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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 packages Nemo and AbstractAlgebra. Hecke is part of the OSCAR project and the development is supported by the Deutsche Forschungsgemeinschaft DFG within the Collaborative Research Center TRR 195.

• https://github.com/thofma/Hecke.jl (Source code)
• https://thofma.github.io/Hecke.jl/dev/ (Online documentation)

So far, Hecke provides the following features:

• Number fields (absolute, relative, simple and non-simple)
• Orders and ideals in number fields
• Class and unit group computations of orders
• Lattice enumeration
• Sparse linear algebra
• Class field theory
• Abelian groups
• Associative algebras
• Ideals and orders in (semisimple) associative algebras
• Locally free class groups of orders in semisimple algebras
• Quadratic and Hermitian forms and lattices

To use Hecke, a julia version of 1.0 is necessary (the latest stable julia version will do). Please see https://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")

If your research depends on computations done with Hecke, please consider giving us a formal citation:

• Claus Fieker, William Hart, Tommy Hofmann and Fredrik Johansson, Nemo/Hecke: Computer Algebra and Number Theory Packages for the Julia Programming Language. In: Proceedings of ISSAC '17, pages 157–164, New York, NY, USA, 2017. ACM.

@inproceedings{nemo,
    author = {Fieker, Claus and Hart, William and Hofmann, Tommy and Johansson, Fredrik},
     title = {Nemo/Hecke: Computer Algebra and Number Theory Packages for the Julia Programming Language},
 booktitle = {Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation},
    series = {ISSAC '17},
      year = {2017},
     pages = {157--164},
  numpages = {8},
       url = {https://doi.acm.org/10.1145/3087604.3087611},
       doi = {10.1145/3087604.3087611},
 publisher = {ACM},
   address = {New York, NY, USA},
}

Here is a quick example of using Hecke:

julia> using Hecke

Welcome to

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

Version 0.22.8...
 ... which comes with absolutely no warranty whatsoever
(c) 2015-2024 by Claus Fieker, Tommy Hofmann and Carlo Sircana


julia> Qx, x = polynomial_ring(FlintQQ, "x");

julia> f = x^3 + 2;

julia> K, a = number_field(f, "a");

julia> O = maximal_order(K);

julia> O
Maximal order of Number field of degree 3 over QQ
with basis AbsSimpleNumFieldElem[1, a, a^2]

The online documentation can be found here:

• stable
• dev