# Number of complex embeddings of a certain number field

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 $$r_2(F) =1$$?

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