This is a question from Silverman's The arithmetic of elliptic curves, exercise 10.7.
Prove $WC(E/ \Bbb R)$ is isomorphic to $\Bbb Z/2\Bbb Z$ when discriminant $Δ＞0$ and $0$ when $Δ＜0$.
From Silverman's Theorem $3.6$, $WC(E/K)$ is isomorphic to $H^1(G,E)$, where $G$ is absolute Galois group of $K$.
So in this case, I only need to calculate $H^1(\Bbb Z/2\Bbb Z ,E)$.
I wonder why sign of determinant have to do with this question. Thank you for your help.