Where can I found some resources to learn how to determine the integer points of given elliptic curve? I would like to learn a method based on computing the rank and the torsion group of given curve. Also, how can I determine the integer points if the curve is not on its Weierstrass form?
4 Answers
There are precisely two available "serious" implementations of the standard algorithm for computing integral points on an elliptic curve: a nonfree one in Magma (http://magma.maths.usyd.edu.au/magma/) and a free one in Sage (http://sagemath.org). The one in Sage was done by Cremona and two German masters students a few years ago, and when refereeing the Sage code, I compared the answers with Magma, and uncovered and reported numerous bugs in Magma, which were subsequently fixed. Here's how to use Sage to find all integral (or Sintegral!) points on a curve over Q:
sage: E = EllipticCurve([1,2,3,4,5])
sage: E.integral_points()
[(1 : 2 : 1)]
sage: E.S_integral_points([2])
[(103/64 : 233/512 : 1), (1 : 2 : 1)]
and here is how to use Magma:
> E := EllipticCurve([1,2,3,4,5]);
> IntegralPoints(E);
[ (1 : 2 : 1) ]
> SIntegralPoints(E, [2]);
[ (1 : 2 : 1), (103/64 : 233/512 : 1) ]
Note that in both cases by default the points are only returned up to sign. In Sage you get both signs like this:
sage: E.integral_points(both_signs=True)
[(1 : 6 : 1), (1 : 2 : 1)]
Finally, you can use Magma for free online here: http://magma.maths.usyd.edu.au/calc/ and you can use Sage free here: https://sagecell.sagemath.org/. With Sage, you can also just download it for free and install it on your computer. With Magma, you have to pay between $100 and a few thousand dollars, depending on who you are, and deal with copy protection.
NOTE: Technically a system called SIMATH (http://tnt.math.se.tmu.ac.jp/simath/) had an implementation of computing integral points. But it was killed by our friends at Siemens Corp.

3$\begingroup$ I forgot to mention that the integral_points command in Sage is documented here: sagemath.org/doc/reference/sage/schemes/elliptic_curves/… $\endgroup$ Nov 20, 2010 at 21:55

4$\begingroup$ WARNING: Once the Magma integral points code was fixed, the Magma developers found bugs in the Sage integral_points code, as of Sage4.6.1. See trac.sagemath.org/sage_trac/ticket/10152 $\endgroup$ Mar 7, 2011 at 0:01

2$\begingroup$ Link to documentation is now at doc.sagemath.org/html/en/reference/arithmetic_curves/sage/… $\endgroup$ Nov 13, 2020 at 18:53
Just in case anyone is still reading: the tool of the trade are elliptic logarithms. A lot of people have worked on making this effective, but two recent articles pointing you in the right direction are
A Pethö, H.G. Zimmer, J. Gebel, E. Herrmann, Computing all $S$integral points on elliptic curves, Math. Proc. Camb. Philos. Soc. 127 (1999), No.3, 383402
R.J. Stroeker, N. Tzanakis, Computing all integer solutions of a genus 1 equation Math. Comput. 72 (2003), No. 244, 19171933
Gebel, Pethö and Zimmert have used this method for finding all integer points on BachetMordell curves $y^2 = x^3+k$ for all small values of $k$, for example.
There are problems ahead if the curve is not in Weierstrass form since the transformation from a genus 1 curve to a curve in Weierstrass form does not preserve integrality. I do not remember whether you can find anything useful in the textbook
 S. Schmitt, H.G. Zimmer, Elliptic curves. A computational approach , de Gruyter (2003)
but would be surprised if you couldn't.

3$\begingroup$ The nonWeierstrass form is taken care of in Stroeker and de Weger's "Solving elliptic diophantine equations: The general cubic case." I recently had cause to work through it carefully, and the trick to deal with the non integralitypreservingness of the transform is pretty slick. $\endgroup$ Sep 27, 2010 at 1:01

$\begingroup$ Hm, I had a look at that paper and see that when we transform the equation from general cubic to Weierstrass form then the coefficients become rational, not integer. If I then try to apply Magma or SageMath code to find integer points it returns error and states that coefficients should be integers. So, how to find integer points in general nonWeierstrass model remains unclear. $\endgroup$ Oct 28, 2021 at 13:38

$\begingroup$ @BogdanGrechuk It is easy to change the Weierstrass equation into an equation with integer coefficients, just change with $Y=d^3y$ and $X=d^2x$. $\endgroup$ Oct 29, 2021 at 13:05
I would recommend Silverman & Tate's "Rational Points on Elliptic Curves", I'm pretty sure you'll find what you're looking for there.

2$\begingroup$ Chapter 5 of SilvermanTate is about integral points, and about why there are finitely many. It doesn't give much help in actually determining them, which was the question. $\endgroup$ Nov 20, 2010 at 21:41

$\begingroup$ Oh okay. It's been a while since I've worked with that book. Thanks for the clarification. $\endgroup$ Nov 21, 2010 at 19:17

$\begingroup$ by the by, is Lemermeyer still active in his mathematics? I haven't heard much about him since 2004 or so. $\endgroup$ Nov 24, 2009 at 14:06

$\begingroup$ His website links to a school where he apparently teaches, more i don't know. BTW, could one read your "How to play like Alekhine"? Some of my friends are excellent chess players and surely would like to read it. $\endgroup$ Nov 24, 2009 at 21:18

9$\begingroup$ Yes, I'm currently teaching at a gymnasium (the German equivalent of a high school). I spent a lot of my time since 2003 on projects in German (correspondence Hasse  Noether with Roquette, Jacobi's lectures in number theory from 1836/37 with H. Pieper). $\endgroup$ Feb 21, 2010 at 20:46