Q:

Trivial Weil-Châtelet group

Does there exist an elliptic curve over a number field $K$ such that $WC(E/K)\cong H^1(G_K, E)$ is trivial?

A:

Weil-Chatelet groups are huge. A theorem of Shafarevich states that if $n \ge 2$ and if $E$ is an elliptic curve (or an abelian variety) over a number field $k$ then $H^1(G_k,E)$ has infinitely many elements of order $n$. See Section 5 of Pete Clark's lecture on WC groups:
http://math.uga.edu/~pete/wcnotes.pdf

