# Lattices over Number Fields

- [Introduction](introduction.md)

- [Number Field Lattices](numberfieldlattices.md)

  - [Creation of Number Field Lattices](numberfieldlattices.md#creation-of-number-field-lattices)

    - [`NumberFieldLattice(K, d): FldNum, RngIntElt → LatNF`](numberfieldlattices.md#function-numberfieldlattice-fldnum-rngintelt)

    - [`NumberFieldLattice(S): [ModTupFldElt] → LatNF`](numberfieldlattices.md#function-numberfieldlattice-modtupfldelt)

    - [`NumberFieldLattice(D): ModDed → LatNF`](numberfieldlattices.md#function-numberfieldlattice-modded)

    - [`NumberFieldLattice(L): Lat → LatNF`](numberfieldlattices.md#function-numberfieldlattice-lat)

    - [`NumberFieldLatticeWithGram(F): Mtrx → LatNF`](numberfieldlattices.md#function-numberfieldlatticewithgram-mtrx)

    - [`StandardLattice(V): SpcPlr → LatNF`](numberfieldlattices.md#function-standardlattice-spcplr)

    - [`StandardLattice(V): SpcPlr → Lat`](numberfieldlattices.md#function-standardlattice-spcplr-2)

    - [`LatticeWithBasis(V, B): SpcPlr, Mtrx[RngOrd] → LatNF`](numberfieldlattices.md#function-latticewithbasis-spcplr-mtrx-rngord)

    - [`LatticeWithBasis(V, B): SpcPlr, Mtrx[FldAlg] → LatNF`](numberfieldlattices.md#function-latticewithbasis-spcplr-mtrx-fldalg)

    - [`LatticeWithBasis(V, B): SpcPlr, Mtrx[RngInt] → Lat`](numberfieldlattices.md#function-latticewithbasis-spcplr-mtrx-rngint)

    - [`LatticeWithBasis(V, B): SpcPlr, Mtrx[FldRat] → Lat`](numberfieldlattices.md#function-latticewithbasis-spcplr-mtrx-fldrat)

    - [`LatticeWithBasis(V, B, J): SpcPlr, AlgMatElt[RngOrd], [ RngOrdFracIdl ] → LatNF`](numberfieldlattices.md#function-latticewithbasis-spcplr-algmatelt-rngord-rngordfracidl)

    - [`LatticeWithBasis(V, B, J): SpcPlr, AlgMatElt[FldAlg], [ RngOrdFracIdl ] → LatNF`](numberfieldlattices.md#function-latticewithbasis-spcplr-algmatelt-fldalg-rngordfracidl)

    - [`LatticeWithBasis(V, B, J): SpcPlr, AlgMatElt[FldQuad], [ RngOrdFracIdl ] → LatNF`](numberfieldlattices.md#function-latticewithbasis-spcplr-algmatelt-fldquad-rngordfracidl)

    - [`LatticeWithBasis(V, B, J): SpcPlr, AlgMatElt[RngInt], [ RngIntFracIdl ] → Lat`](numberfieldlattices.md#function-latticewithbasis-spcplr-algmatelt-rngint-rngintfracidl)

    - [`LatticeWithBasis(V, B, J): SpcPlr, AlgMatElt[RngInt], [ RngInt ] → Lat`](numberfieldlattices.md#function-latticewithbasis-spcplr-algmatelt-rngint-rngint)

    - [`LatticeWithBasis(V, B, J): SpcPlr, AlgMatElt[FldRat], [ RngIntFracIdl ] → Lat`](numberfieldlattices.md#function-latticewithbasis-spcplr-algmatelt-fldrat-rngintfracidl)

    - [`LatticeWithBasis(V, B, J): SpcPlr, AlgMatElt[FldRat], [ RngInt ] → Lat`](numberfieldlattices.md#function-latticewithbasis-spcplr-algmatelt-fldrat-rngint)

    - [`LatticeWithPseudobasis(V, P): SpcPlr, PMat → LatNF`](numberfieldlattices.md#function-latticewithpseudobasis-spcplr-pmat)

    - [`Lattice(V, L): SpcPlr, ModDed → LatNF`](numberfieldlattices.md#function-lattice-spcplr-modded)

