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

Let $\mathrm{Sha}(E/K)$ denote the Tate-Shafarevich group of $E/K$. Can we explicitly write down the norm map of the Tate-Shafarevich group $\mathrm{Sha}(E/K) \to \mathrm{Sha}(E/\mathbb{Q})$?

(Reference: What's the Hilbert class field of an elliptic curve?)

At first, I thought that the map $[C] \mapsto [C] + [C]^{\sigma}$ would provide the desired map. However, it does not commute with the Galois action, meaning that $([C] + [D])^{\sigma} \neq [C]^{\sigma} + [D]^{\sigma}$ in general.

Thank you for your help.