<s>[INST] Abelian torsion group of rational points of an elliptic curve I want to find the abelian group of rational points $E(\mathbb{Q})_{\text{torsion}}$ of the elliptic curve $y^2=x^3+8$. 
$$E(\mathbb{Q})_{\text{torsion}}=\{P \in E(\mathbb{Q}) | P \text{ of finite order }\}$$ 
From Lutz-Nagell theorem we have the following: 
Let $E|_{\mathbb{Q}} , y^2=x^3+ax+b, a, b \in \mathbb{Z}$ and $P=(x, y) \in  E(\mathbb{Q})$. 
We suppose that $P$ is of finite order. 
Then $x, y \in \mathbb{Z}$ and $y=0$ (that corresponds to the points of order $2$ ) or $y^2 \mid D(f)=4a^3+27b^2$. 
We have also the following: 
If $P(x, y) \in E(\mathbb{Q})$ and $x \notin \mathbb{Z}$ or $y \notin \mathbb{Z}$, then $P$ has infinite order. 
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So, to find the points $P$ of finite order of the curve $y^2=x^3+8$ we do the following: 
For $y=0 \Rightarrow x^3+8=0 \Rightarrow x=-2$, $$P=(-2, 0) \in E(\mathbb{Q})$$ of order $2$
Let $f(x)=x^3+8$, then $D(f)=27 \cdot 8^2=2^6 \cdot 3^3$. 
$$y^2 \mid 2^6 \cdot 3^3$$ 
$$2^i, i=0, 2, 4, 6 \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3^j, j=0, 2$$
$$y^2 \mid D(f) \Rightarrow y^2=1, 3^2, 2^2, 2^2 \cdot 3^2, 2^4, 2^4 \cdot 3^2, 2^6, 2^6 \cdot 3^2$$ 
$$ \Rightarrow y=\pm 1 , \pm 3, \pm 2, \pm 2 \cdot 3, \pm 2^2 , \pm 2^2 \cdot 3 , \pm 2^3, \pm 2^3 \cdot 3 \\ \Rightarrow y=\pm 1 , \pm 3, \pm 2, \pm 6, \pm 4 , \pm 12 , \pm 8, \pm 24 $$ 
For $y=\pm 1 \Rightarrow x^3+7=0 \Rightarrow x \notin \mathbb{Z}$. 
For $y=\pm 3 \Rightarrow x^3-1=0 \Rightarrow x=1$, so there are two points with integer coordinates, $P_1=(1, 3), P_2=(1, -3)$. 
Does this mean from Lutz-Nagell theorem that these two points are of finite order?? 
For $y=\pm 2 \Rightarrow x^3+4=0 \Rightarrow x \notin \mathbb{Z}$. 
For $y=\pm 6 \Rightarrow x^3-28=0 \Rightarrow x \notin \mathbb{Z}$. 
For $y=\pm 4 \Rightarrow x^3-8=0 \Rightarrow x=2$, so there are two points with integer coordinates, $P_3=(2, 4), P_4=(2, -4)$. 
Which is the order of these points??
For $y=\pm 12 \Rightarrow x^3-136=0 \Rightarrow x \notin \mathbb{Z}$. 
For $y=\pm 8 \Rightarrow x^3-56=0 \Rightarrow x \notin \mathbb{Z}$. 
For $y=\pm 24 \Rightarrow x^3-568=0 \Rightarrow x \notin \mathbb{Z}$. 
 [/INST] For this curve the torsion points are $P = (-2, 0)$ of order $2$ and the point at infinity. The reason why your points $P_1, ..., P_4$ are not torsion points are due to the fact that the Nagell-Lutz Theorem only gives you the possible candidates for the torsion points. In other words, Nagell-Lutz is not a if and only if-theorem.
When checking the list of possible torsion points, then if a point $P$ has order greater than $12$ then that point has infinite order. So you need to check if $nP = \mathcal{O}$, for $2 \leq n \leq 12$. If not, then $P$ has infinite order. This is due to the celebrated theorem of Barry Mazur, see wikipedia.
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