Possible strategies:

- 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. - 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.
- 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.

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

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

I emailed you about this on 20 November 2005....

-so your Problem 4.1.1 needs to be changed....but 4.2 still remains, of course!

John