Integer points of an elliptic curve 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?
 A: 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.
A: 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.
A: Lemermeyer's resources list of links etc. on elliptic curves.
A: I would recommend Silverman & Tate's "Rational Points on Elliptic Curves", I'm pretty sure you'll find what you're looking for there.
