Hecke

About

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

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

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
• Class and Unit group computation, S-units, PID testing
• Lattice enumeration
• Sparse linear algebra
• Normal forms for modules over maximal orders
• Extensions of number fields, non-simple extensions of number fields
• Orders and ideals in extensions of fields
• Abelian groups
• Ray class groups, quotients of ray class groups
• Invariant subgroups
• Class Field Theory
• Associative Algebras

Installation

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 

  _    _           _        
 | |  | |         | |       
 | |__| | ___  ___| | _____ 
 |  __  |/ _ \/ __| |/ / _ \
 | |  | |  __/ (__|   <  __/
 |_|  |_|\___|\___|_|\_\___|
  
Version 0.5.0 ... 
 ... which comes with absolutely no warranty 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 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.