Reduced Quadratic Form

A primitive positive definite quadratic form a*xx + b*xy + c*yy is reduced if the absolute value of b is less than or equal to a and a is less than or equal to c and b is greater than or equal to zero if either the absolute value of b is equal to a or a is equal to c.

Examples:

28xx - 7xy + 293yy is reduced since abs(b) < a < c.
31xx + 31xy + 272yy is reduced since abs(b) <= a < c and b > 0.
92xx + 33xy + 92yy is reduced since abs(b) < a <= c and b > 0.
31xx - 31xy + 272yy is not reduced since abs(b) = a but b < 0.
92xx - 33xy + 92yy is not reduced since a = c but b < 0.

You can check that the above forms are primitive. They are positive definite since their discriminants are negative and the coefficient of the xx term is greater than zero.

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