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.