Elements

  (O::NumFieldOrder)(a::NumFieldElem, check::Bool = true) -> NumFieldOrderElem

Given an element $a$ of the ambient number field of $\mathcal O$, 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::NumFieldOrder)(a::NumFieldOrderElem, check::Bool = true) -> NumFieldOrderElem

Given an element $a$ of some order in the ambient number field of $\mathcal O$, 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::NumFieldOrder)(a::IntegerUnion) -> NumFieldOrderElem

Given an element $a$ of type ZZRingElem or Integer, this function coerces the element into $\mathcal O$.

  (O::AbsNumFieldOrder)(arr::Vector{ZZRingElem})

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

  (O::AbsNumFieldOrder)(arr::Vector{Integer})

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

parent(a::NumFieldOrderElem) -> NumFieldOrder

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

parent(a::MatRingElem{T}) where T <: NCRingElement

Return the matrix algebra containing a.

This is the matrix algebra over the base ring of a whose degree is the number of rows (equivalently columns) of a.

Examples

julia> S = matrix_ring(ZZ, 2)
Matrix ring of degree 2
  over integers

julia> A = S([1 2; 3 4])
[1   2]
[3   4]

julia> parent(A) == S
true

elem_in_nf(a::NumFieldOrderElem) -> NumFieldElem

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

coordinates(v::FPModuleElem{T}, i::Int) where T <: RingElement

Return the coordinates of the module element $v$ as a Vector{T}.

coordinates(a::AbsNumFieldOrderElem) -> Vector{ZZRingElem}

Returns the coefficient vector of $a$ with respect to the basis of the order.

discriminant(B::Vector{NumFieldOrderElem})

Returns the discriminant of the family $B$ of algebraic numbers, i.e. $det((tr(B[i]*B[j]))_{i, j})^2$.

discriminant(C::ConCrv{T}) -> T

Compute the discriminant of $C$.

discriminant(C::HypellCrv{T}) -> T

Compute the discriminant of $C$.

discriminant(E::EllipticCurve) -> FieldElem

Return the discriminant of $E$.

discriminant(O::AlgssRelOrd)

Returns the discriminant of $O$.

discriminant(g::Vector)

Compute the product of all differences of distinct elements in the array.

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

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

representation_matrix(a::AbsNumFieldOrderElem) -> ZZMatrix

Returns the representation matrix of the element $a$.

representation_matrix(a::AbsNumFieldOrderElem, K::AbsSimpleNumField) -> 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::NumFieldOrderElem)

Returns the trace of $a$ as an element of the base ring.

norm(a::NumFieldOrderElem)

Returns the norm of $a$ as an element in the base ring.

absolute_norm(a::NumFieldOrderElem) -> ZZRingElem

Return the absolute norm as an integer.

absolute_tr(a::NumFieldOrderElem) -> ZZRingElem

Return the absolute trace as an integer.

rand(O::AbsSimpleNumFieldOrder, n::IntegerUnion) -> AbsNumFieldOrderElem

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::NumFieldOrderElem, abs_tol::Int) -> Vector{ArbFieldElem}

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

conjugates_arb(x::NumFieldOrderElem, abs_tol::Int) -> Vector{AcbFieldElem}

Compute the conjugates of $x$ as elements of type AcbFieldElem. 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::NumFieldOrderElem, abs_tol::Int) -> Vector{ArbFieldElem}

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 ArbFieldElem radius less then 2^-abs_tol.

t2(x::NumFieldOrderElem, abs_tol::Int = 32) -> ArbFieldElem

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

minpoly([ZZPolyRing], a::AbsNumFieldOrderElem) -> ZZPolyRingElem

The minimal polynomial of $a$. The parent of the polynomial can be supplied as a first argument.

charpoly([ZZPolyRing], a::AbsNumFieldOrderElem) -> ZZPolyRingElem

The characteristic polynomial of $a$. The parent of the polynomial can be supplied as a first argument.

factor(a::AbsSimpleNumFieldOrderElem) -> Fac{AbsSimpleNumFieldOrderElem}

Computes a factorization of $a$ into irreducible elements. The return value is a factorization fac, which satisfies a = unit(fac) * prod(p^e for (p, e) in fac).

The function requires that $a$ is non-zero and that all prime ideals containing $a$ are principal, which is for example satisfied if class group of the order of $a$ is trivial.

denominator(a::NumFieldElem, O::AbsSimpleNumFieldOrder) -> ZZRingElem

Returns the smallest positive integer $k$ such that $k \cdot a$ is contained in $\mathcal O$.

discriminant(B::Vector{NumFieldOrderElem})

Returns the discriminant of the family $B$ of algebraic numbers, i.e. $det((tr(B[i]*B[j]))_{i, j})^2$.

discriminant(C::ConCrv{T}) -> T

Compute the discriminant of $C$.

discriminant(C::HypellCrv{T}) -> T

Compute the discriminant of $C$.

discriminant(E::EllipticCurve) -> FieldElem

Return the discriminant of $E$.

discriminant(O::AlgssRelOrd)

Returns the discriminant of $O$.

discriminant(g::Vector)

Compute the product of all differences of distinct elements in the array.