Number field K such that A factors through the Galois group of the normal closure of K.
Degree (=dimension) of an Artin representation A.
The Galois group of the field through which A factors.
Character of an Artin representation A, represented as a complex-valued character of Group(A).
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.
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).
Decompose an Artin representation A into constituents with rational-valued characters, each corresponding to a Galois conjugacy class of irreducibles. Returns a sequence of tuples [...<Ai,ni>...] with Ai having rational-valued character and ni its exponent in A (nonzero but possibly negative).
> K := NumberField(PolynomialWithGaloisGroup(12,15)); > DefiningPolynomial(K); x^12 - 24*x^10 + 216*x^8 - 896*x^6 + 1680*x^4 - 1152*x^2 + 48 > GroupName(GaloisGroup(K)); C3:D4 > c := PermutationCharacter(K); > Decomposition(c); [ <Artin representation C3:D4: (1,1,1,1,1,1,1,1,1) of K, 1>, <Artin representation C3:D4: (1,1,1,-1,1,-1,1,1,1) of K, 1>, <Artin representation C3:D4: (2,2,2,0,-1,0,-1,-1,-1) of K, 2>, <Artin representation C3:D4: (2,-2,0,0,2,0,0,0,-2) of K, 1>, <Artin representation C3:D4: (2,-2,0,0,-1,0,-1-2*J,1+2*J,1) of K, 1>, <Artin representation C3:D4: (2,-2,0,0,-1,0,1+2*J,-1-2*J,1) of K, 1> ] > RationalDecomposition(c); [ <Artin representation C3:D4: (1,1,1,1,1,1,1,1,1) of K, 1>, <Artin representation C3:D4: (1,1,1,-1,1,-1,1,1,1) of K, 1>, <Artin representation C3:D4: (2,2,2,0,-1,0,-1,-1,-1) of K, 2>, <Artin representation C3:D4: (2,-2,0,0,2,0,0,0,-2) of K, 1>, <Artin representation C3:D4: (4,-4,0,0,-2,0,0,0,2) of K, 1>];
Returns the polynomial whose roots Group(A) permutes.
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.
Returns the same Artin representation, but over an isomorphic version of the field that has had its representation optimized.
Smallest Galois extension K of the rationals through which A factors. Note that this field may be enormous and incomputable.
> 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 S3: (2,0,-1) of ext<Q|x^3+8*x+81> > Kernel(B); Number Field with defining polynomial x^6 + 48*x^4 + 576*x^2 + 179195 over the Rational Field
Return true iff a given Artin representation is irreducible as a complex representation.
Return true iff a given Artin representation is ramified at p.
Return true iff a given Artin representation is wildly ramified at p.
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.
Global epsilon-factor ε(A) of an Artin representation. Currently only implemented in a few basic cases, and returns 0 otherwise. See Example Example: Local and Global Epsilon Factors for Dirichlet Characters.
Global root number ε(A)/|ε(A)| of an Artin representation. Currently only implemented in a few basic cases, and returns 0 otherwise. See Example Example: Local and Global Epsilon Factors for Dirichlet Characters.
Local epsilon-factor ε(A) of an Artin representation at p. Currently only implemented in a few basic cases, and returns 0 otherwise. See Example Example: Local and Global Epsilon Factors for Dirichlet Characters.
Local root number εp(A)/|εp(A)| of an Artin representation at p. Currently only implemented in a few basic cases, and returns 0 otherwise. See Example Example: Local and Global Epsilon Factors for Dirichlet Characters.
Local root number w_∞(A) of an Artin representation at infinity. See Example Example: Local and Global Epsilon Factors for Dirichlet Characters.
> R<x> := PolynomialRing(Rationals()); > K := NumberField(x^4-3); > A := ArtinRepresentations(K); > Degree(Kernel(A[5]),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, 12, 3, 4, 576 ] > [IsRamified(a,3): a in A]; [ false, true, true, false, true ] > [IsWildlyRamified(a,3): a in A]; [ false, false, false, false, false ] > EulerFactor(A[5],5); x^2 + 1 > EpsilonFactor(A[5],3); -3
Convert a one-dimensional Artin representation to a Dirichlet character.
Convert a one-dimensional Artin representation A to a Hecke character. This is more natural than the previous, as in general Hecke characters will have L-functions matching that of the Artin representation, while Dirichlet characters only necessarily have L-functions when defined over the rationals.
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).
> 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, 2, 1, 1, 2, 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 C4: (1,-1,-I,I) of ext<Q|x^4+x^3-54*x^2-54*x+551> > Factorization(Discriminant(Integers(Field($1)))); [ <5, 3>, <43, 2> ]