WISDE

Code accompanying "Minimal degree of an isogeny between a supersingular elliptic curve and its conjugate" by Yves Aubry, Roger Oyono and Christelle Vincent

The folders bases and results contain, respectively, files with names bases[primenumber].sage and results[primenumber].sage, respectively, where [primenumber] is a prime.

1. For a given prime p, basesp.sage contains a list containing a basis for each maximal order of the quaternion algebra over QQ ramified at p and infinity. This list is readable by SageMath, and the value of i^2 and j^2 are either those chosen by SageMath upon instantiating B.<i,j,k> = QuaternionAlgebra(p) in SageMath version 10.0.rc3 (in this case, B.<i,j,k> = QuaternionAlgebra(p) is the first line of the file) or those chosen by Magma V2.28-23 or V2.29-2 (in this case, the first line of the file reads B.<i,j,k> = QuaternionAlgebra(a,b) where i^2 = a and j^2 = b).
2. For a given prime p, resultsp.sage contains a list containing a basis for each maximal order of the quaternion algebra over QQ ramified at p and infinity as well as the tuple (N1,N2,N3) giving the successive minima of the quadratic module R of Section 5.1.

The folder delta_p dictionaries contains three files each containing a dictionary whose keys are primes and whose values are the associated value of delta_p. The exact content of each dictionary is given as the first line of each file.

Finally, the folder code contains the code used to obtain the results of this article.

1. The file basiscomputation.mag contains a single Magma procedure to compute a basis for each maximal order of the quaternion algebra over QQ ramified at a prime p and at infinity. To produce the timing of this function given in Section 5.2, for example, we used the following commands:

      p := 263429;
      output_file := "bases" cat Sprint(p) cat ".sage";
      Load("basiscomputation.mag");
      T := Time(); compute_bases(p,output_file); Time(T);
   

2. The file deltacomputation.sage contains the functions necessary to compute delta_p for a prime p. The two main functions are find_dp which calls Magma internally to create the file containing a basis for each maximal order of the quaternion algebra ramified at p and infinity, and find_dp_magma which assumes the existence of a file containing a basis for each maximal order of the quaternion algebra ramified at p and infinity. To produce the timing of the second function given in Section 5.2, for example, we used the following commands:

      p = 263429
      input_file = 'bases' + str(p) + '.sage'
      output_file = 'results' + str(p) + '.sage'
      load('deltacomputation.sage')
      %time find_dp_magma(p, input_file, output_file)
   

3. The file sieving.mag contains a single function that verifies a necessary condition for a prime p to admit a supersingular elliptic curve E defined over FFbar_p with delta(E) = delta_p. To produce the timing of this function given in Section 5.2, for example, we used the commands:

      Load("sieving.mag");
      T := Time(); verify(265231);  Time(T);
   

   To pass the verbose flag, one can type, for example:

       verify(264949: verbose:=true);