From Mazur's theorem we have the following:
If $E |_{\mathbb{Q}}: y^2=x^3+ax+b, a, b \in \mathbb{Z}$ an elliptic curve, then $$E(\mathbb{Q})_{\text{torsion}} \cong \mathbb{Z}/n\mathbb{Z}, \text{ for } n=1, 2, \dots ,10, 12$$ or $$E(\mathbb{Q})_{\text{torsion}} \cong \mathbb{Z}/2m\mathbb{Z} \oplus \mathbb{Z}/ 2\mathbb{Z}, \text{ for } m=1, 2, 3, 4$$
A special case is the following:
$$E|_{\mathbb{Q}}, y^2=x^3+ax, a \in \mathbb{Z}$$ We suppose that $a$ is not divisible by $4^{th}$power $\neq 1$. Then $$E(\mathbb{Q})_{\text{torsion}} \cong \left\{\begin{matrix} \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} & \text{ if } -a=\square \\ \mathbb{Z}/4\mathbb{Z} & \text{ if } a=4\\ \mathbb{Z}/2\mathbb{Z} & \text{ otherwise } \end{matrix}\right.$$
Could you explain it to me??
I guess your question is about the special case of Mazur's theorem for the curves $y^2=x^3+ax$, as the proof of the full theorem is much involved.
Anyway, you could look at Silverman's The Arithmetic of Elliptic Curves, GTM 106, pages 310+311 for an idea how to prove this.