Let $E/\Bbb{Q}$ be an elliptic curve and $\phi: E\to E'$ be an isogeny of degree 2 and $\hat{\phi}$ be its dual isogeny.
How can we calculate the group $E(\Bbb{Q})/\hat{\phi}(E'(\Bbb{Q}))$?
In the case of $E:y^2=x^3+DX$, page $350$ $9$th line from the bottom of SIlverman's book 'The arithmetic of elliptic curves', he reads $E(\Bbb{Q})/\hat{\phi}(E'(\Bbb{Q}))\cong \Bbb{Z}/2\Bbb{Z}$ without explanations. But group $E(\Bbb{Q})/\hat{\phi}(E'(\Bbb{Q}))$ looks difficult in general. Do you know any strategy to figure out this group ? (I also have the same quetion for $E'(\Bbb{Q})/{\phi}(E(\Bbb{Q}))$)