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Let $K$ be an imaginary quadratic field with class number 1 and $E/K$ be an elliptic curve with CM by $\mathcal{O}_K$. Consider the number field $F=K(E[p])$. Can $r_2(F) =1$?

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    $\begingroup$ $K$ doesn't have any real embedding so $F$ has $[F:Q]/2= [F:K]$ pairs of complex embeddings $\endgroup$ – reuns Jun 26 at 13:29
  • $\begingroup$ So it is obvious that [F:Q] is a totally imaginary extension? $\endgroup$ – debanjana Jun 26 at 13:36

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