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I am reading a paper on elliptic curve cryptography. In the paper (Section 4), the authors claim that when the characteristic of the field is represented as $$p=a^2+|D|b^2\\ \\ or\\ p=\frac{|D|+1}{4}a^2+|D|ab+|D|b^2,$$

then the trace of the Frobenius morphism can be written as $$T=\pm 2a\\ \\ or\\ T=\pm a$$ respectively. I have tried in vain to prove this claim. Could anyone suggest a reference or a strategy to prove this result?

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    $\begingroup$ From your description it is not possible to help you as you do not explain the notations. What is $D$? The problem of course comes from the paper you link to. It does not explain this either. I suggest that you look at a better text on isogenies for elliptic curve over finite flieds. Like Kohel's thesis, cited as [6] in the linked paper. $\endgroup$ Mar 24, 2017 at 9:49

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