Let $E$ be an elliptic curve defined by $E:y^2＝x^3-x$. I want to figure out all points $P∈E(\bar{\Bbb Q })$ which satisfies $[2]P∈E({\Bbb Q })$.In other word, I would like to explicitly write down the elements of a set
$K'＝${${P∈E(\bar{\Bbb Q })｜[2]P∈E({\Bbb Q })}$}$＝[2]^{-1}E( \Bbb Q)$. Is this hand manageable?
Motivation of this question:$K'/K$ is finite and it's Galois group is known to be abelian exponent $2$. This kind of extension is used in the proof of weak model Weil theorem, and I want to check with my hand concrete examples of such ($[m]^{-1}E(K)$) extensions.
Thank you in advance.
For this particular elliptic curve, the Mordell-Weil rank is equal to $0$. This is equivalent to the fact that $1$ is not a congruent number, which can be proved with elementary methods.
Thus we can completely determine: $E(\Bbb Q) = \{(-1, 0), (0, 0), (1, 0), \infty\} = E[2]$.
Thus the set of all points $P$ with $[2]P \in E(\Bbb Q)$ is simply $E[4]$, which has $16$ points and can be totally determined by solving the equation $[4]P = 0$.
We can compute $E[4]$ by solving equations $[2]P = Q$ for each $Q \in E[2]$. For $Q = \infty$ we get just $E[2]$ as four solutions.
The doubling formula for this curve is $x([2]P) = \frac{x^4 + 2x^2 + 1}{4(x^3 - x)}$, where $x = x(P)$.
Thus for example the $x(P)$ values for the four solutions to $[2]P = (-1, 0)$ are the roots of the polynomial $(x^4 + 2x^2 + 1) + 4(x^3 - x) = 0$, which are $-1 \pm \sqrt 2$, each with multiplicity $2$ since there are two $y(P)$ values for each $x(P)$ value.
Calculations for the other two are similar.