    - [`Lattice(V, L): SpcPlr, Lat → Lat`](numberfieldlattices.md#function-lattice-spcplr-lat)

    - [`Module(L): LatNF → ModDed`](numberfieldlattices.md#function-module-latnf)

    - [`ChangeRing(L, R): LatNF, Rng → LatNF`](numberfieldlattices.md#function-changering-latnf-rng)

    - [`sub<L | RHS>: LatNF, Any → LatNF, Map`](numberfieldlattices.md#constructor-constructor-sub-latnf-any-latnf-map)

    - [`ext<L | RHS>: LatNF, Any → LatNF, Map`](numberfieldlattices.md#constructor-constructor-ext-latnf-any-latnf-map)

    - [`A + B: LatNF, LatNF → LatNF`](numberfieldlattices.md#operation-op-plus-latnf-latnf)

    - [`A meet B: LatNF, LatNF → LatNF`](numberfieldlattices.md#operation-op-meet-latnf-latnf)

    - [`r * L: RngElt, LatNF → LatNF`](numberfieldlattices.md#operation-op-times-rngelt-latnf)

    - [`L * r: Lat NF, LatNF → LatNF`](numberfieldlattices.md#operation-op-times-lat-nf-latnf)

    - [`L / r: Lat NF, LatNF → LatNF`](numberfieldlattices.md#operation-op-div-lat-nf-latnf)

    - [`BasisScaling(L, r): LatNF, RngElt → LatNF`](numberfieldlattices.md#function-basisscaling-latnf-rngelt)

    - [`InnerProductScaling(L, r): LatNF, RngElt → LatNF`](numberfieldlattices.md#function-innerproductscaling-latnf-rngelt)

    - [`ScaledLattice(L, r): LatNF, FldElt → LatNF`](numberfieldlattices.md#function-scaledlattice-latnf-fldelt)

    - [`J * L: RngOrdFracIdl, LatNF → LatNF`](numberfieldlattices.md#operation-op-times-rngordfracidl-latnf)

    - [`L * J: LatNF, RngOrdFracIdl → LatNF`](numberfieldlattices.md#operation-op-times-latnf-rngordfracidl)

    - [`L / J: LatNF, RngOrdFracIdl → LatNF`](numberfieldlattices.md#operation-op-div-latnf-rngordfracidl)

    - [`T * L: Mtrx, LatNF → LatNF`](numberfieldlattices.md#operation-op-times-mtrx-latnf)

    - [`TJ * L: PMat, LatNF → LatNF`](numberfieldlattices.md#operation-op-times-pmat-latnf)

    - [`L * T: LatNF, Mtrx → LatNF`](numberfieldlattices.md#operation-op-times-latnf-mtrx)

    - [`DirectSum(A, B): LatNF, LatNF → LatNF`](numberfieldlattices.md#function-directsum-latnf-latnf)

    - [`DirectSum(A): SeqEnum[LatNF] → LatNF`](numberfieldlattices.md#function-directsum-seqenum-latnf)

    - [`OrthogonalComplement(L, v): LatNF, LatNFElt → LatNF`](numberfieldlattices.md#function-orthogonalcomplement-latnf-latnfelt)

    - [`OrthogonalComplement(L, S): LatNF, LatNF → LatNF`](numberfieldlattices.md#function-orthogonalcomplement-latnf-latnf)

    - [`Dual(L): LatNF → LatNF`](numberfieldlattices.md#function-dual-latnf)

    - [`SimpleLattice(L): LatNF → LatNF`](numberfieldlattices.md#function-simplelattice-latnf)

    - [`ZLattice(L): LatNF → Lat`](numberfieldlattices.md#function-zlattice-latnf)

    - [`RestrictionOfScalarsToZ(L): LatNF → Lat`](numberfieldlattices.md#function-restrictionofscalarstoz-latnf)

    - [`MaximalSublattices(L, p): LatNF, RngOrdIdl → [LatNF], [RngIntElt]`](numberfieldlattices.md#function-maximalsublattices-latnf-rngordidl)

