Elliptic Curves - Identifying a Torsion Point, with some Rank thrown in I'm a totally amateur mathematician who discovered elliptic curves this past summer and am quite fascinated by them. I am trying to learn more about them, but I am hampered by having studied Abstract Algebra 35 years and not remembering such things as what a field or Galois Theory is. I'm trying to fill in gaps by YouTube videos, but it is slow-going now. I do know what a group is and in particular what an abelian group is, but that's about all I understand in some of the highly technical materials I've come across.
I have read about something called a Torsion Point. However, almost everything I find about them throws in all of these advanced math concepts such as fields. But instead of coming right out here and asking all of you for a plain English definition of a Torsion Point, please let me ask my own question and see if I have the right idea.
The elliptic curve in question is $y^2=x^3-3x+34$. (Correction of 18 to 34 in that equation - I copied the wrong one from my source document.) The points that are both rational and integers are $(-3,4)$, $(-3,-4)$, $(-1,6)$, $(-1,-6)$, $(2,6)$, $(2,-6)$, $(5,12)$, $(5,-12)$, $(15,58)$, $(15,-58)$, $(29,156)$, and $(29,-156)$. I found that if I started with $P:(5,12)$, the next points became $2P:(-1,6)$, $3P:(-3,-4)$, $4P:(2,-6)$, $5P:(29,-156)$, $6P:(15,58)$. At this point, the multiples of $P$ cease being integers and probably continue infinitely.
Here's my question: is the starting point $P:(5,12)$ the Torsion Point of this elliptic curve? I happened upon it as the first one of the sequence of integer points by trial and error, but would there be an easier way to find it? Also, would this elliptic curve be considered to have rank $= 1$, since it appears that all of the rational points turn up with just one starting point? I have seen some examples of elliptic curves with rank given, but again, sometimes the language of Abstract Algebra and other advanced math disciplines gets in the way of understanding how this is computed.
Thanks for any plain English help that anyone can give me on this.
 A: If you understand what $nP$ means, then here is what torsion means: a point $P$ is torsion if some multiple $nP$ of it is equal to the identity (that is, the point at infinity). It is a highly nontrivial fact which was proven by Mazur that for elliptic curves over the rationals, to check that $P$ is not torsion it suffices to check that none of the multiples $2P, 3P, 4P, \dots 12P$ are the point at infinity. So if you compute these multiples for your point $P$ and none of them are the point at infinity then $P$ is not torsion.
An elliptic curve over $\mathbb{Q}$ has rank $1$ if all of its rational points can be generated by a single point $P$, which is not torsion, together with some other points which are torsion. I don't know whether that's true in your case because I don't know what all the rational points are. You say you've found all the integer points, but you need to be able to generate all the rational points too.
(As a historical note, "torsion" means "twist." It's not at all obvious what the above definition has to do with twisting; this is actually quite a long story and involves algebraic topology, specifically some groups that appear in that subject which sometimes have nontrivial torsion in a way that is related to "twists" in some topological space. It would take quite awhile to explain what this all means but in any case it's not a random word that comes out of nowhere, there are specific historical reasons for it.)
If you haven't found it yet then I can warmly recommend Silverman and Tate's Rational Points on Elliptic Curves, which was written to be accessible to undergraduates and I think does a good job at conveying the essential ideas without getting bogged down in technical details.
