Let $E:y^2=x(x-a)(x-b)(a\neq b\in \Bbb{Q})$ be an elliptic curve over a field of rational numbers.
Let $d\in \Bbb{Z}$ be a square free integer and $E_d:dy^2=x(x-a)(x-b)$ be a quadratic twist of $E$.
It is clear that $\#E_d(\Bbb{Q})[2]=4$ because $y=0$ has 3 roots. .
My question is, what is $\#E_d(\Bbb{Q})_{tor}$ ?
If $\#E_d(\Bbb{Q})_{tor}=\#E_d(\Bbb{Q})[2]・・・①$ holds, it's easy, but I don't have confident about ①.
This one is related https://mathoverflow.net/questions/76413/torsion-subgroups-in-families-of-twists-of-elliptic-curves question in Mathoverflow, but this page mainly deals with odd part, not $2-$part.