Let $K$ be a quadratic number field.Let $E$ be an elliptic curve defined over $\Bbb{Q}$.

$Sha(E/K)$ be Tate-Shafarevich group of $E/K$. Can we explicitly write down norm map of Tate-Shafarevich group $Sha(E/K)\to Sha(E/\Bbb{Q})$ ?

(cf. What's the Hilbert class field of an elliptic curve?)

I first thought that the map $[C]\to [C]+[C]^{\sigma}$ gives the map. But it does not commute with Galois action, that is, $([C]+[D])^{\sigma}=[C]^{\sigma}+[D]^{\sigma}$ does not hold in general.

Thank you for your help.