Speed up calculation of left kernel

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First, calculations over the integers will be slow, and the best way to speed them up is, if possible, work over a field (or several fields) instead. Can you get the information you need from A.change_ring(QQ) and A.change_ring(GF(2))?

If that won't work, then you should try some of the different available algorithms for left_kernel. I tried this with much smaller matrices, just to test things:

sage: %timeit random_matrix(ZZ, 123, 74).left_kernel()
1 loop, best of 3: 8.25 s per loop
sage: %timeit random_matrix(ZZ, 123, 74).left_kernel(algorithm='default')
1 loop, best of 3: 8.28 s per loop
sage: %timeit random_matrix(ZZ, 123, 74).left_kernel(algorithm='pari')
1 loop, best of 3: 1.07 s per loop
sage: %timeit random_matrix(ZZ, 123, 74).left_kernel(algorithm='flint')
1 loop, best of 3: 7.77 s per loop
sage: %timeit random_matrix(ZZ, 123, 74).left_kernel(algorithm='padic')
1 loop, best of 3: 7.93 s per loop

The different algorithms may not scale equally well, though, so experiment with different algorithms and with different sizes of matrices.