1911 Encyclopædia Britannica/Number/Characters of a form or class

36. ''Characters of a form or class. Genera.''&mdash;Let $$(a, b, c)$$ be any primitive form; we have seen above (&sect; 32) that if $$\alpha, \beta, \gamma, \delta$$ are any integers
 * $$4(a\alpha^2+b\alpha\gamma+c\gamma^2)(a\beta^2+b\beta\delta+c\delta^2) = b'^2-(\alpha\delta-\beta\gamma)^2\mathrm{D}$$

where $$b' = 2a\alpha\beta+b(\alpha\delta+\beta\gamma)+2c\gamma\delta$$. Now the expressions in brackets on the left hand may denote any two numbers $$m, n$$ representable by the form $$(a, b, c)$$; the formula shows that $$4mn$$ is a residue of $$\mathrm{D}$$, and hence $$mn$$ is a residue of every odd prime factor of $$\mathrm{D}$$, and if $$p$$ is any such factor the symbols $$\left(\frac{m}{p}\right)$$ and $$\left(\frac{n}{p}\right)$$ will have the same value. Putting $$(a, b, c)=f$$, this common value is denoted by $$\left(\frac{f}{p}\right)$$ and called a quadratic character (or simply character) of $$f$$ with respect to $$p$$. Since $$a$$ is representable by $$f$$ ($$x=1, y=0$$) the value $$\left(\frac{f}{p}\right)$$ is the same as $$\left(\frac{a}{p}\right)$$. For example, if $$\mathrm{D} = -140$$, the scheme of characters for the six reduced primitive forms, and therefore for the classes they represent, is
 * $$\begin{array}{lccc }

& \left(\frac{f}{5}\right) & \left(\frac{f}{7}\right) & \\ (1, 0, 35) & + & + \\ (4, \pm 2, 9) & & \\ \hline (5, 0, 7) & - & - \\ (3, \pm 2, 12) & & \\ \hline \end{array}$$. In certain cases there are supplementary characters of the type $$\left(\frac{-1}{f}\right)$$ and $$\left(\frac{2}{f}\right)$$, and the characters $$\left(\frac{f}{p}\right)$$ are discriminated according as an odd or even power of $$p$$ is contained in $$\mathrm{D}$$; but in every case there are certain combinations of characters (in number one-half of all possible combinations) which form the total characters of actually existing classes. Classes which have the same total character are said to belong to the same genus. Each genus of the same order contains the same number of classes.

For any determinant $$\mathrm{D}$$ we have a principal primitive class for which all the characters are $$+$$; this is represented by the principal form $$(1, 0, -n)$$ or $$(1, 1, -n)$$ according as $$\mathrm{D}$$ is of the form $$4n$$ or $$4n+1$$. The corresponding genus is called the principal genus. Thus, when $$\mathrm{D} = -140$$, it appears from the table above that in the primitive order there are two genera, each containing three classes; and the non-existent total characters are $$+-$$ and $$-+$$.