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# Compute Every Elliptic Curve of Conductor

Problem 4.1.1   Find with computational proof every elliptic curve over of conductor .

Possible strategies:

1. Run Cremona's program (i.e., use modular symbols and linear algebra). This program is freely available, but I don't think anybody but Cremona actually uses it, so... This may or may not work, depending on constraints in the program. Use sage.math.washington.edu which has 64GB RAM.
2. Use the Mestre method of graphs applied to level to compute a sparse matrix for . This will result in a very sparse matrix. Find the dimensions of the eigenspaces of with eigenvalues , possibly using Wiedemann's algorithm, or sparse linear algebra (or ???). Do the same for , , etc., if necessary.
3. Create an algorithm based on Dembelle-Stein-Kohel's ideas, i.e., compute in the quaternion algebra ramified at and with auxiliary level . This will lead to the same linear algebra problem as we get with Mestre's method.

I'm certain 2 or 3 above will succeed, since Andrei Jorza, Jen Balakrishnan, and I did something similar (for prime level) 2 years ago successfully.

Remark 4.1.2 (From Mark Watkins:)

Problem 4.1.1 was done by Cremona when I visited him last November. There are 3 curves of this conductor, two of rank 3, and one of rank 4. All are in the SW ECDB.

234446 [2,117223] 4 8.943847 1 +334976
[1,-1,0,-79,289] [2,1] X 1
234446 [2,117223] 3 9.848943 1 +82752
[1,1,0,-696,6784] [6,1] X 1
234446 [2,117223] 3 19.244917 1 +229824
[1,1,1,-949,-7845] [18,1] X 1


Remark 4.1.3 (From John Cremona:)

I ran level 234446 some time ago! It is not true that the curve of rank 4 is the only one! There are two others and they both have rank 3!

234446  a       1       [1,1,0,-696,6784]       3       1       0
234446  b       1       [1,-1,0,-79,289]        4       1       0
234446  c       1       [1,1,1,-949,-7845]      3       1       0