MAGMA Computational Algebra System

Class Group Coercions

The class group of a nonmaximal quadratic order R of discriminant m² D_K, are related to the class group of the maximal order O_K of fundamental discriminant D_K by an exact sequence.

1 -> ((O_K/mO_K)^ * /O_K^ * (Z/mZ)^ * ) -> Cl(O) -> Cl(O_K) -> 1

Similar maps exist between quadratic orders O₁ and O₂ in a field K, with conductors m₁ and m₂, respectively, such that m₁ | m₂. The corresponding maps on quadratic forms are implemented on quadratic forms. The homomorphism is returned as a map object, or can be called directly via the coercion operator.

FundamentalQuotient(Q) : QuadBin -> Map

The quotient homomorphism from the class group of Q to the class group of fundamental discriminant.

QuotientMap(Q1, Q2) : QuadBin, QuadBin -> Map

Given two structures of quadratic forms Q₁ and Q₂, such that the discriminant of Q₂ equals a square times the discriminant of Q₁, the quotient homomorphism from Q₁ to Q₂ is returned as a map object.

Q ! f : QuadBin, QuadBinElt -> QuadBinElt

The ! operator applies the quotient homomorphism for automatic coercion of forms f of discriminant m²D into the structure Q of forms of discriminant D.