Elements
Elements in orders have two representations: they can be viewed as elements in the giving the coefficients wrt to the order basis where they are elements in. On the other hand, as every order is in a field, they also have a representation as number field elements. Since, asymptotically, operations are more efficient in the field (due to fast polynomial arithmetic) than in the order, the primary representation is that as a field element.
Creation
Elements are constructed either as linear combinations of basis elements or via explicit coercion. Elements will be of type AbsNumFieldOrderElem
, the type if actually parametrized by the type of the surrounding field and the type of the field elements. E.g. the type of any element in any order of an absolute simple field will be AbsSimpleNumFieldOrderElem
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AbsNumFieldOrder
— Type.
(O::NumFieldOrder)(a::NumFieldElem, check::Bool = true) -> NumFieldOrderElem
Given an element of the ambient number field of , this function coerces the element into . It will be checked that is contained in if and only if check
is true
.
(O::NumFieldOrder)(a::NumFieldOrderElem, check::Bool = true) -> NumFieldOrderElem
Given an element of some order in the ambient number field of , this function coerces the element into . It will be checked that is contained in if and only if check
is true
.
(O::NumFieldOrder)(a::IntegerUnion) -> NumFieldOrderElem
Given an element of type ZZRingElem
or Integer
, this function coerces the element into .
(O::AbsNumFieldOrder)(arr::Vector{ZZRingElem})
Returns the element of with coefficient vector arr
.
(O::AbsNumFieldOrder)(arr::Vector{Integer})
Returns the element of with coefficient vector arr
.
Basic properties
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parent
— Method.
parent(a::NumFieldOrderElem) -> NumFieldOrder
Returns the order of which is an element.
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elem_in_nf
— Method.
elem_in_nf(a::NumFieldOrderElem) -> NumFieldElem
Returns the element considered as an element of the ambient number field.
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coordinates
— Method.
coordinates(a::AbsNumFieldOrderElem) -> Vector{ZZRingElem}
Returns the coefficient vector of with respect to the basis of the order.
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discriminant
— Method.
discriminant(B::Vector{NumFieldOrderElem})
Returns the discriminant of the family of algebraic numbers, i.e. .
discriminant(E::EllipticCurve) -> FieldElem
Return the discriminant of .
discriminant(C::HypellCrv{T}) -> T
Compute the discriminant of .
discriminant(O::AlgssRelOrd)
Returns the discriminant of .
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==
— Method.
==(x::NumFieldOrderElem, y::NumFieldOrderElem) -> Bool
Returns whether and are equal.
Arithmetic
All the usual arithmetic operatinos are defined:
-(::NUmFieldOrdElem)
+(::NumFieldOrderElem, ::NumFieldOrderElem)
-(::NumFieldOrderElem, ::NumFieldOrderElem)
*(::NumFieldOrderElem, ::NumFieldOrderElem)
^(::NumFieldOrderElem, ::Int)
mod(::AbsNumFieldOrderElem, ::Int)
mod_sym(::NumFieldOrderElem, ::ZZRingElem)
powermod(::AbsNumFieldOrderElem, ::ZZRingElem, ::Int)
Miscellaneous
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representation_matrix
— Method.
representation_matrix(a::AbsNumFieldOrderElem) -> ZZMatrix
Returns the representation matrix of the element .
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representation_matrix
— Method.
representation_matrix(a::AbsNumFieldOrderElem, K::AbsSimpleNumField) -> FakeFmpqMat
Returns the representation matrix of the element considered as an element of the ambient number field . It is assumed that is the ambient number field of the order of .
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tr
— Method.
tr(a::NumFieldOrderElem)
Returns the trace of as an element of the base ring.
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norm
— Method.
norm(a::NumFieldOrderElem)
Returns the norm of as an element in the base ring.
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absolute_norm
— Method.
absolute_norm(a::NumFieldOrderElem) -> ZZRingElem
Return the absolute norm as an integer.
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absolute_tr
— Method.
absolute_tr(a::NumFieldOrderElem) -> ZZRingElem
Return the absolute trace as an integer.
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rand
— Method.
rand(O::AbsSimpleNumFieldOrder, n::IntegerUnion) -> AbsNumFieldOrderElem
Computes a coefficient vector with entries uniformly distributed in and returns the corresponding element of the order .
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minkowski_map
— Method.
minkowski_map(a::NumFieldOrderElem, abs_tol::Int) -> Vector{ArbFieldElem}
Returns the image of under the Minkowski embedding. Every entry of the array returned is of type ArbFieldElem
with radius less then 2^-abs_tol
.
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conjugates_arb
— Method.
conjugates_arb(x::NumFieldOrderElem, abs_tol::Int) -> Vector{AcbFieldElem}
Compute the conjugates of as elements of type AcbFieldElem
. Recall that we order the complex conjugates such that for .
Every entry of the array returned satisfies radius(real(y)) < 2^-abs_tol
, radius(imag(y)) < 2^-abs_tol
respectively.
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conjugates_arb_log
— Method.
conjugates_arb_log(x::NumFieldOrderElem, abs_tol::Int) -> Vector{ArbFieldElem}
Returns the elements as elements of type ArbFieldElem
radius less then 2^-abs_tol
.
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t2
— Method.
t2(x::NumFieldOrderElem, abs_tol::Int = 32) -> ArbFieldElem
Return the -norm of . The radius of the result will be less than 2^-abs_tol
.
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minpoly
— Method.
minpoly(a::AbsNumFieldOrderElem) -> ZZPolyRingElem
The minimal polynomial of .
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charpoly
— Method.
charpoly(a::AbsNumFieldOrderElem) -> ZZPolyRingElem
charpoly(a::AbsNumFieldOrderElem, FlintZZ) -> ZZPolyRingElem
The characteristic polynomial of .
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factor
— Method.
factor(a::AbsSimpleNumFieldOrderElem) -> Fac{AbsSimpleNumFieldOrderElem}
Computes a factorization of 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 is non-zero and that all prime ideals containing are principal, which is for example satisfied if class group of the order of is trivial.
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denominator
— Method.
denominator(a::NumFieldElem, O::AbsSimpleNumFieldOrder) -> ZZRingElem
Returns the smallest positive integer such that is contained in .
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discriminant
— Method.
discriminant(B::Vector{NumFieldOrderElem})
Returns the discriminant of the family of algebraic numbers, i.e. .
discriminant(E::EllipticCurve) -> FieldElem
Return the discriminant of .
discriminant(C::HypellCrv{T}) -> T
Compute the discriminant of .
discriminant(O::AlgssRelOrd)
Returns the discriminant of .