MAGMA Computational Algebra System

Introduction

A binary quadratic form is an integral form ax² + bxy + cy² which is represented in Magma by a tuple < (a, b, c) >. Binary quadratic forms play an central role in the ideal theory of quadratic fields, the classical theory of complex multiplication, and the theory of modular forms. Algorithms for binary quadratic forms provide efficient means of computing in the ideal class group of orders in a quadratic field. By using the explicit relation of definite quadratic forms with lattices with nontrivial endomorphism ring in the complex plane, one can apply modular and elliptic functions to forms, and exploit the analytic theory of complex multiplication.

The structures of quadratic forms of a given discriminant D correspond to ordered bases of ideals in an order in a quadratic number field, defined up to scaling by the rationals. A form is primitive if the coefficients a, b, and c are coprime. For negative discriminants the primitive reduced forms in this structure are in bijection with the class group of projective or invertible ideals. For positive discriminants, the reduced orbits of forms are used for this purpose. Magma holds efficient algorithms for composition, enumeration of reduced forms, class group computations, and discrete logarithms. A significant novel feature is the treatment of nonfundamental discriminants, corresponding to nonmaximal orders, and the collections of homomorphisms between different class groups coming from the inclusions of these orders.

The functionality for binary quadratic forms is rounded out with various functions for applying modular and elliptic functions to forms, and for class polynomials associated to class groups of definite forms.