 代码 本数据集通过Sagemath与Lithops生成，相关代码托管于我们的GitHub仓库：github.com/gfinol/IsogenyGraph。