Computational algebraic number theory




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Hecke is a software package for algebraic number theory maintained by Claus Fieker, Tommy Hofmann and Carlo Sircana. It is written in julia and is based on the computer algebra packages Nemo and AbstractAlgebra.

So far, Hecke provides the following features:

  • Orders (including element and ideal arithmetic) in relative and absolute number fields
  • Class and Unit group computation
  • Computation of maximal orders
  • Extensions of Number Fields (relative fields)
  • non-simple fields
  • Lattice enumeration
  • Sparse linear algebra
  • Class field theory
  • Abelian Groups


To use Hecke, a julia version of 1.0 is necessary (the latest stable julia version will do). Please see http://julialang.org/downloads for instructions on how to obtain julia for your system. Once a suitable julia version is installed, use the following steps at the julia prompt to install Hecke:

julia> using Pkg
julia> Pkg.add("Hecke")

Quick start

Here is a quick example of using Hecke:

julia> using Hecke

Welcome to 

  _    _           _        
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 | |__| | ___  ___| | _____ 
 |  __  |/ _ \/ __| |/ / _ \
 | |  | |  __/ (__|   <  __/
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Version 0.5 ... 
 ... which comes with absolutely no warrant whatsoever
(c) 2015-2018 by Claus Fieker, Tommy Hofmann and Carlo Sircana

julia> Qx, x = PolynomialRing(FlintQQ, "x");
julia> f = x^3 + 2;
julia> K, a = NumberField(f, "a");
julia> O = maximal_order(K);
julia> O
Maximal order of Number field over Rational Field with defining polynomial x^3 + 2 
with basis [1,a,a^2]


The online documentation can be found here: http://thofma.github.io/Hecke.jl/latest/

The documentation of the single functions can also be accessed at the julia prompt. Here is an example:

help?> signature
search: signature


  signature(O::NfMaximalOrder) -> Tuple{Int, Int}

  |  Returns the signature of the ambient number field of \mathcal O.

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