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Jan 3 at 15:04 comment added KStar Thanks for this! I had realised the things mentioned in your last paragraph, except the result of Mazur and Swinnerton-Dyer which is interesting. I decided to not remark on the parity considerations in my question in case it wasn’t particularly relevant. I think it would be interesting if there is an explicit example where $Z\neq r$.
Jan 3 at 11:48 history undeleted Chris Wuthrich
Jan 3 at 11:47 history edited Chris Wuthrich CC BY-SA 4.0
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Jan 3 at 10:29 history deleted Chris Wuthrich via Vote
Jan 3 at 10:29 history edited Chris Wuthrich CC BY-SA 4.0
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Jan 3 at 10:11 history answered Chris Wuthrich CC BY-SA 4.0