TITLE: Integer points of an elliptic curve
QUESTION [21 upvotes]: 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?

REPLY [29 votes]: 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.