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?

  • $\begingroup$ The problem is mentioned in Chapter 1 of Larry Washington's textbook, Elliptic Curves: Number Theory and Cryptography. I know that doesn't answer the question - it just gives you a place to look. $\endgroup$ – Gerry Myerson May 6 '12 at 3:56
  • $\begingroup$ Another non-answer: Zagier takes a different approach to solving the cannonball problem at people.mpim-bonn.mpg.de/zagier/files/doi/10.2307/2007900/… $\endgroup$ – Gerry Myerson May 6 '12 at 4:04
  • $\begingroup$ Well actually Washington is where I found the problem, he translates into elliptic curves and says that the problem can then be solved using some of the theorems but only gives a reference and never returns to the problem. The reference he gives is a journal entry and solves the problem in some other way (elliptic curves are never mentioned). I have just read Silverman so reckon I should know all I need to know to solve this. $\endgroup$ – fretty May 6 '12 at 8:42
  • $\begingroup$ Just curious, do you need the equation to be a global minimal model for some reason, or would it be enough to have an integral model? $\endgroup$ – Álvaro Lozano-Robledo May 7 '12 at 3:54
  • $\begingroup$ Probably an integral model will be enough. Even better if we get an isogeny between the two curves. $\endgroup$ – fretty May 7 '12 at 8:03

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


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