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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?

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There are precisely two available "serious" implementations of the standard algorithm for computing integral points on an elliptic curve: a non-free 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 S-integral!) 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.

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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, 383-402

  • R.J. Stroeker, N. Tzanakis, Computing all integer solutions of a genus 1 equation Math. Comput. 72 (2003), No. 244, 1917-1933

Gebel, Pethö and Zimmert have used this method for finding all integer points on Bachet-Mordell 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.

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    $\begingroup$ The non-Weierstrass 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 integrality-preservingness of the transform is pretty slick. $\endgroup$
    – Cam McLeman
    Commented 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 non-Weierstrass model remains unclear. $\endgroup$
    – Bogdan Grechuk
    Commented Oct 28, 2021 at 13:38
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    $\begingroup$ @BogdanGrechuk It is easy to change the Weierstrass equation into an equation with integer coefficients, just change with $Y=d^3y$ and $X=d^2x$. $\endgroup$
    – Chris Wuthrich
    Commented Oct 29, 2021 at 13:05
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I would recommend Silverman & Tate's "Rational Points on Elliptic Curves", I'm pretty sure you'll find what you're looking for there.

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    $\begingroup$ Chapter 5 of Silverman-Tate 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. $\endgroup$
    – William Stein
    Commented 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. $\endgroup$
    – Aaron Mazel-Gee
    Commented Nov 21, 2010 at 19:17
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Lemermeyer's resources list of links etc. on elliptic curves.

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  • $\begingroup$ by the by, is Lemermeyer still active in his mathematics? I haven't heard much about him since 2004 or so. $\endgroup$
    – Jose Capco
    Commented 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. $\endgroup$
    – Thomas Riepe
    Commented Nov 24, 2009 at 21:18
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    $\begingroup$ 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). $\endgroup$
    – Franz Lemmermeyer
    Commented Feb 21, 2010 at 20:46

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