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elements.md

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Elements

CurrentModule = Hecke

Elements in orders have two representations: they can be viewed as elements in the $\mathbf Z^n$ 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

AbsNumFieldOrder

Basic properties

parent(::AbsSimpleNumFieldOrderElem)
elem_in_nf(::AbsSimpleNumFieldOrderElem)
coordinates(::AbsSimpleNumFieldOrderElem)
discriminant(::Vector{AbsSimpleNumFieldOrderElem})
==(::AbsSimpleNumFieldOrderElem, ::AbsSimpleNumFieldOrderElem)

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

representation_matrix(::AbsNumFieldOrderElem)
representation_matrix(::AbsSimpleNumFieldOrderElem, ::AbsSimpleNumField)
tr(::NumFieldOrderElem)
norm(::NumFieldOrderElem)
absolute_norm(::AbsNumFieldOrderElem)
absolute_tr(::AbsNumFieldOrderElem)
rand(::AbsSimpleNumFieldOrder, ::Int)
minkowski_map(::AbsSimpleNumFieldOrderElem, ::Int)
conjugates_arb(::AbsSimpleNumFieldOrderElem, ::Int)
conjugates_arb_log(::AbsSimpleNumFieldOrderElem, ::Int)
t2(::AbsSimpleNumFieldOrderElem, ::Int)
minpoly(::AbsSimpleNumFieldOrderElem)
charpoly(::AbsSimpleNumFieldOrderElem)
factor(::AbsSimpleNumFieldOrderElem)
denominator(a::NumFieldElem, O::RelNumFieldOrder)
discriminant(::Vector{AbsNumFieldOrderElem})