# Mod P Galois Representations

- [Introduction](Introduction.md)

  - [Motivation](Introduction.md#motivation)

  - [Definitions](Introduction.md#definitions)

  - [Classification of $\varphi$-modules](Introduction.md#classification-of-varphi-modules)

  - [Connection with Galois Representations](Introduction.md#connection-with-galois-representations)

- [$\varphi$-modules and Galois Representations in Magma](modules-Galois-representations.md)

  - [$\varphi$-modules](modules-Galois-representations.md#varphi-modules)

    - [Category](modules-Galois-representations.md#category)

    - [Creation Functions](modules-Galois-representations.md#creation-functions)

      - [`PhiModule(M): AlgMatElt → PhiMod`](modules-Galois-representations.md#function-phimodule-algmatelt)

      - [`ElementaryPhiModule(S,d,h): RngSerLaur, RngIntElt, RngIntElt → PhiMod`](modules-Galois-representations.md#function-elementaryphimodule-rngserlaur-rngintelt-rngintelt)

      - [`PhiModuleElement(x,D): AlgMatElt, PhiMod → PhiModElt`](modules-Galois-representations.md#function-phimoduleelement-algmatelt-phimod)

    - [Attributes of $\varphi$-modules](modules-Galois-representations.md#attributes-of-varphi-modules)

      - [`Dimension(D): PhiMod → RngIntElt`](modules-Galois-representations.md#function-dimension-phimod)

      - [`CoefficientRing(D): PhiMod → RngSerLaur`](modules-Galois-representations.md#function-coefficientring-phimod)

      - [`FrobeniusMatrix(D): PhiMod → AlgMatElt`](modules-Galois-representations.md#function-frobeniusmatrix-phimod)

    - [Basic Operations and Properties of $\varphi$-modules](modules-Galois-representations.md#basic-operations-and-properties-of-varphi-modules)

      - [`IsEtale(D): PhiMod → BoolElt`](modules-Galois-representations.md#function-isetale-phimod)

      - [`ChangePrecision(~D, prec): ~PhiMod, RngIntElt`](modules-Galois-representations.md#function-changeprecision-phimod-rngintelt-ref)

      - [`DirectSum(D1, D2): PhiMod, PhiMod → PhiMod`](modules-Galois-representations.md#function-directsum-phimod-phimod)

      - [`BaseChange(~D, P): ~~PhiMod, AlgMatElt`](modules-Galois-representations.md#function-basechange-phimod-algmatelt-ref)

      - [`RandomBaseChange(~D): PhiMod`](modules-Galois-representations.md#function-randombasechange-phimod-ref)

      - [`Phi(D, x): PhiMod, PhiModElt → PhiModElt`](modules-Galois-representations.md#function-phi-phimod-phimodelt)

    - [Reduction of $\varphi$-modules and Galois Representations](modules-Galois-representations.md#reduction-of-varphi-modules-and-galois-representations)

      - [`SemisimpleDecomposition(D): PhiMod → AlgMatElt, AlgMatElt, SeqEnum, SeqEnum`](modules-Galois-representations.md#function-semisimpledecomposition-phimod)

      - [`Slopes(D): PhiMod → SeqEnum`](modules-Galois-representations.md#function-slopes-phimod)

      - [`SSGaloisRepresentation(D): PhiMod → SSGalRep`](modules-Galois-representations.md#function-ssgaloisrepresentation-phimod)

  - [Semisimple Galois Representations](modules-Galois-representations.md#semisimple-galois-representations)

    - [Category](modules-Galois-representations.md#id1)

    - [Creation Functions](modules-Galois-representations.md#id2)

      - [`SSGaloisRepresentation(E,K,w,P): FldFin, RngSerLaur, SeqEnum, SeqEnum → SSGalRep`](modules-Galois-representations.md#function-ssgaloisrepresentation-fldfin-rngserlaur-seqenum-seqenum)

    - [Basic Operations](modules-Galois-representations.md#basic-operations)

      - [`CoefficientRing(V): SSGalRep → FldFin`](modules-Galois-representations.md#function-coefficientring-ssgalrep)

      - [`FixedField(V): SSGalRep → RngSerLaur`](modules-Galois-representations.md#function-fixedfield-ssgalrep)

      - [`Weights(V): SSGalRep → SeqEnum`](modules-Galois-representations.md#function-weights-ssgalrep)

    - [Representation Associated to a $\varphi$-Module](modules-Galois-representations.md#representation-associated-to-a-varphi-module)

      - [`SSGaloisRepresentation(D): PhiMod → SSGalRep`](modules-Galois-representations.md#function-ssgaloisrepresentation-phimod-2)

- [Examples](Examples.md)

  - [`Example: generic`](Examples.md#example-ex-b604c9)