TITLE: A question on BSD conjecture
QUESTION [7 upvotes]: If $E$ is an elliptic curve over $\mathbb{Q}$, and $K$ is an imaginary quadratic field. If $rank E(K)\leq 1$ and both $E$ and the quadratic twist of $E$ by $K$ satisfy the full BSD conjecture, does the base change of $E$ to $K$ satisfy the full BSD conjecture?

REPLY [10 votes]: Yes.
This follows from the fact that BSD is invariant under Weil restriction and isogeny (the Weil restriction of $E/K$ to $\mathbb{Q}$ is isogenous to the product of $E$ with its quadratic twist).
Note that you don't need to assume here that $K$ is imaginary nor anything about the rank of $E/K$.