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Sep 15 at 7:44 comment added kindasorta This is literally what I was searching for, thanks.
Sep 12 at 5:23 comment added Satan's Minion When $n=2j$ is even, the Tate twist $S^{2j}(V)(j)$ has weight zero and is self-dual, and it is known that its BK Selmer group vanishes if $E$ is non-CM by Theorem 5.6 here (arxiv.org/pdf/1912.11265). For most other twists, this problem is very hard and there is some extensive literature on the cases $n=2$ and (more recently) $n=3$. BTW, no need to assume good reduction at $p$.
Sep 11 at 19:51 history asked kindasorta CC BY-SA 4.0