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Number field K such that A factors through the Galois group of the normal closure of K.
Dimension(A) : ArtRep -> RngIntElt
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).
We give an example to show the difference between these decompositions.
> 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.
We take an S 4-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 S 3-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.
RootNumber(A,infty) : ArtRep, Infty -> FldComElt
Local root number w_∞(A) of an Artin representation at infinity.
See Example Example: Local and Global Epsilon Factors for Dirichlet Characters.
Here are the invariants of Artin representations that factor through
the splitting field of x 4 - 3, a D 4-extension of Q.
> 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).
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, 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> ]
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