    - [`MinimalSuperlattices(L, p): LatNF, RngOrdIdl → [LatNF], [RngIntElt]`](numberfieldlattices.md#function-minimalsuperlattices-latnf-rngordidl)

    - [`MaximalIntegralLattice(Q): Mtrx → LatNF`](numberfieldlattices.md#function-maximalintegrallattice-mtrx)

    - [`MaximalIntegralLattice(L): LatNF → LatNF`](numberfieldlattices.md#function-maximalintegrallattice-latnf)

    - [`MaximalIntegralLattice(L): Lat → Lat`](numberfieldlattices.md#function-maximalintegrallattice-lat)

    - [`MaximalIntegralLattice(L, p): LatNF, RngOrdIdl → LatNF`](numberfieldlattices.md#function-maximalintegrallattice-latnf-rngordidl)

    - [`MaximalIntegralLattice(L, p): Lat, RngInt → LatNF`](numberfieldlattices.md#function-maximalintegrallattice-lat-rngint)

    - [`MaximalIntegralLattice(L, p): Lat, RngIntElt → LatNF`](numberfieldlattices.md#function-maximalintegrallattice-lat-rngintelt)

    - [`MaximalIntegralLattice(V): SpcPlr → LatNF`](numberfieldlattices.md#function-maximalintegrallattice-spcplr)

    - [`Example: Creation Examples`](numberfieldlattices.md#example-ex-d20734)

  - [Attributes of Number Field Lattices](numberfieldlattices.md#attributes-of-number-field-lattices)

    - [`Basis(L): LatNF → [ModTupFldElt]`](numberfieldlattices.md#function-basis-latnf)

    - [`BasisMatrix(L): LatNF → Mtrx`](numberfieldlattices.md#function-basismatrix-latnf)

    - [`FreeBasis(L): LatNF → [ModTupFldElt]`](numberfieldlattices.md#function-freebasis-latnf)

    - [`LocalBasis(L, p): LatNF, RngOrdIdl → [ ModTupFldElt ]`](numberfieldlattices.md#function-localbasis-latnf-rngordidl)

    - [`LocalBasis(L, p): Lat, RngInt → [ ModTupFldElt ]`](numberfieldlattices.md#function-localbasis-lat-rngint)

    - [`PseudoBasis(L): LatNF → [ModTupFldElt]`](numberfieldlattices.md#function-pseudobasis-latnf)

    - [`PseudoBasisMatrix(L): LatNF → Mtrx`](numberfieldlattices.md#function-pseudobasismatrix-latnf)

    - [`PseudoMatrix(L): LatNF → PMat`](numberfieldlattices.md#function-pseudomatrix-latnf)

    - [`CoefficientIdeals(L): LatNF → SeqEnum`](numberfieldlattices.md#function-coefficientideals-latnf)

    - [`Involution(L): LatNF → FldAut`](numberfieldlattices.md#function-involution-latnf)

    - [`Generators(L): LatNF → SeqEnum`](numberfieldlattices.md#function-generators-latnf)

    - [`GeneratorMatrix(L): LatNF → Mtrx`](numberfieldlattices.md#function-generatormatrix-latnf)

    - [`InnerProductMatrix(L): LatNF → Mtrx`](numberfieldlattices.md#function-innerproductmatrix-latnf)

    - [`AmbientSpace(L): LatNF → SpcPlr`](numberfieldlattices.md#function-ambientspace-latnf)

    - [`EmbeddingSpace(L): LatNF → Mod`](numberfieldlattices.md#function-embeddingspace-latnf)

    - [`MakeAmbientInnerProduct(~L, IP): LatNF`](numberfieldlattices.md#function-makeambientinnerproduct-latnf-ref)

    - [`GramMatrix(L): LatNF → Mtrx`](numberfieldlattices.md#function-grammatrix-latnf)

    - [`PseudoGramMatrix(L): LatNF → Mtrx`](numberfieldlattices.md#function-pseudogrammatrix-latnf)

    - [`GramMatrix(L, S): LatNF, [ ModTupFldElt ] → AlgMatElt`](numberfieldlattices.md#function-grammatrix-latnf-modtupfldelt)

