Unformatted text preview: E in terms of r and s . For example, decide what E ought to be when p = 144169 = (315) 2 + (212) 2 . 7. Let χ be a character of order 4 on F * p , so that χ 2 is the quadratic symbol ( · p ) . Show that E =2 Re J , where J = J ( χ, χ 2 ). Check this general formula by calculating E and J explicitly in the case where p = 5 and χ is the character mapping 2 to i . 8. Regard J as an element of Z [ i ]. Show that J + 1 is divisible by (2 + 2 i ). (See page 168 of the book if you get stuck.) 9. Suppose that J = α + iβ where α and β are integers. Explain why α is odd, β is even, α + β + 1 is divisible by 4 and α 2 + β 2 = p . Recapitulate what you have learned in the form of a rule for calculating N p when p is congruent to 1 mod 4....
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 Spring '08
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 Prime number, K. A. Ribet

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