The topic of integer points on elliptic curves is a big and interesting one, which should probably be discussed in sage-nt rather than sage-support (if one sage-* at all!). So I will be brief.
This is NOT about the Nagel-Lutz Theorem. That says that points of finite order on elliptic curves (defined over Q by Weierstrass equations with a mild restriction) are integral and also satisfy a simple condition on the y-coordinate. but this does NOT tell you how to find all integral points, since in general integral points may have infinite order. The fact that the number of integral points is finite is a special case of a theorem of Siegel. There are essentially two ways to tackle it: as a purely Diophantine problem (which leads to Thue equations and the like); or, using Mordell-Weil bases and elliptic logarithms + LLL. the latter works very well provided that the MW basis can be found, and that is what has been implemented in Sage. (There is also a more general method for finding all S-integral points.) There is a huge literature on all this, which is not at all elementary; the theory was mainly worked out in the 1990s, and is mainly for elliptic curves given by integral Weierstrass models (with also some work on equations of the form y^2=quartic). Note that it only makes sense to ask about integral points on an affine curve, as opposed to a projective one. For a very clear treatment of the simplest case of integral points over Q I would recommend section 8.7 in volume 1 of Henri Cohen's book Number Theory (Vol 1: Tools and Diophantine Equations), GTM 239. John Cremona On Dec 8, 2:10 am, Yann <yannlaiglecha...@gmail.com> wrote: > If you want solution for this precise equation, look for "thue > equation". > The thue equations are some of the few for which there exists > efficient methods. > > for example in PARI/GP (from sage with gp_console()) > > sage: gp_console() > GP/PARI CALCULATOR Version 2.3.3 (released) > [snip] > PARI/GP is free software, covered by the GNU General Public License, > and > comes WITHOUT ANY WARRANTY WHATSOEVER. > > Type ? for help, \q to quit. > Type ?12 for how to get moral (and possibly technical) support. > > parisize = 8000000, primelimit = 500000 > ? t = thueinit(x^3-3*x-1) \\ corresponds to the homogenized > x^3-3*x*y^2-y^3 > ? thue(t,1) \\ solve with rhs = 1 > %2 = [[-3, 2], [-1, 1], [0, -1], [1, -3], [2, 1], [1, 0]] > > PS: this might be wrapped in SAGE, I didn't checked. -- To post to this group, send email to sage-support@googlegroups.com To unsubscribe from this group, send email to sage-support-unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-support URL: http://www.sagemath.org
