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I am basically trying to solve the cannonball problem using elliptic curves.

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

Now my plan is to find the torsion group via Nagell-Lutz and then show (somehow) that there is no integral point with infinite order (maybe even find that there is no point of infinite order).

My problem is with the fact that when written in standard form the curve is defined over $\mathbb{Q}$. How do I find a global minimal model for this curve?

Also does my strategy sound about right?

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    Well originally it was both, but after reading Washington's exercise on getting the minimal model I know this bit now.2012-05-07

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Tate's algorithm for computing the conductors was shown by Michael Laska to be able to be adapted for computing the minimal Weierstrass equation.