    - [`GramMatrix(S): [ LatNFElt ] → AlgMatElt`](numberfieldlattices.md#function-grammatrix-latnfelt)

    - [`Rank(L): LatNF → RngIntElt`](numberfieldlattices.md#function-rank-latnf)

    - [`Dimension(L): LatNF → RngIntElt`](numberfieldlattices.md#function-dimension-latnf)

    - [`Degree(L): LatNF → RngIntElt`](numberfieldlattices.md#function-degree-latnf)

    - [`BaseRing(L): LatNF → FldNum`](numberfieldlattices.md#function-basering-latnf)

    - [`CoordinateRing(L): LatNF → RngOrd`](numberfieldlattices.md#function-coordinatering-latnf)

    - [`Order(L): LatNF → RngOrd`](numberfieldlattices.md#function-order-latnf)

    - [`Determinant(L): LatNF → FldNumElt`](numberfieldlattices.md#function-determinant-latnf)

    - [`Discriminant(L): LatNF → RngOrdFracIdl`](numberfieldlattices.md#function-discriminant-latnf)

    - [`Volume(L): LatNF → RngOrdFracIdl`](numberfieldlattices.md#function-volume-latnf)

    - [`Norm(L): LatNF → RngOrdFracIdl`](numberfieldlattices.md#function-norm-latnf)

    - [`Scale(L): LatNF → RngOrdFracIdl`](numberfieldlattices.md#function-scale-latnf)

    - [`BadPrimes(L): LatNF → Set`](numberfieldlattices.md#function-badprimes-latnf)

    - [`AuxiliaryForms(L): LatNF → [ AlgMatElt ]`](numberfieldlattices.md#function-auxiliaryforms-latnf)

    - [`ElementaryDivisors(A, B): LatNF, LatNF → [ RngOrdFracIdl ]`](numberfieldlattices.md#function-elementarydivisors-latnf-latnf)

    - [`Discriminant(A, B): LatNF, LatNF → RngOrdFracIdl`](numberfieldlattices.md#function-discriminant-latnf-latnf)

    - [`Index(A, B): LatNF, LatNF → RngOrdFracIdl`](numberfieldlattices.md#function-index-latnf-latnf)

    - [`JordanDecomposition(L, p): LatNF, RndOrdIdl → List, List, SeqEnum`](numberfieldlattices.md#function-jordandecomposition-latnf-rndordidl)

    - [`MaximalNormSplitting(L, p): LatNF, RngOrdIdl → SeqEnum, List`](numberfieldlattices.md#function-maximalnormsplitting-latnf-rngordidl)

    - [`GoodBasisOfNormGenerators(L, p): LatNF, RngOrdIdl → SeqEnum, SeqEnum`](numberfieldlattices.md#function-goodbasisofnormgenerators-latnf-rngordidl)

    - [`GenusSymbol(L, p): LatNF, RngOrdIdl → SeqEnum, Any`](numberfieldlattices.md#function-genussymbol-latnf-rngordidl)

    - [`LocalGenus(L, p): LatNF, RngOrdIdl → SymGenLoc`](numberfieldlattices.md#function-localgenus-latnf-rngordidl)

    - [`Genus(L): LatNF → SymGen`](numberfieldlattices.md#function-genus-latnf)

    - [`HasseInvariant(M, p): AlgMatElt, RngOrdIdl → RngIntElt`](numberfieldlattices.md#function-hasseinvariant-algmatelt-rngordidl)

    - [`HasseInvariant(L, p): LatNF, RngOrdIdl → RngIntElt`](numberfieldlattices.md#function-hasseinvariant-latnf-rngordidl)

    - [`WittInvariant(M, p): AlgMatElt, RngOrdIdl → RngIntElt`](numberfieldlattices.md#function-wittinvariant-algmatelt-rngordidl)

    - [`WittInvariant(L, p): LatNF, RngOrdIdl → RngIntElt`](numberfieldlattices.md#function-wittinvariant-latnf-rngordidl)

    - [`SpinorNorm(L, p): LatNF, RngOrdIdl → ModTupFld, Map, BoolElt`](numberfieldlattices.md#function-spinornorm-latnf-rngordidl)

