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Basic Invariants

Field(A) : ArtRep -> FldNum
Number field K such that A factors through the Galois group of the normal closure of K.
Degree(A) : ArtRep -> RngIntElt
Dimension(A) : ArtRep -> RngIntElt
Degree (=dimension) of an Artin representation A.
Group(A) : ArtRep -> GrpPerm
The Galois group of the through which A factors.
Character(A) : ArtRep -> AlgChtrElt
Character of an Artin representation A, represented as a complex-valued character of Group(A).
Conductor(A) : ArtRep -> RngIntElt
Conductor of an Artin representation A (which must be a true representation, i.e. its character is not allowed to be a generalized character). Computes all the necessary local information if Artin representations were defined with Ramification:=false, so the first call to this function might take some time.
Decomposition(A) : ArtRep -> SeqEnum[Tup]
Decompose an Artin representation A into irreducible constituents. Returns a sequence of tuples [...<Ai,ni>...] with Ai irreducible and ni its exponent in A (nonzero but possibly negative).
DefiningPolynomial(A) : ArtRep -> RngUPolElt
Returns the polynomial whose roots Group(A) permutes.
Minimize(A) : ArtRep -> ArtRep
    Optimize: BoolElt                   Default: true
Returns A attached to the smallest number field K such that A factors through its Galois closure. If Optimize := true, attempts to minimize the defining polynomial of K using OptimizedRepresentation.
Kernel(A) : ArtRep -> FldNum
Smallest Galois extension K of the rationals through which A factors. Note that this field may be enormous and incomputable.

Example ArtRep_artin-minimize (H44E2)

We take an S4-extension of Q and compute its Artin representations.

> R<x> := PolynomialRing(Rationals());
> K := NumberField(x^4+9*x-2);
> A := ArtinRepresentations(K);
> [Dimension(a): a in A];
[ 1, 1, 2, 3, 3 ]
Then we minimize the 2-dimensional one, which factors through an S3-quotient.

> B := Minimize(A[3]); B;
Artin Representation of Number Field with defining polynomial x^3 + 8*x - 81
over the Rational Field with character ( 2, 0, -1 )
> Kernel(B);
Number Field with defining polynomial x^6 + 48*x^4 + 576*x^2 + 179195 over the 
Rational Field

IsIrreducible(A) : ArtRep -> BoolElt
Return true iff a given Artin representation is irreducible as a complex representation.
IsRamified(A, p) : ArtRep, RngIntElt -> BoolElt
Return true iff a given Artin representation is ramified at p.
IsWildlyRamified(A, p) : ArtRep, RngIntElt -> BoolElt
Return true iff a given Artin representation is wildly ramified at p.
EulerFactor(A, p) : ArtRep, RngIntElt -> RngUPolElt
    R: Fld                              Default: ComplexField()
The local polynomial (Euler factor) of an Artin representation A at the prime p. It is a polynomial with coefficients in the field R, which is complex numbers by default, and it is the inverse characteristic polynomial of (arithmetic) Frobenius at p on the inertia invariant subspace of A.

Example ArtRep_artin-invariants (H44E3)

Here are the invariants of Artin representations that factor through the splitting field of x4 - 3, a D4-extension of Q.

> R<x> := PolynomialRing(Rationals());
> K := NumberField(x^4-3);
> A := ArtinRepresentations(K);
> Degree(Field(A[1]),Rationals());
8
> [Dimension(a): a in A];
[ 1, 1, 1, 1, 2 ]
> Character(A[5]);
( 2, -2, 0, 0, 0 )
> [Conductor(a): a in A];
[ 1, 3, 4, 12, 576 ]
> [IsRamified(a,3): a in A];
[ false, true, false, true, true ]
> [IsWildlyRamified(a,3): a in A];
[ false, false, false, false, false ]
> EulerFactor(A[5],5);
$.1^2 + 1.000000000000000000000000000000

DirichletCharacter(A) : ArtRep -> GrpDrchElt
Convert a one-dimensional Artin representation to a Dirichlet character.
ArtinRepresentation(ch) : GrpDrchElt -> ArtRep
    field: FldNum                       Default: 
Convert a Dirichlet character ch to a one-dimensional Artin representation A. To avoid recomputation, the minimal field through which A factors may be supplied by the field parameter. This now uses class field theory (thanks to C. Fieker).

Example ArtRep_one-dim-artin-reps (H44E4)

An example that goes back and forth between the Dirichlet character and the Artin representation.

> load galpols;
> f := PolynomialWithGaloisGroup(8,46); // order 576
> K := NumberField(f); // octic field
> A := ArtinRepresentations(K);
> [Degree(a) : a in A];
[ 1, 1, 1, 1, 4, 4, 6, 6, 9, 9, 9, 9, 12 ]
> [Order(Character(Determinant(a))) : a in A];
[ 1, 2, 4, 4, 2, 2, 1, 2, 2, 1, 4, 4, 2 ]
> chi := DirichletCharacter(A[3]); // order 4
> Conductor(chi), Conductor(chi^2);
215 5
> Minimize(ArtinRepresentation(chi)); // disc = N(chi)^2*N(chi^2)
Artin representation of Number Field with defining polynomial
 x^4 - x^3 - 54*x^2 + 54*x + 551 with character ( 1, -1, I, -I )
> Factorization(Discriminant(Integers(Field($1))));
[ <5, 3>, <43, 2> ]

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Version: V2.19 of Mon Dec 17 14:40:36 EST 2012