Classification algorithms for positive definite quadratic forms of the form aX² + b√nXY + cY² with n = 2, 3

Abstract

We investigate quadratic forms of the form aX² + b√nXY + cY² with a > 0 and b²n − 4ac < 0 in the cases n = 2 and n = 3. For these forms, we develop a reduction framework leading to an effective classification algorithm with respect to a natural equivalence relation. In particular, for each discriminant we obtain a finite set of reduced representatives, allowing explicit enumeration of equivalence classes and computation of the associated class numbers. Numerical experiments based on this algorithm suggest relations with classical class numbers.