On genera of binary quadratic forms
Irving Reiner · Bulletin of the American Mathematical Society · 1945
Let fi~ax 2 + 2bxy+cy 2 be a properly primitive form with integral coefficients, and let the determinant D=ac -b 2 be written as JD= ±2*A, where A is odd and positive, and the factorization of A into distinct primes is A~qi al • • • q r ar .Let us suppose that a is positive and prime to 2D.The genus of j8 is then completely determined by the Legendre symbols (a\qi), • • • , (a\q r ), and (~l|a) if D = 0 or 1 (mod 4), (2|a) if D^O or 6 (mod 8), and (-2|a) if £> = 0 or 2 (mod 8). 1 These characters are not independent, however, since -D = b 2 -ac and (a, ô) = l imply that (-Z)ja) = l; from this, using the law of quadratic reciprocity, we get