    - [`Mass(L): LatNF → FldRatElt`](numberfieldlattices.md#function-mass-latnf)

    - [`Mass(L): Lat → FldRatElt`](numberfieldlattices.md#function-mass-lat)

    - [`LocalFactor(L, p): LatNF, RngOrdIdl → FldRatElt`](numberfieldlattices.md#function-localfactor-latnf-rngordidl)

    - [`Neighbours(L, p): LatNF, RngOrdIdl → [LatNF], [RngIntElt]`](numberfieldlattices.md#function-neighbours-latnf-rngordidl)

    - [`IteratedNeighbours(L, p): LatNF, RngOrdIdl → [LatNF]`](numberfieldlattices.md#function-iteratedneighbours-latnf-rngordidl)

    - [`GenusRepresentatives(L): LatNF → [LatNF], Assoc`](numberfieldlattices.md#function-genusrepresentatives-latnf)

    - [`Example: Attr Examples`](numberfieldlattices.md#example-ex-58cb4b)

    - [`NumberOfIsotropicSubspaces(L, P, k): LatNF, RngOrdIdl, RngIntElt → RngIntElt`](numberfieldlattices.md#function-numberofisotropicsubspaces-latnf-rngordidl-rngintelt)

    - [`NumberOfIsotropicSubspaces(L, P, k): Lat, RngInt, RngIntElt → RngIntElt`](numberfieldlattices.md#function-numberofisotropicsubspaces-lat-rngint-rngintelt)

    - [`NumberOfIsotropicSubspaces(L, P, k): Lat, RngIntElt, RngIntElt → RngIntElt`](numberfieldlattices.md#function-numberofisotropicsubspaces-lat-rngintelt-rngintelt)

    - [`NumberOfNeighbors(L, P, k): LatNF, RngOrdIdl, RngIntElt → RngIntElt`](numberfieldlattices.md#function-numberofneighbors-latnf-rngordidl-rngintelt)

    - [`NumberOfNeighbors(L, P, k): Lat, RngInt, RngIntElt → RngIntElt`](numberfieldlattices.md#function-numberofneighbors-lat-rngint-rngintelt)

    - [`NumberOfNeighbors(L, P, k): Lat, RngIntElt, RngIntElt → RngIntElt`](numberfieldlattices.md#function-numberofneighbors-lat-rngintelt-rngintelt)

    - [`NeighborProcess(L, P, k): LatNF, RngOrdIdl, RngIntElt → NeighborProc`](numberfieldlattices.md#function-neighborprocess-latnf-rngordidl-rngintelt)

    - [`NeighborProcess(L, P, k): Lat, RngInt, RngIntElt → NeighborProc`](numberfieldlattices.md#function-neighborprocess-lat-rngint-rngintelt)

    - [`Neighbor(nProc): NeighborProc → LatNF`](numberfieldlattices.md#function-neighbor-neighborproc)

    - [`Advance(nProc): NeighborProc`](numberfieldlattices.md#function-advance-neighborproc)

    - [`SkipTo(nProc, V, S): NeighborProc, SeqEnum[ModTupFldElt[FldFin]], AlgMatElt[FldFin]`](numberfieldlattices.md#function-skipto-neighborproc-seqenum-modtupfldelt-fldfin-algmatelt-fldfin)

  - [Predicates on Number Field Lattices](numberfieldlattices.md#predicates-on-number-field-lattices)

    - [`IsSimple(L): LatNF → BoolElt`](numberfieldlattices.md#function-issimple-latnf)

    - [`IsFree(L): LatNF → BoolElt`](numberfieldlattices.md#function-isfree-latnf)

    - [`IsZero(L): LatNF → BoolElt`](numberfieldlattices.md#function-iszero-latnf)

    - [`IsFull(L): LatNF → BoolElt`](numberfieldlattices.md#function-isfull-latnf)

    - [`IsHermitian(L): LatNF → BoolElt`](numberfieldlattices.md#function-ishermitian-latnf)

    - [`IsQuadratic(L): LatNF → BoolElt`](numberfieldlattices.md#function-isquadratic-latnf)

    - [`IsTotallyPositiveDefinite(L): LatNF → BoolElt`](numberfieldlattices.md#function-istotallypositivedefinite-latnf)

