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Dou you need your matrix M to stay in the Integer Ring ? If not, you could try to work on the Rational Field:

sage: M = M.change_ring(QQ)
sage: M.echelonize()


or even on the the Real Double Field (which will do some roundings, but may save more space and time):

sage: M = M.change_ring(RDF)
sage: M.echelonize()


No idea whether it will solve your problem.

Dou you need your matrix M to stay in the Integer Ring ? If not, you could try to work on the Rational Field: (Sage will use an algotithm that allow division):

sage: M = M.change_ring(QQ)
sage: M.echelonize()


or even on the the Real Double Field (which will do some roundings, but may save more space and time):

sage: M = M.change_ring(RDF)
sage: M.echelonize()


No idea whether it will solve your problem.

Dou you need your matrix M to stay in the Integer Ring ? If not, you could try to work on the Rational Field (Sage (so that Sage will be allowed to use an algotithm that allow uses division):

sage: M = M.change_ring(QQ)
sage: M.echelonize()


or even on the the Real Double Field (which will do some roundings, but may save more space and time):

sage: M = M.change_ring(RDF)
sage: M.echelonize()


No idea whether it will solve your problem.