Let $E/\Bbb{Q}$ be an elliptic curve. Let $K=\Bbb{Q}(\sqrt{D})$ be an quadratic number field.
There is a trace map $E(\Bbb{Q}(\sqrt{D}))\to E(\Bbb{Q})$ given by $P\to P+P^{\sigma}$, $\sigma$ is a generator of $Gal(K/\Bbb{Q})$.
On the other hand, let $trace'$ be a map $E(\Bbb{Q}(\sqrt{D})\to E_D(\Bbb{Q})$ given by $P\to P+P^{\sigma}$.
What is the value of $\#coker(trace)/\#coker(trace')$ ?
My observation: If $rank(E/\Bbb{Q})=rank(E_D/\Bbb{Q})=0$ , two exact sequence $0\to E_D(\Bbb{Q})\to E(\Bbb{Q}(\sqrt{D}))\to E(\Bbb{Q})\to coker(tracę)\to 0$ and $0\to E(\Bbb{Q})\to E(\Bbb{Q}(\sqrt{D}))\to E_D(\Bbb{Q})\to coker(tracę')\to 0$ implies the ratio $\#coker(trace)/\#coker(trace')$ is exactly $1$. But what about other cases ? Thank you for your help.