    - [`IsDefinite(L): LatNF → BoolElt, RngOrdElt`](numberfieldlattices.md#function-isdefinite-latnf)

    - [`Signature(F): Mtrx[FldAlg] → ModTupRngElt[RngInt], ModTupRngElt[RngInt], RngIntElt`](numberfieldlattices.md#function-signature-mtrx-fldalg)

    - [`A eq B: LatNF, LatNF → BoolElt`](numberfieldlattices.md#operation-op-eq-latnf-latnf)

    - [`A ne B: LatNF, LatNF → BoolElt`](numberfieldlattices.md#operation-op-ne-latnf-latnf)

    - [`IsIdentical(A, B): LatNF, LatNF → BoolElt`](numberfieldlattices.md#function-isidentical-latnf-latnf)

    - [`IsSublattice(S, L): LatNF, LatNF → BoolElt, Mtrx`](numberfieldlattices.md#function-issublattice-latnf-latnf)

    - [`S subset L: LatNF, LatNF → BoolElt`](numberfieldlattices.md#operation-op-subset-latnf-latnf)

    - [`IsMaximal(L): LatNF → BoolElt, LatNF`](numberfieldlattices.md#function-ismaximal-latnf)

    - [`IsMaximal(L, p): LatNF → BoolElt, LatNF`](numberfieldlattices.md#function-ismaximal-latnf-2)

    - [`IsIntegral(L): LatNF → BoolElt`](numberfieldlattices.md#function-isintegral-latnf)

    - [`IsIntegral(L): Lat → BoolElt`](numberfieldlattices.md#function-isintegral-lat)

    - [`IsIntegral(L, p): LatNF, RngOrdIdl → BoolElt`](numberfieldlattices.md#function-isintegral-latnf-rngordidl)

    - [`IsIntegral(L, p): Lat, RngInt → BoolElt`](numberfieldlattices.md#function-isintegral-lat-rngint)

    - [`IsIntegral(L, p): Lat, RngIntElt → BoolElt`](numberfieldlattices.md#function-isintegral-lat-rngintelt)

    - [`IsEven(L): LatNF → BoolElt`](numberfieldlattices.md#function-iseven-latnf)

    - [`IsMaximalIntegral(L): LatNF → BoolElt, LatNF`](numberfieldlattices.md#function-ismaximalintegral-latnf)

    - [`IsMaximalIntegral(L): Lat → BoolElt, Lat`](numberfieldlattices.md#function-ismaximalintegral-lat)

    - [`IsMaximalIntegral(L, p): LatNF, RngOrdIdl → BoolElt, LatNF`](numberfieldlattices.md#function-ismaximalintegral-latnf-rngordidl)

    - [`IsMaximalIntegral(L, p): Lat, RngInt → BoolElt, LatNF`](numberfieldlattices.md#function-ismaximalintegral-lat-rngint)

    - [`IsMaximalIntegral(L, p): Lat, RngIntElt → BoolElt, Lat`](numberfieldlattices.md#function-ismaximalintegral-lat-rngintelt)

    - [`IsModular(L): LatNF → BoolElt, RngOrdFracIdl`](numberfieldlattices.md#function-ismodular-latnf)

    - [`IsModular(L, p): LatNF, RngOrdIdl → BoolElt, RngIntElt`](numberfieldlattices.md#function-ismodular-latnf-rngordidl)

    - [`IsIsotropic(L, p): LatNF, RngOrdIdl → BoolElt`](numberfieldlattices.md#function-isisotropic-latnf-rngordidl)

    - [`IsIsotropic(L, p): LatNF, PlcNumElt → BoolElt`](numberfieldlattices.md#function-isisotropic-latnf-plcnumelt)

    - [`IsLocallyIsometric(L1, L2, p): LatNF, LatNF, RngOrdIdl → BoolElt`](numberfieldlattices.md#function-islocallyisometric-latnf-latnf-rngordidl)

    - [`IsSameGenus(L1, L2): LatNF, LatNF → BoolElt`](numberfieldlattices.md#function-issamegenus-latnf-latnf)

