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If I am considering an elliptic curve, for example $$y^2=x^3-2$$ $$\text{Edit: and } y^2=x^3+2$$ over $\mathbb Q$, how to find rational points?

What possibilities do we have to calculate some of the rational points on it? Are there even possibilities for calculating integer points on the curve?

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  • $\begingroup$ Sorry, I wrote down another elliptic curve, then intendet. $\endgroup$ – Marc Jun 18 '15 at 14:47
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    $\begingroup$ How about (3,5)? $\endgroup$ – Álvaro Lozano-Robledo Jun 18 '15 at 14:48
  • $\begingroup$ I'm afraid not. Sometimes the "first" rational point on the curve (the smallest in absolute value) can be enormous. Obviously I answer in a general context. $\endgroup$ – Piquito Jun 18 '15 at 14:51
  • $\begingroup$ I added another elliptic curve. Thats the curve I am dealing with (I wrote "-" instead of "+", I am sorry for that) $\endgroup$ – Marc Jun 18 '15 at 15:35
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    $\begingroup$ Well... P=(-1,-1) works for the second one... That's a point of infinite order, and generates the group of rational points. For instance, 2P = (17/4, 71/8). $\endgroup$ – Álvaro Lozano-Robledo Jun 18 '15 at 15:47
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The simplest way is to use existing methods in computer algebra systems, e.g. if you use the online Magma calculator here there are now awfully sophisticated algorithms there for this sort of thing. To learn more you could read the relevant section in the Magma handbook here

In the case of the first of your curves, if I put in the following

E:=EllipticCurve([0,0,0,0,-2]);

MordellWeilGroup(E);

RationalPoints(E : Bound:=1000);

then the output is

Abelian Group isomorphic to Z

Defined on 1 generator (free)

Mapping from: Abelian Group isomorphic to Z

Defined on 1 generator (free) to Set of points of E with coordinates in Rational Field given by a rule [no inverse]

true true

{@ (0 : 1 : 0), (3 : 5 : 1), (3 : -5 : 1), (129/100 : 383/1000 : 1), (129/100 : -383/1000 : 1) @}

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Yes, there are methods to calculate points on elliptic curves. There are books dedicated to this topic... I'd recommend Silverman and Tate's "Rational Points on Elliptic Curves", for instance.

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  • $\begingroup$ I got this book. This was actually an exercise from this book. Could you tell me where the methods are explained? I cant find them.. $\endgroup$ – Marc Jun 18 '15 at 15:36
  • $\begingroup$ I also added another elliptic curve above. In my opinion the new one doesnt have any integer points on it. I am also not able to find any rational points.. $\endgroup$ – Marc Jun 18 '15 at 15:37
  • $\begingroup$ Chapter II for the points of finite order, and Chapter III for the points of infinite order (see Section III.6 in particular for examples). In general, finding all rational points is not a simple calculation. In many cases, it is beyond our computing capabilities... $\endgroup$ – Álvaro Lozano-Robledo Jun 18 '15 at 15:45

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