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?
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() [(-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, ); [ (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.
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.