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

So far, Hecke provides the following features:

  • Orders (including element and ideal arithmetic) in number fields
  • Computation of maximal orders
  • Verified residue computations of Dedekind zeta functions
  • Factor base creation and relations search in number fields
  • Lattice enumeration
  • Sparse linear algebra


To use Hecke, a julia version of 0.6 or higher 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> Pkg.add("Hecke")

Quick start

Here is a quick example of using Hecke:

julia> using Hecke

Welcome to 

  _    _           _        
 | |  | |         | |       
 | |__| | ___  ___| | _____ 
 |  __  |/ _ \/ __| |/ / _ \
 | |  | |  __/ (__|   <  __/
 |_|  |_|\___|\___|_|\_\___|

Version 0.3.0 ... 
 ... which comes with absolutely no warrant whatsoever
(c) 2015 by Claus Fieker and Tommy Hofmann

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.