Torsion points in elliptic curve I was going through a research paper based on elliptic divisibility sequences. In that paper the author has taken an elliptic curve over the rational field and a non-torsion point in it, for example the curve is given by $y^{2} = x^{3} + 80$ and the non-torsion point is $P=(4,12)$. By looking at the point and the curve how can we claim the point to be torsion or non-torsion point? What are some methods of determining a point to be torsion/ non-torsion can you explain with some examples.
 A: One way to check that a rational point $P$ on an elliptic curve $E$ defined over $\Bbb Q$ is not torsion is to apply the famous theorem of B. Mazur that lists all the possible torsion groups for $E$, namely $E_{\rm tors}$ must be one of the following:

*

*${\Bbb Z}/n{\Bbb Z}$ with $1\leq n\leq 10$ or $n=12$,

*${\Bbb Z}/2{\Bbb Z}\times{\Bbb Z}/n{\Bbb Z}$ with $2\leq n\leq4$.

This reduces the checking to a finite number of computations.
A: *

*Mazur famously classified all possible torsion subgroups of $E(\mathbb{Q})$, see https://en.wikipedia.org/wiki/Torsion_conjecture. In particular, he showed that the order of a point can not exceed 12. Therefore, to check if a point is non-torsion, one can compute the points $P,2P,\dots, 12P$. If they are distinct, then $P$ has infinite order.

*The canonical height is a function $\hat{h}:E(\bar{\mathbb{Q}})\to [0,\infty)$ which vanishes precisely at the torsion points. Therefore, to check if a point is non-torsion, one can compute $\hat{h}(P)$. If it is non-zero, the point is of infinite order. There are algorithms for computing $\hat{h}$. For example, Magma has the function CanonicalHeight, http://magma.maths.usyd.edu.au/magma/.

A: There are answers here which invoke Mazur's theorem, but there are effective methods for computing torsion subgroups of elliptic curves over any number field (where we have Merel's generalisation of Mazur's result, but it is not so kind as to list all possible subgroup structures).
First we have a theorem of Nagell-Lutz which (over $\mathbb{Q}$) says that if your Weierstrass equation
$$y^2 = f(x)$$
has integer coefficients, then torsion points have $y$-coordinate either $0$ (i.e., $2$-torsion) or the $y$-coordinate is an integer and divides the discriminant of $f(x)$. This gives one a finite list of polynomials for which one needs to compute roots. (This result has a generalisation over arbitrary number fields, see Silverman The Arithmetic of Elliptic Curves Theorem VIII.7.1). In particular, if there are any denominators, you know that the point is not torsion.
Another trick which is often extremely useful is to note that if a finite place $v$ of $K$ at which $E$ has good reduction does not divide $n$ then the $n$-torsion subgroup of $E$ must embed in $E(\mathbb{F}_v)$. Computing the points on our curve over a few finite fields is then often enough to disprove the existence of $K$-rational torsion points.
Your example $E : y^2 = x^3 + 80$ is a clear candidate for both of these approaches. Note that your point $P = (4,12)$ has $5P = (244/25 , 3972/125)$, so can't be torsion.
The second approach, computing $\#E(\mathbb{F}_7)$ and $\#E(\mathbb{F}_{11})$ are $13$ and $12$ respectively. The first implies (by the above) that $\#E(\mathbb{Q})_{tors} = 7^a \cdot 13$ and the second implies $\#E(\mathbb{Q})_{tors} = 11^b \cdot 12$ for some $a,b$. But then clearly $a = b = 0$ so $E$ has trivial torsion over $\mathbb{Q}$.
Note that this second calculation involves only local calculations and no reference to Mazur's theorem - it was known before Mazur how one can compute torsion subgroups.
