The Class Group

Abstract

We come again to the subject of the representation of integers by forms. Let us assume that, for a form f = (a, b, c) of discriminant Δ, integers x and y exist so that f represents r, that is, r = ax² + bxy + cy². This is a primitive representation if gcd(x,y) = 1. If the representation is primitive, then integers z and w exist so that xw − yz = 1. Then f is equivalent to a form f′ = (r, s, t), where f′ is obtained from f by using the transformation

$$ \left( {\begin{array}{*{20}{c}} x&y\\ z&w \end{array}} \right), $$

and equations (1.2). We note that the choice of f′ is not unique, but that different values of s differ by multiples of 2r, and thus the different choices lead to equivalent forms. That is, modulo 2r, a unique s is determined from (x,y) such that

$$ {s^2} - 4rt = \Delta ,{\text{ }}for\,\,some\,\,integer\,\,t. $$