# Local Galois Representations

- [Overview](overview.md)

  - [Notation and Printing](overview.md#notation-and-printing)

    - [`Example: Galrep Printing`](overview.md#example-ex-8ff07a)

  - [Conventions](overview.md#conventions)

  - [Implementation Notes](overview.md#implementation-notes)

- [Creating Galois Representations](creating.md)

  - [`ZeroRepresentation(K): FldPad → GalRep`](creating.md#function-zerorepresentation-fldpad)

  - [`Example: Galrep Zerorepresentation`](creating.md#example-ex-153a0b)

  - [`PrincipalCharacter(K): FldPad → GalRep`](creating.md#function-principalcharacter-fldpad)

  - [`Example: Galrep Principalcharacter`](creating.md#example-ex-48649a)

  - [`CyclotomicCharacter(K): FldPad → GalRep`](creating.md#function-cyclotomiccharacter-fldpad)

  - [`Example: Galrep Cyclotomiccharacter`](creating.md#example-ex-4a5d9f)

  - [`UnramifiedCharacter(K,c): FldPad, Any → GalRep`](creating.md#function-unramifiedcharacter-fldpad-any)

  - [`Example: Galrep Unramifiedcharacter`](creating.md#example-ex-6bab13)

  - [`UnramifiedRepresentation(K,CharPoly): FldPad, RngUPolElt → GalRep`](creating.md#function-unramifiedrepresentation-fldpad-rngupolelt)

  - [`Example: Galrep Unramifiedrepresentation`](creating.md#example-ex-b1231d)

  - [`UnramifiedRepresentation({K,dim,}{dimcomputed,CharPoly}): FldPad, RngIntElt, RngIntElt, RngUPolElt → GalRep`](creating.md#function-unramifiedrepresentation-fldpad-rngintelt-rngintelt-rngupolelt)

  - [`Example: Galrep Notfullycomputed`](creating.md#example-ex-f653e8)

  - [`SP(K,n): FldPad, RngIntElt → GalRep`](creating.md#function-sp-fldpad-rngintelt)

  - [`Example: Galrep Sp`](creating.md#example-ex-8d3d78)

  - [`SP(K,f,n): FldPad, RngUPolElt, RngIntElt → GalRep`](creating.md#function-sp-fldpad-rngupolelt-rngintelt)

  - [`Example: Galrep Sp 1`](creating.md#example-ex-3bd7e7)

  - [Representations from Finite Extensions](creating.md#representations-from-finite-extensions)

    - [`GaloisRepresentations(F,K): FldPad, FldPad → SeqEnum[GalRep]`](creating.md#function-galoisrepresentations-fldpad-fldpad)

    - [`Example: Galrep Galoisrepresentations 1`](creating.md#example-ex-314f0e)

    - [`GaloisRepresentations(f): RngUPolEltFldPad → SeqEnum[GalRep]`](creating.md#function-galoisrepresentations-rngupoleltfldpad)

    - [`Example: Galrep Galoisrepresentations`](creating.md#example-ex-da3085)

    - [`PermutationCharacter(F,K): FldPad, FldPad → GalRep`](creating.md#function-permutationcharacter-fldpad-fldpad)

    - [`Example: Galrep Permutationcharacter`](creating.md#example-ex-ba722b)

    - [`A !! chi: GalRep, AlgChtrElt → GalRep`](creating.md#operation-op-galrep-algchtrelt)

    - [`A !! chi: GalRep, SeqEnum → GalRep`](creating.md#operation-op-galrep-seqenum)

    - [`Example: Galrep Smash`](creating.md#example-ex-b9793f)

  - [Local Representations of Global Objects](creating.md#local-representations-of-global-objects)

    - [`GaloisRepresentation(chi,p): GrpDrchElt, RngIntElt → GalRep`](creating.md#function-galoisrepresentation-grpdrchelt-rngintelt)

    - [`Example: Galrep Dirichlet`](creating.md#example-ex-a9f044)

    - [`GaloisRepresentation(A,p): ArtRep, RngIntElt → GalRep`](creating.md#function-galoisrepresentation-artrep-rngintelt)

    - [`Example: Galrep Artin`](creating.md#example-ex-e01700)

    - [`GaloisRepresentation(E): CrvEll → GalRep`](creating.md#function-galoisrepresentation-crvell)

    - [`Example: Galrep E`](creating.md#example-ex-29c470)

    - [`GaloisRepresentation(E,p): CrvEll, RngIntElt → GalRep`](creating.md#function-galoisrepresentation-crvell-rngintelt)