    - [`IsRationallyEquivalent(L1, L2, p): LatNF, LatNF, RngOrdIdl → BoolElt`](numberfieldlattices.md#function-isrationallyequivalent-latnf-latnf-rngordidl)

    - [`IsRationallyEquivalent(L1, L2, p): LatNF, LatNF, PlcNumElt → BoolElt`](numberfieldlattices.md#function-isrationallyequivalent-latnf-latnf-plcnumelt)

    - [`IsRationallyEquivalent(L1, L2): LatNF, LatNF → BoolElt`](numberfieldlattices.md#function-isrationallyequivalent-latnf-latnf)

  - [Totally Positive Definite Lattices](numberfieldlattices.md#totally-positive-definite-lattices)

    - [`AutomorphismGroup(L): LatNF → GrpMat`](numberfieldlattices.md#function-automorphismgroup-latnf)

    - [`ProperAutomorphismGroup(L): LatNF → GrpMat`](numberfieldlattices.md#function-properautomorphismgroup-latnf)

    - [`IsIsometric(A, B): LatNF, LatNF → BoolElt, Mtrx`](numberfieldlattices.md#function-isisometric-latnf-latnf)

    - [`IsProperlyIsometric(A, B): LatNF, LatNF → BoolElt, Mtrx`](numberfieldlattices.md#function-isproperlyisometric-latnf-latnf)

    - [`IsSimilar(A, B): LatNF, LatNF → BoolElt, Mtrx, FldNumElt`](numberfieldlattices.md#function-issimilar-latnf-latnf)

    - [`Sphere(L, e): LatNF, RngElt → Setq`](numberfieldlattices.md#function-sphere-latnf-rngelt)

    - [`IsRepresented(L, e): LatNF, RngElt → BoolElt, LatNFElt`](numberfieldlattices.md#function-isrepresented-latnf-rngelt)

- [Number Field Lattice Elements](numberfieldlatticeelements.md)

  - [Creation](numberfieldlatticeelements.md#creation)

    - [`Zero(L): LatNF → LatNFElt`](numberfieldlatticeelements.md#function-zero-latnf)

    - [`L ! e: LatNF, Any → LatNFElt`](numberfieldlatticeelements.md#operation-op-latnf-any)

    - [`L . i: LatNF, RngIntElt → LatNFElt`](numberfieldlatticeelements.md#operation-operation-latnf-rngintelt-latnfelt)

    - [`CoordinatesToLattice(L, S): LatNF, SeqEnum → LatNFElt`](numberfieldlatticeelements.md#function-coordinatestolattice-latnf-seqenum)

    - [`CoordinatesToLattice(L, v): LatNF, ModTupFldElt → LatNFElt`](numberfieldlatticeelements.md#function-coordinatestolattice-latnf-modtupfldelt)

  - [Parent and Element Relations](numberfieldlatticeelements.md#parent-and-element-relations)

    - [`v in L: ModTupRngElt, LatNF → BoolElt, ModTupFldElt`](numberfieldlatticeelements.md#operation-op-in-modtuprngelt-latnf)

    - [`v in L: LatNFElt, LatNF → BoolElt, ModTupFldElt`](numberfieldlatticeelements.md#operation-op-in-latnfelt-latnf)

    - [`Parent(v): LatNFElt → LatNF`](numberfieldlatticeelements.md#function-parent-latnfelt)

  - [Arithmetic](numberfieldlatticeelements.md#arithmetic)

    - [`v + w: LatNFElt, LatNFElt → LatNFElt`](numberfieldlatticeelements.md#operation-op-plus-latnfelt-latnfelt)

    - [`v - w: LatNFElt, LatNFElt → LatNFElt`](numberfieldlatticeelements.md#operation-op-minus-latnfelt-latnfelt)

    - [`- v: LatNFElt → LatNFElt`](numberfieldlatticeelements.md#operation-operation-latnfelt-latnfelt)

    - [`v eq w: LatNFElt, LatNFElt → BoolElt`](numberfieldlatticeelements.md#operation-op-eq-latnfelt-latnfelt)

    - [`v ne w: LatNFElt, LatNFElt → BoolElt`](numberfieldlatticeelements.md#operation-op-ne-latnfelt-latnfelt)

