Explicit Construction of the Hilbert Class Fields of Imaginary Quadratic Fields by Integer Lattice Reduction

Motivated by a constructive realization of generalized dihedral groups as Galois groups over Q and by Atkin’s primality test, we present an explicit construction of the Hilbert class fields (ring class fields) of imaginary quadratic fields (orders). This is done by first evaluating the singular moduli of level one for an imaginary quadratic order, and then constructing the “genuine” (i.e., level one) class equation. The equation thus obtained has integer coefficients of astronomical size, and this phenomenon leads us to the construction of the “reduced” class equations, i.e., the class equations of the singular moduli of higher levels. These, for certain levels, turn out to define the same Hilbert class field (ring class field) as the level one class equation, and to have coefficients of small size (e.g., seven digits). The construction of the “reduced” class equations was carried out on MACSYMA, using a refinement of the integer lattice reduction algorithm of Lenstra-Lenstra-Lavász, implemented on the Symbolics 3670 at Rensselaer Polytechnic Institute.