In passing, we will mention that there is no known method to determine in a finite number of steps whether a given rational cubic has a rational point. There is no analogue of Hasse's theorem for cubics. That question is still open, and it is a very important question. The idea of looking modulo m for all integers m is not sufficient. Selmer gave the example
3X3 + 4Y3 + 5Z3 = 0.
This is a cubic, and Selmer shows by an ingenious argument that it has no integer solutions other than
(0, 0, 0). However, one can check that for every integer m, the congruence
3X3 + 4Y3 + 5Z3 ≡ 0 (mod m)
has a solution in integers with no common factor. So for general cubics, the existence of a solution modulo m for all m does not ensure that a rational solution exists. We will leave this difficult problem aside, and assume that we have a cubic which has a rational point O.