    - [`IsZero(v): LatNFElt → BoolElt`](numberfieldlatticeelements.md#function-iszero-latnfelt)

    - [`s * v: RngElt, LatNFElt → LatNFElt`](numberfieldlatticeelements.md#operation-op-times-rngelt-latnfelt)

    - [`v * s: LatNFElt, RngElt → LatNFElt`](numberfieldlatticeelements.md#operation-op-times-latnfelt-rngelt)

    - [`v / s: LatNFElt, RngElt → LatNFElt`](numberfieldlatticeelements.md#operation-op-div-latnfelt-rngelt)

    - [`T * v: Mtrx, LatNFElt → LatNFElt`](numberfieldlatticeelements.md#operation-op-times-mtrx-latnfelt)

    - [`v * T: LatNFElt, Mtrx → LatNFElt`](numberfieldlatticeelements.md#operation-op-times-latnfelt-mtrx)

    - [`v ^ M: LatNFElt, Mtrx → LatNFElt`](numberfieldlatticeelements.md#operation-op-pow-latnfelt-mtrx)

    - [`v ^ G: LatNFElt, GrpMat → Setq[LatNFElt]`](numberfieldlatticeelements.md#operation-op-pow-latnfelt-grpmat)

    - [`Orbit(G, v): GrpMat, LatNFElt → Setq[LatNFElt]`](numberfieldlatticeelements.md#function-orbit-grpmat-latnfelt)

    - [`Stabilizer(G, v): GrpMat, LatNFElt → GrpMat`](numberfieldlatticeelements.md#function-stabilizer-grpmat-latnfelt)

    - [`Norm(v): LatNFElt → FldNumElt`](numberfieldlatticeelements.md#function-norm-latnfelt)

    - [`InnerProduct(v, w): LatNFElt, LatNFElt → FldNumElt`](numberfieldlatticeelements.md#function-innerproduct-latnfelt-latnfelt)

    - [`Example: Nflatelt Ex`](numberfieldlatticeelements.md#example-ex-9da2f4)

  - [Access Functions](numberfieldlatticeelements.md#access-functions)

    - [`Vector(v): LatNFElt → ModTupFldElt`](numberfieldlatticeelements.md#function-vector-latnfelt)

    - [`Eltseq(v): LatNFElt → SeqEnum`](numberfieldlatticeelements.md#function-eltseq-latnfelt)

    - [`Coordinates(v): LatNFElt → ModTupFldElt`](numberfieldlatticeelements.md#function-coordinates-latnfelt)

- [Examples](examples.md)

  - [`Example: Simple Nflat Example`](examples.md#example-ex-3a8daf)

  - [`Example: Nf Second Ex`](examples.md#example-ex-22b211)

  - [`Example: Nf Lat Sqrt5`](examples.md#example-ex-dda13e)

  - [`Example: Nflat Notsimple`](examples.md#example-ex-077d76)

  - [`Example: Nflat Maass`](examples.md#example-ex-4a3ec7)

- [Lorentzian Lattices](lorentzian-lattices.md)

  - [Special Intrinsics](lorentzian-lattices.md#special-intrinsics)

    - [`IsLorentzian(L): LatNF → BoolElt, ModTupFldElt, RngIntElt`](lorentzian-lattices.md#function-islorentzian-latnf)

    - [`IsTimelike(v): LatNFElt → BoolElt`](lorentzian-lattices.md#function-istimelike-latnfelt)

    - [`IsSpacelike(v): LatNFElt → BoolElt`](lorentzian-lattices.md#function-isspacelike-latnfelt)

    - [`AutomorphismGroup(L, v): LatNF, LatNFElt → GrpMat, GrpMat`](lorentzian-lattices.md#function-automorphismgroup-latnf-latnfelt)

    - [`IsIsometric(L, v, w): LatNF, LatNFElt, LatNFElt → BoolElt, Mtrx`](lorentzian-lattices.md#function-isisometric-latnf-latnfelt-latnfelt)

    - [`Example: Simple Lorentz Lat Ex`](lorentzian-lattices.md#example-ex-ac8528)