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I am trying to understand the proof of assertion (i) in Proposition 3.10 (page 14) of this paper http://arxiv.org/pdf/1312.3884v3.pdf

$M$ stands for a square free integer which is prime to $7$, $A$ denotes the curve $X_0(49)$, $Y^2=X^3+21X^2+112X$ in Weierstrass form and $A^{(M)}: y^2=x^3+21XMx^2+112M^2x $ is the quadratic twist of $A$ by $\mathbb Q(\sqrt{M})/ \mathbb Q$. We know that $A$ has complex multiplication by $\mathbb Q(\sqrt{-7})$ and that $\mathbb Q(\sqrt{-7})= \mathbb Q(A[2])$.

The authors say that the last equality implies that $A^{(M)}(\mathbb R)$ has one connected component. Could somebody explain to me why this is true, or preferably point me towards relevant literature? Thank you very much

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  1. $2$ torsion as a Galois module doesn't change under quadratic twists,

  2. An elliptic curve over the real numbers has two real connected components if and only if all 2-torsion points are real. More generally, the group of real points is isomorphic as a group to S^1 or S^1 * Z/2Z, which are distinguished by their 2-torsion.

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