Elements

  (O::NfOrd)(a::Union{fmpz, Integer}) -> NfAbsOrdElem

Given an element $a$ of type fmpz or Integer, this function coerces the element into $\mathcal O$. It will be checked that $a$ is contained in $\mathcal O$ if and only if check is true.

  (O::NfOrd)(arr::Array{fmpz, 1})

Returns the element of $\mathcal O$ with coefficient vector arr.

  (O::NfOrd)() -> NfAbsOrdElem

This function constructs a new element of $\mathcal O$ which is set to $0$.

parent(a::NfAbsOrdElem) -> NfOrd

Returns the order of which $a$ is an element.

parent(g::perm{T}) where T = PermGroup

Return the parent of the permutation g.

julia> G = PermutationGroup(5); g = perm([3,4,5,2,1])
(1,3,5)(2,4)

julia> parent(g) == G
true

parent(a::AbstractAlgebra.MatElem{T}, cached::Bool = true) where T <: RingElement

Return the parent object of the given matrix.

parent(a::AbstractAlgebra.MatAlgElem{T}, cached::Bool = true) where T <: RingElement

Return the parent object of the given matrix.

elem_in_nf(a::NfAbsOrdElem) -> nf_elem

Returns the element $a$ considered as an element of the ambient number field.

elem_in_basis(a::NfAbsOrdElem) -> Array{fmpz, 1}

Returns the coefficient vector of $a$.

  elem_in_basis(a::NfRelOrdElem{T}) -> Vector{T}

Returns the coefficient vector of $a$.

discriminant(B::Array{NfAbsOrdElem, 1}) -> fmpz

Returns the discriminant of the family $B$.

==(x::NfAbsOrdElem, y::NfAbsOrdElem) -> Bool

Returns whether $x$ and $y$ are equal.

zero(O::NfOrd) -> NfAbsOrdElem

Returns the zero element of $\mathcal O$.

one(O::NfOrd) -> NfAbsOrdElem

Returns the one element of $\mathcal O$.

iszero(a::NfOrd) -> Bool

Tests if $a$ is zero.

isone(a::NfOrd) -> Bool

Tests if $a$ is one.

-(x::NfAbsOrdElem) -> NfAbsOrdElem

Returns the additive inverse of $x$.

-(A::SMat, B::SMat) -> SMat

Return the difference $A - B$.

-(A::SRow, B::SRow) -> SRow

Returns the difference of $A$ and $B$.

-(a::AbstractAlgebra.ResElem{T}, b::AbstractAlgebra.ResElem{T}) where {T <: RingElement}

Return $a - b$.

-(a::AbstractAlgebra.ResElem{T}, b::T) where {T <: RingElem}

Return $a - b$.

-(a::T, b::AbstractAlgebra.ResElem{T}) where {T <: RingElem}

Return $a - b$.

-(a::AbstractAlgebra.ResFieldElem{T}, b::AbstractAlgebra.ResFieldElem{T}) where {T <: RingElement}

Return $a - b$.

-(a::AbstractAlgebra.ResFieldElem{T}, b::T) where {T <: RingElem}

Return $a - b$.

-(a::T, b::AbstractAlgebra.ResFieldElem{T}) where {T <: RingElem}

Return $a - b$.

-(a::AbstractAlgebra.PolyElem{T}, b::AbstractAlgebra.PolyElem{T}) where {T <: RingElement}

Return $a - b$.

-(a::AbstractAlgebra.NCPolyElem{T}, b::AbstractAlgebra.NCPolyElem{T}) where {T <: NCRingElem}

Return $a - b$.

-(a::AbstractAlgebra.RelSeriesElem{T}, b::AbstractAlgebra.RelSeriesElem{T}) where {T <: RingElement}

Return $a - b$.

-(a::AbstractAlgebra.AbsSeriesElem{T}, b::AbstractAlgebra.AbsSeriesElem{T}) where {T <: RingElement}

Return $a - b$.

-(a::Generic.LaurentSeriesElem{T}, b::Generic.LaurentSeriesElem{T}) where {T <: RingElement}

Return $a - b$.

-(x::Generic.MatrixElem{T}, y::Generic.MatrixElem{T}) where {T <: RingElement}

Return $a - b$.

-(x::T, y::Generic.MatrixElem{T}) where {T <: RingElem}

Return $S(x) - y$ where $S$ is the parent of $y$.

-(x::Generic.MatrixElem{T}, y::T) where {T <: RingElem}

Return $x - S(y)$, where $S$ is the parent of $a$.

-(a::AbstractAlgebra.FracElem{T}, b::AbstractAlgebra.FracElem{T}) where {T <: RingElem}

Return $a - b$.

-(a::AbstractAlgebra.FracElem{T}, b::T) where {T <: RingElem}

Return $a - b$.

-(a::T, b::AbstractAlgebra.FracElem{T}) where {T <: RingElem}

Return $a - b$.

