# Artin Representations

- [Overview](overview.md)

- [Constructing Artin Representations](functions.md)

  - [`ArtinRepresentations(K): FldNum → SeqEnum`](functions.md#function-artinrepresentations-fldnum)

  - [`K !! ch: FldNum, AlgChtrElt → ArtRep`](functions.md#operation-op-fldnum-algchtrelt)

  - [`K !! ch: FldNum, SeqEnum → ArtRep`](functions.md#operation-op-fldnum-seqenum)

  - [`PermutationCharacter(K): FldNum → ArtRep`](functions.md#function-permutationcharacter-fldnum)

  - [`Determinant(A): ArtRep → ArtRep`](functions.md#function-determinant-artrep)

  - [`ChangeField(A,K): ArtRep, FldNum → ArtRep, BoolElt`](functions.md#function-changefield-artrep-fldnum)

  - [`K !! A: FldNum, ArtRep → ArtRep, BoolElt`](functions.md#operation-op-fldnum-artrep)

  - [`Example: Artin Const`](functions.md#example-ex-81e6bd)

- [Basic Invariants](invariants.md)

  - [`Field(A): ArtRep → FldNum`](invariants.md#function-field-artrep)

  - [`BaseField(A): ArtRep → Fld`](invariants.md#function-basefield-artrep)

  - [`Degree(A): ArtRep → RngIntElt`](invariants.md#function-degree-artrep)

  - [`Dimension(A): ArtRep → RngIntElt`](invariants.md#function-dimension-artrep)

  - [`Group(A): ArtRep → GrpPerm`](invariants.md#function-group-artrep)

  - [`Character(A): ArtRep → AlgChtrElt`](invariants.md#function-character-artrep)

  - [`Conductor(A): ArtRep → RngIntElt`](invariants.md#function-conductor-artrep)

  - [`Decomposition(A): ArtRep → SeqEnum[Tup]`](invariants.md#function-decomposition-artrep)

  - [`RationalDecomposition(A): ArtRep → SeqEnum[Tup]`](invariants.md#function-rationaldecomposition-artrep)

  - [`Example: Artin Decompose`](invariants.md#example-ex-12f333)

  - [`DefiningPolynomial(A): ArtRep → RngUPolElt`](invariants.md#function-definingpolynomial-artrep)

  - [`Minimize(A): ArtRep → ArtRep`](invariants.md#function-minimize-artrep)

  - [`OptimizedRepresentation(A): ArtRep → ArtRep`](invariants.md#function-optimizedrepresentation-artrep)

  - [`Kernel(A): ArtRep → FldNum`](invariants.md#function-kernel-artrep)

  - [`Example: Artin Minimize`](invariants.md#example-ex-9129cc)

  - [`IsIrreducible(A): ArtRep → BoolElt`](invariants.md#function-isirreducible-artrep)

  - [`IsRamified(A, p): ArtRep, RngIntElt → BoolElt`](invariants.md#function-isramified-artrep-rngintelt)

  - [`IsWildlyRamified(A, p): ArtRep, RngIntElt → BoolElt`](invariants.md#function-iswildlyramified-artrep-rngintelt)

  - [`EulerFactor(A, p): ArtRep, RngIntElt → RngUPolElt`](invariants.md#function-eulerfactor-artrep-rngintelt)

  - [`EpsilonFactor(A): ArtRep → FldComElt`](invariants.md#function-epsilonfactor-artrep)

  - [`RootNumber(A): ArtRep → FldComElt`](invariants.md#function-rootnumber-artrep)

  - [`EpsilonFactor(A,p): ArtRep, RngIntElt → FldComElt`](invariants.md#function-epsilonfactor-artrep-rngintelt)

  - [`RootNumber(A,p): ArtRep, RngIntElt → FldComElt`](invariants.md#function-rootnumber-artrep-rngintelt)

  - [`EpsilonFactor(A,infty): ArtRep, Infty → FldComElt`](invariants.md#function-epsilonfactor-artrep-infty)

  - [`RootNumber(A,infty): ArtRep, Infty → FldComElt`](invariants.md#function-rootnumber-artrep-infty)

  - [`Example: Artin Invariants`](invariants.md#example-ex-2bf31b)

  - [`DirichletCharacter(A): ArtRep → GrpDrchElt`](invariants.md#function-dirichletcharacter-artrep)

  - [`HeckeCharacter(A): ArtRep → GrpHeckeElt`](invariants.md#function-heckecharacter-artrep)

  - [`ArtinRepresentation(ch): GrpDrchElt → ArtRep`](invariants.md#function-artinrepresentation-grpdrchelt)

  - [`Example: One Dim Artin Reps`](invariants.md#example-ex-db71ab)

- [Arithmetic](arithmetic.md)

  - [`A1 + A2: ArtRep, ArtRep → ArtRep`](arithmetic.md#operation-op-plus-artrep-artrep)

  - [`A1 - A2: ArtRep, ArtRep → ArtRep`](arithmetic.md#operation-op-minus-artrep-artrep)

  - [`A1 * A2: ArtRep, ArtRep → ArtRep`](arithmetic.md#operation-op-times-artrep-artrep)

  - [`A1 eq A2: ArtRep, ArtRep → BoolElt`](arithmetic.md#operation-op-eq-artrep-artrep)

  - [`A1 ne A2: ArtRep, ArtRep → BoolElt`](arithmetic.md#operation-op-ne-artrep-artrep)

  - [`Example: Artin Arith1`](arithmetic.md#example-ex-ba9cf8)

  - [`Example: Artin Arith2`](arithmetic.md#example-ex-401593)

- [Implementation Notes](notes.md)