Cannonball problem using elliptic curves

I am basically trying to solve the cannonball problem using elliptic curves (see Ch 1 of Washington's book).

In other words I have to show that the only integer points on the "elliptic curve" $6y^2 = 2x^3 + 3x^2 + x$ are $(0,0), (1,\pm 1), (24,\pm 70)$.

I asked this question on stack exchange and so far no luck so turn to overflow.

Washington says that the problem is solvable using the theory of elliptic curves and gives a reference. However the reference does not solve the problem using elliptic curves...neither does any other source I can find on the problem.

How do I go about solving this? I feel it should be just a simple use of the theory but it is proving difficult.

flag	asked May 7 2012 at 9:12 fretty 48●3

With $Y=72y$ and $X=12x+6$ your equation becomes $Y^2 = X^3 - 36 X$. Now use an implemented algorithm for finding $S$-integral points with $S=\{2,3\}$, e.g. in sage, and you are done. There is plenty of literature on such problems, for instance in Smart's book on "The Algorithmic Resolution of Diophantine Equations". LMS Student Text, 41. – Chris Wuthrich May 7 2012 at 10:36

Oh I see, I did have this idea but got a little confused about whether integer points would translate perfectly between the two curves. Now I have just realised that this doesn't matter, ANY integer point on the original curve must give an integer point on the second and we do not care about the converse. – fretty May 7 2012 at 11:49