+(x::NfAbsOrdElem, y::NfAbsOrdElem) -> NfAbsOrdElem

Returns $x + y$.

-(x::NfAbsOrdElem, y::NfAbsOrdElem) -> NfAbsOrdElem

Returns $x - y$.

*(x::NfAbsOrdElem, y::NfAbsOrdElem) -> NfAbsOrdElem

Returns $x \cdot y$.

^(x::NfAbsOrdElem, y::Union{fmpz, Int})

Returns $x^y$.

mod(a::NfAbsOrdElem, m::Union{fmpz, Int}) -> NfAbsOrdElem

Reduces the coefficient vector of $a$ modulo $m$ and returns the corresponding element. The coefficient vector of the result will have entries $x$ with $0 \leq x \leq m$.

powermod(a::NfAbsOrdElem, i::fmpz, m::Integer) -> NfAbsOrdElem

Returns the element $a^i$ modulo $m$.

representation_matrix(a::NfAbsOrdElem) -> fmpz_mat

Returns the representation matrix of the element $a$.

representation_matrix(a::NfAbsOrdElem, K::AnticNumberField) -> FakeFmpqMat

Returns the representation matrix of the element $a$ considered as an element of the ambient number field $K$. It is assumed that $K$ is the ambient number field of the order of $a$.

representation_matrix(a::NfAbsOrdElem, K::AnticNumberField) -> FakeFmpqMat

Returns the representation matrix of the element $a$ considered as an element of the ambient number field $K$. It is assumed that $K$ is the ambient number field of the order of $a$.

tr(a::NfAbsOrdElem) -> fmpz

Returns the trace of $a$.

norm(a::NfAbsOrdElem) -> fmpz

Returns the norm of $a$.

norm(A::NfAbsOrdIdl) -> fmpz

Returns the norm of $A$, that is, the cardinality of $\mathcal O/A$, where $\mathcal O$ is the order of $A$.

norm(a::NfRelOrdIdl) -> NfOrdIdl

Returns the norm of $a$.

norm(a::NfRelOrdFracIdl{T, S}) -> S

Returns the norm of $a$

rand(O::NfOrd, n::Union{Integer, fmpz}) -> NfAbsOrdElem

Computes a coefficient vector with entries uniformly distributed in $\{-n,\dotsc,-1,0,1,\dotsc,n\}$ and returns the corresponding element of the order $\mathcal O$.

minkowski_map(a::NfAbsOrdElem, abs_tol::Int) -> Array{arb, 1}

Returns the image of $a$ under the Minkowski embedding. Every entry of the array returned is of type arb with radius less then 2^-abs_tol.

conjugates_arb(x::NfAbsOrdElem, abs_tol::Int) -> Array{acb, 1}

Compute the the conjugates of x as elements of type acb. Recall that we order the complex conjugates $\sigma_{r+1}(x),...,\sigma_{r+2s}(x)$ such that $\sigma_{i}(x) = \overline{\sigma_{i + s}(x)}$ for $r + 2 \leq i \leq r + s$.

Every entry y of the array returned satisfies radius(real(y)) < 2^-abs_tol, radius(imag(y)) < 2^-abs_tol respectively.

conjugates_arb_log(x::NfAbsOrdElem, abs_tol::Int) -> Array{arb, 1}

Returns the elements $(\log(\lvert \sigma_1(x) \rvert),\dotsc,\log(\lvert\sigma_r(x) \rvert), \dotsc,2\log(\lvert \sigma_{r+1}(x) \rvert),\dotsc, 2\log(\lvert \sigma_{r+s}(x)\rvert))$ as elements of type arb radius less then 2^-abs_tol.

t2(x::NfAbsOrdElem, abs_tol::Int = 32) -> arb

Return the $T_2$-norm of $x$. The radius of the result will be less than 2^-abs_tol.

minpoly(a::NfAbsOrdElem) -> fmpz_poly

minpoly(a::NfAbsOrdElem, FlintZZ) -> fmpz_poly

The minimal polynomial of $a$.

minpoly(S::Ring, M::MatElem{T}, charpoly_only::Bool = false) where {T <: RingElement}

Returns the minimal polynomial $p$ of the matrix $M$. The polynomial ring $S$ of the resulting polynomial must be supplied and the matrix must be square.

minpoly(S::Ring, M::MatAlgElem{T}, charpoly_only::Bool = false) where {T <: RingElement}

Returns the minimal polynomial $p$ of the matrix $M$. The polynomial ring $S$ of the resulting polynomial must be supplied and the matrix must be square.

charpoly(a::NfAbsOrdElem) -> fmpz_poly

charpoly(a::NfAbsOrdElem, FlintZZ) -> fmpz_poly

The characteristic polynomial of $a$.

charpoly(V::Ring, Y::Generic.MatrixElem{T}) where {T <: RingElement}

Returns the characteristic polynomial $p$ of the matrix $M$. The polynomial ring $R$ of the resulting polynomial must be supplied and the matrix is assumed to be square.