Algorithms for Finding Orientations of Supersingular Elliptic Curves and Quaternion Orders.

This code accompanies the paper "Finding Orientations of Supersingular Elliptic Curves and Quaternion Orders", available on arXiv and eprint. See the paper for the list of authors.

It includes the following:

1. A brute force algorithm for solving the $\mathfrak{O}$-Orienting problem - Given a curve $E$ which can be oriented by a quadratic order $\mathfrak{O}$, find the orientation.
2. Algorithms 5.1 and 5.2 for the Quaternion Embedding Problem - Finding embeddings of a quadratic order within a maximal quaternion order in $B_{p,\infty}$.
   • Includes rerandomization of the basis of the order, making it very fast for any quadratic order with discriminant less than $O(p)$.
   • Algorithm 5.3 for checking if a solution gives a primitive embedding.
   • Experimental results on finding primitive embeddings (i.e. orientations) as described in Section 5.5.
3. Examples for Lemma 5.3 - Checking properties of the Hermite normal form of basis of quaternion orders.

The code is explained with examples in jupyter notebook files which can be found in the Notebooks directory. However, if you just need the code in .py files, it is in the repository root. Tested in Sagemath 10.0.