# Basic Invariants ## `Field(A): ArtRep -> FldNum` Number field $K$ such that $A$ factors through the Galois group of the normal closure of $K$. ## `BaseField(A): ArtRep -> Fld` The field $F$ over which $A$ is a representation of ${\operatorname{Gal}}(\bar F/F)$. This is ${\mathbb{Q}}$ unless $A$ was constructed as a relative Artin representation, in which case it is the base field of the defining relative extension. ## `Degree(A): ArtRep -> RngIntElt` ## `Dimension(A): ArtRep -> RngIntElt` Degree (=dimension) of an Artin representation $A$. ## `Group(A): ArtRep -> GrpPerm` The Galois group of the field 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 `[......]` with $A_i$ irreducible and $n_i$ its exponent in $A$ (nonzero but possibly negative). ## `RationalDecomposition(A): ArtRep -> SeqEnum[Tup]` 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 `[......]` with $A_i$ having rational-valued character and $n_i$ its exponent in $A$ (nonzero but possibly negative). ## `Example: Artin Decompose (ex-12f333)` We give an example to show the difference between these decompositions. ```magma > 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); [ , , , , , ] > RationalDecomposition(c); [ , , , , ]; ``` ## `DefiningPolynomial(A): ArtRep -> RngUPolElt` Returns the polynomial whose roots `Group(A)` permutes. ## `Minimize(A): ArtRep -> ArtRep` ```magma 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`. ## `OptimizedRepresentation(A): ArtRep -> ArtRep` Returns the same Artin representation, but over an isomorphic version of the field that has had its representation optimized. ## `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: Artin Minimize (ex-9129cc)` We take an $S_4$-extension of ${\mathbb{Q}}$ and compute its Artin representations. ```magma > R := 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. ```magma > B := Minimize(A[3]); B; Artin representation S3: (2,0,-1) of ext > 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` ```magma 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 $A$. ## `EpsilonFactor(A): ArtRep -> FldComElt` Global epsilon-factor $\epsilon(A)$ of an Artin representation. Currently only implemented in a few basic cases, and raises an error otherwise. See Example [Example: Local and Global Epsilon Factors for Dirichlet Characters](../../LocalFields/LocalGaloisRepresentations/fields-2.md#galrep-ex1). ## `RootNumber(A): ArtRep -> FldComElt` Global root number $\epsilon(A)/|\epsilon(A)|$ of an Artin representation. Currently only implemented in a few basic cases, and raises an error otherwise. See Example [Example: Local and Global Epsilon Factors for Dirichlet Characters](../../LocalFields/LocalGaloisRepresentations/fields-2.md#galrep-ex1). ## `EpsilonFactor(A, p): ArtRep, RngIntElt -> FldComElt` Local epsilon-factor $\epsilon(A)$ of an Artin representation at $p$. Currently only implemented in a few basic cases, and raises an error otherwise. See Example [Example: Local and Global Epsilon Factors for Dirichlet Characters](../../LocalFields/LocalGaloisRepresentations/fields-2.md#galrep-ex1). ## `RootNumber(A, p): ArtRep, RngIntElt -> FldComElt` Local root number $\epsilon_p(A)/|\epsilon_p(A)|$ of an Artin representation at $p$. Currently only implemented in a few basic cases, and raises an error otherwise. See Example [Example: Local and Global Epsilon Factors for Dirichlet Characters](../../LocalFields/LocalGaloisRepresentations/fields-2.md#galrep-ex1). ## `EpsilonFactor(A, infty): ArtRep, Infty -> FldComElt` ## `RootNumber(A, infty): ArtRep, Infty -> FldComElt` Local root number $w_\infty(A)$ of an Artin representation at infinity. See Example [Example: Local and Global Epsilon Factors for Dirichlet Characters](../../LocalFields/LocalGaloisRepresentations/fields-2.md#galrep-ex1). ## `Example: Artin Invariants (ex-2bf31b)` Here are the invariants of Artin representations that factor through the splitting field of $x^4-3$, a $D_4$-extension of ${\mathbb{Q}}$. ```magma > R := 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 ``` ## `DirichletCharacter(A): ArtRep -> GrpDrchElt` Convert a one-dimensional Artin representation to a Dirichlet character. ## `HeckeCharacter(A): ArtRep -> GrpHeckeElt` 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. ## `ArtinRepresentation(ch): GrpDrchElt -> ArtRep` ```magma 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: One Dim Artin Reps (ex-db71ab)` An example that goes back and forth between the Dirichlet character and the Artin representation. ```magma > 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 > Factorization(Discriminant(Integers(Field($1)))); [ <5, 3>, <43, 2> ] ```