    - [`Example: Galrep E P`](creating.md#example-ex-94d575)

    - [`GaloisRepresentation(E,P): CrvEll, RngOrdIdl → GalRep`](creating.md#function-galoisrepresentation-crvell-rngordidl)

    - [`Example: Galrep E P`](creating.md#example-ex-f7aec3)

    - [`GaloisRepresentation(C): CrvHyp[FldPad] → GalRep`](creating.md#function-galoisrepresentation-crvhyp-fldpad)

    - [`Example: Galrep Crvhyp Qp`](creating.md#example-ex-1fdb67)

    - [`GaloisRepresentation(C,p): CrvHyp[FldRat], RngIntElt → GalRep`](creating.md#function-galoisrepresentation-crvhyp-fldrat-rngintelt)

    - [`Example: Galrep Crvhyp Q`](creating.md#example-ex-1f2ef0)

    - [`GaloisRepresentation(C,P): CrvHyp[FldNum], RngOrdIdl → GalRep`](creating.md#function-galoisrepresentation-crvhyp-fldnum-rngordidl)

    - [`Example: Galrep Crvhyp Nf`](creating.md#example-ex-e4e4fd)

    - [`GaloisRepresentation(f,p): ModFrmElt, RngIntElt → GalRep`](creating.md#function-galoisrepresentation-modfrmelt-rngintelt)

    - [`Example: Galrep F P`](creating.md#example-ex-e7791f)

- [Basic Invariants](invariants.md)

  - [`BaseField(A): GalRep → FldPad`](invariants.md#function-basefield-galrep)

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

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

  - [`Example: Galrep Degree`](invariants.md#example-ex-890f18)

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

  - [`GaloisGroup(A): GalRep → GrpPerm`](invariants.md#function-galoisgroup-galrep)

  - [`Example: Galrep Group`](invariants.md#example-ex-baedd9)

  - [`FrobeniusElement(A): GalRep → GrpPermElt`](invariants.md#function-frobeniuselement-galrep)

  - [`Example: Galrep Frobeniuselement`](invariants.md#example-ex-abc80d)

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

  - [`Example: Galrep Character`](invariants.md#example-ex-ce7d20)

  - [`Field(A): GalRep → FldPad`](invariants.md#function-field-galrep)

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

  - [`Example: Galrep Definingpolynomial`](invariants.md#example-ex-216918)

  - [`Automorphism(A,g): GalRep, GrpPermElt → Map`](invariants.md#function-automorphism-galrep-grppermelt)

  - [`Example: Galrep Automorphism`](invariants.md#example-ex-3bde84)

  - [`EulerFactor(A): GalRep → RngUPolElt`](invariants.md#function-eulerfactor-galrep)

  - [`Example: Galrep Eulerfactor`](invariants.md#example-ex-641625)

  - [`IsZero(A): GalRep → BoolElt`](invariants.md#function-iszero-galrep)

  - [`IsOne(A): GalRep → BoolElt`](invariants.md#function-isone-galrep)

  - [`Factorization(A): GalRep → List, GalRep`](invariants.md#function-factorization-galrep)

  - [`Example: Galrep Factorization`](invariants.md#example-ex-d8fc15)

  - [Ramification](invariants.md#ramification)

    - [`InertiaGroup(A): GalRep → GrpPerm`](invariants.md#function-inertiagroup-galrep)

    - [`Example: Galrep Inertia`](invariants.md#example-ex-e156fa)

    - [`InertiaGroup(A,n): GalRep, RngIntElt → GrpPerm`](invariants.md#function-inertiagroup-galrep-rngintelt)

    - [`Example: Galrep Higherinertia`](invariants.md#example-ex-92ac63)

    - [`IsUnramified(A): GalRep → BoolElt`](invariants.md#function-isunramified-galrep)

    - [`Example: Galrep Isunramified`](invariants.md#example-ex-57fb0d)

    - [`IsRamified(A): GalRep → BoolElt`](invariants.md#function-isramified-galrep)

    - [`Example: Galrep Isramified`](invariants.md#example-ex-27c5ab)

    - [`IsTamelyRamified(A): GalRep → BoolElt`](invariants.md#function-istamelyramified-galrep)

    - [`IsWildlyRamified(A): GalRep → BoolElt`](invariants.md#function-iswildlyramified-galrep)

    - [`Example: Galrep Iswildlyramified`](invariants.md#example-ex-1ea277)

    - [`InertiaInvariants(A): GalRep → GalRep`](invariants.md#function-inertiainvariants-galrep)

