Torsion subgroup of $y^2=x^3+4x$

Question

I am trying to find the torsion subgroup $E(\mathbb{Q})$ of the elliptic curve $E: y^2=x^3+4x$ over $\mathbb{Q}$ which apparently is $\mathbb{Z}/4\mathbb{Z}$ according to exercise 4.9 of the book "Rational Points on Elliptic curves" by Silverman and Tate. But the only integer points of $E$ that I could find are $(0,0), (2,4)$ and $(2,-4)$. Apparently, $P=(2,4)$ is the generator of $E(\mathbb{Q})$ but the $x$-coordinate of $P+P$ is $x([2]P)=0$. So, how is $P$ a generator of the torsion subgroup of $E(\mathbb{Q})$ as mentioned here? If someone could point out what I am missing here and what is the fourth point of the torsion subgroup of $E(\mathbb{Q})$ I would really appreciate it.