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Hi,

20,000 x 20,000 is not large and you should be able to do it in a few seconds on a modern CPU:

http://m4ri.sagemath.org/performance.html

However, Sage doesn't expose the fast code we have for this in M4RI:, cf.

http://m4ri.sagemath.org/doxygen/solve_8h.html

That's why it's not as fast as it should be. Sorry, we should fix this.