Several invariants of an abelian extension can easily be obtained from the ideal groups without first computing defining equations for the field.
Let A be an abelian extension. Based on the conductor-discriminant relation made explicit by [Coh00, Section 3.5.2], the discriminant of the class field A is computed. This does not involve the computation of defining equations. The second return value is the signature of the resulting field.
The absolute discriminant of A as a number field over Q.
Computes the conductor of the abelian extension A, i.e. the smallest ideal and the smallest set of infinite places that are necessary to define A. The algorithm used is based on [Pau96], [HPP97].
The degree of the abelian extension A.
The degree of the abelian extension A over Q.
The base field of the abelian extension A, that is FieldOfFractions(BaseRing(A)).
The base ring of the abelian extension A, that is the maximal order used to define the underlying ray class group.
The norm group (see the definition of norm group) used to define the abelian extension A.
The decomposition field of the finite prime p in the abelian extension A as an abelian (sub)extension.
The decomposition field of the place p in the abelian extension A as an abelian extension.
The decomposition group of the finite prime p in the abelian extension A. The abelian group returned is a subgroup of the norm group.
The decomposition group of the place p in the abelian extension A. The abelian group returned is a subgroup of the NormGroup.
The "type" of the decomposition of the finite prime ideal p in the abelian extension A as a sequence of pairs < f, e > giving the degrees and the ramification indices.
The "type" of the decomposition of the place p in the abelian extension A as a sequence of pairs < f, e > giving the degrees and the ramification indices.
Normal: BoolElt Default: false
The "type" of the decomposition over Q of the prime number p in the abelian extension A as a sequence of pairs < f, e > giving the degrees and the ramification indices. If Normal is set to true then the algorithm assumes that the base field of A is normal. This is used to speed up the computations.
Normal: BoolElt Default: false
Computes the decomposition type of all elements in l and returns them as a multi-set. The list l must only contain objects for which DecompositionType is defined. If Normal eq true then the underlying DecompositionType function must be able to deal with it too. If Normal is set to true then the algorithm assumes that the base field of the abelian extension A is normal. This is used to speed up the computations.
Normal: BoolElt Default: false
Computes the decomposition type over Q in the abelian extension A of all prime numbers a ≤p ≤b and returns them as a multi set.If Normal is set to true then the algorithm assumes that the base field of A is normal. This is used to speed up the computations.