Let $E/\Bbb{Q}$ be an elliptic curve. Let $K=\Bbb{Q}(\sqrt{d})$ be a quadratic field.If $E:y^2=x^3+ax+b$, let $E_D:Dy^2=X^3+ax+b$. In this page, professor Silverman reads there is a natural map $E_D(\Bbb{Q})\times E(\Bbb{Q})\to E(\Bbb{Q}(\sqrt{D}))$. But how can we define this map ? https://mathoverflow.net/questions/76413/torsion-subgroups-in-families-of-twists-of-elliptic-curves/76464?noredirect=1#comment1182446_76464
The map occurred to my mind is $(P,Q)\to P'+Q$($P'$ is a point corresponding to $P$ by an isomorphism $E(\Bbb{Q}(\sqrt{D}))\cong E_D(\Bbb{Q}))$, but then it seems difficult to find kernel and cockerel of this map.