Reduced Binary Quadratic Form

A reduced binary quadratic form is the binary quadratic form such that the following conditions hold. Let be the discriminant, then

1. If is negative, is reduced if and if whenever or , and is called real.

2. If is positive, is reduced if , and is called imaginary or positive definite.

Every imaginary binary quadratic form is equivalent to a unique reduced form and every real binary quadratic form is equivalent to a finite number of reduced forms.