    - [`Example: Galrep Inertiainvariants`](invariants.md#example-ex-04fb22)

    - [`ConductorExponent(A): GalRep → RngIntElt`](invariants.md#function-conductorexponent-galrep)

    - [`Conductor(A): GalRep → FldPadElt`](invariants.md#function-conductor-galrep)

    - [`Example: Galrep Conductor`](invariants.md#example-ex-7491ac)

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

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

    - [`Example: Epsilon Rootnumber`](invariants.md#example-ex-3d8d04)

  - [Semisimplicity and Irreducibles](invariants.md#semisimplicity-and-irreducibles)

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

    - [`Example: Galrep Isirreducible`](invariants.md#example-ex-f6575f)

    - [`IsIndecomposable(A): GalRep → BoolElt`](invariants.md#function-isindecomposable-galrep)

    - [`IsSemisimple(A): GalRep → BoolElt`](invariants.md#function-issemisimple-galrep)

    - [`Semisimplification(A): GalRep → GalRep`](invariants.md#function-semisimplification-galrep)

    - [`Decomposition(A): GalRep → SeqEnum[GalRep]`](invariants.md#function-decomposition-galrep)

    - [`Example: Galrep Decomposition`](invariants.md#example-ex-2ab1fc)

- [Arithmetic](arithmetic.md)

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

  - [`Example: Galrep Sum`](arithmetic.md#example-ex-1e67b8)

  - [`A1 - A2: GalRep, GalRep → BoolElt`](arithmetic.md#operation-op-minus-galrep-galrep)

  - [`Example: Galrep Difference`](arithmetic.md#example-ex-2242b4)

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

  - [`Example: Galrep Product`](arithmetic.md#example-ex-eb562c)

  - [`A1 / A2: GalRep, GalRep → GalRep`](arithmetic.md#operation-op-div-galrep-galrep)

  - [`A1 / A2: RngIntElt, GalRep → GalRep`](arithmetic.md#operation-op-div-rngintelt-galrep)

  - [`A ^ n: GalRep, Any → GalRep`](arithmetic.md#operation-op-pow-galrep-any)

  - [`Example: Galrep Power`](arithmetic.md#example-ex-173850)

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

  - [`Example: Galrep Eq`](arithmetic.md#example-ex-e6a517)

  - [`Determinant(A): GalRep → GalRep`](arithmetic.md#function-determinant-galrep)

  - [`Example: Galrep Determinant`](arithmetic.md#example-ex-e59efc)

  - [`TateTwist(A,n): GalRep, RngIntElt → GalRep`](arithmetic.md#function-tatetwist-galrep-rngintelt)

  - [`Example: Galrep Tatetwist`](arithmetic.md#example-ex-a9cca5)

- [Changing Precision](precision.md)

  - [`Precision(A): GalRep → RngIntElt`](precision.md#function-precision-galrep)

  - [`ChangePrecision(~A,Prec): GalRep, RngIntElt → {}`](precision.md#function-changeprecision-galrep-rngintelt-ref)

  - [`ChangePrecision(A,Prec): GalRep, RngIntElt → GalRep`](precision.md#function-changeprecision-galrep-rngintelt)

  - [`Example: Galrep Precision`](precision.md#example-ex-8bdd36)

- [Changing Fields](fields.md)

  - [`Minimize(A): GalRep → GalRep`](fields.md#function-minimize)

  - [`Example: Galrep Minimize`](fields.md#example-ex-b8e822)

  - [`Restriction(A,L): GalRep, FldPad → GalRep`](fields.md#function-restriction-galrep-fldpad)

  - [`BaseChange(A,L): GalRep, FldPad → GalRep`](fields.md#function-basechange-galrep-fldpad)

  - [`Example: Galrep Basechange`](fields.md#example-ex-3dc7d9)

  - [`Induction(A,K0): GalRep, FldPad → GalRep`](fields.md#function-induction-galrep-fldpad)

  - [`Example: Galrep Induction`](fields.md#example-ex-aae099)

- [Advanced Examples](fields-2.md)

  - [Example: Local and Global Epsilon Factors for Dirichlet Characters](fields-2.md#example-local-and-global-epsilon-factors-for-dirichlet-characters)

    - [`Example: Galrep Funeq`](fields-2.md#example-ex-938a3a)

  - [Example: Reconstructing a Galois Representation from its Euler Factors](fields-2.md#example-reconstructing-a-galois-representation-from-its-euler-factors)

    - [`Example: Galrep Reconstruction`](fields-2.md#example-ex-6f8047)