Sage Reference Manual

William Stein
wstein@gmail.com

Date: May 4, 2008

Front Matter
Contents
1. Introduction
2. The Sage Command Line
- 2.1 Attach a file to a running instance of Sage
- 2.2 Interactively tracing execution of a command
- 2.3 Sage: Command Line Arguments
3. The Sage Notebook
- 3.1 The Sage Notebook object
- 3.2 A Cell
- 3.3 A Worksheet
- 3.4 Sage Notebook (Twisted Version)
- 3.5 Javascript (AJAX) Component of Sage Notebook
- 3.6 Customization of the Notebook
- 3.7 Sage Notebook CSS
- 3.8 Support for the Notebook (introspection and setup)
- 3.9 Sage Notebook: Introspection
- 3.10 Wiki Interactive Web Page
4. Symbolic Calculus
- 4.1 Symbolic Computation
- 4.2 Symbolic Equations and Inequalities
- 4.3 Functional notation support for common calculus methods
- 4.4 A Sample Session using Sympy
- 4.5 Calculus Tests and Examples
- 4.6 Further examples from Wester's paper
5. 2D Graphics
- 5.1 2D Plotting
- 5.2 Animated plots
6. 3D Graphics
- 6.1 Introduction
- 6.2 Parametric Plots
- 6.3 List Plots
- 6.4 Plotting Functions
- 6.5 Platonic Solids.
- 6.6 Lines, Frames, Spheres, Points, Dots, and Text
- 6.7 Base classes for 3D Graphics objects and plotting
- 6.8 The Tachyon 3D Ray Tracer
7. Games
- 7.1 Sudoku Solver
8. Graph Theory
- 8.1 Graph Theory
- 8.2 A collection of constructors of common graphs
- 8.3 N.I.C.E. - Nice (as in open source) Isomorphism Check Engine
- 8.4 Graph Database Module
- 8.5 A module for dealing with lists of graphs
9. Constants
- 9.1 Mathematical constants
10. Functions
- 10.1 SAGE Functions Class
- 10.2 Transcendental Functions
- 10.3 Piecewise-defined Functions
- 10.4 Orthogonal Polynomials
- 10.5 Special Functions
11. Basic Structure
- 11.1 Abstract base class for SAGE objects
- 11.2 Base class for parent objects with generators
- 11.3 Formal sums
- 11.4 Factorizations
- 11.5 Elements
  - 11.5.1 The Abstract Element Class Hierarchy
  - 11.5.2 How to Define a New Element Class
- 11.6 Mutability Pyrex Implementation
- 11.7 Sequences
- 11.8 Sets
- 11.9 The set of prime numbers
12. Miscellaneous
- 12.1 Miscellaneous functions
- 12.2 SAGE package management commands
- 12.3 Get resource usage of process
- 12.4 Multidimensional enumeration
- 12.5 Installing shortcut scripts
- 12.6 SAGE Interface to the HG/Mercurial Revision Control System
- 12.7 Functional notation
- 12.8 Latex printing support
- 12.9 Logging of SAGE sessions
- 12.10 Object persistence
- 12.11 Support for persistent functions in .sage files
- 12.12 Evaluating a String in SAGE
- 12.13 Miscellaneous arithmetic functions
13. Databases
- 13.1 Cremona's tables of elliptic curves
- 13.2 The Stein-Watkins table of elliptic curves
- 13.3 John Jones's tables of number fields
- 13.4 Linear codes
- 13.5 Interface to Sloane On-Line Encyclopedia of Integer Sequences
- 13.6 Frank Luebeck's tables of Conway polynomials over finite fields
- 13.7 Tables of zeros of the Riemann-Zeta function
14. Interpreter Interfaces
- 14.1 Common Interface Functionality
- 14.2 Interface to Axiom
- 14.3 Interface to GAP
- 14.4 Interface to GP/Pari
- 14.5 Interface to the Gnuplot interpreter
- 14.6 Interface to KASH
- 14.7 Interface to Magma
- 14.8 Interface to Maple
  - 14.8.1 Tutorial
- 14.9 Interface to MATLAB
  - 14.9.1 Tutorial
- 14.10 Interface to Maxima
- 14.11 Interface to Mathematica
- 14.12 Interface to mwrank
- 14.13 Interface to Octave
  - 14.13.1 Computation of Special Functions
  - 14.13.2 Tutorial
- 14.14 Interface to SAGE
- 14.15 Interface to Singular
- 14.16 The Tachyon Ray Tracer
15. C/C++ Library Interfaces
- 15.1 PARI C-library interface
- 15.2 Victor Shoup's NTL C++ Library
- 15.3 Cremona's mwrank C++ library
16. Networking and Grid Computing
- 16.1 Wiki Interactive Web Page
- 16.2 Distributed Sage
17. Cryptography
- 17.1 Cryptosystems
- 17.2 Ciphers
- 17.3 Classical Cryptosystems
- 17.4 Classical Ciphers
- 17.5 Stream Cryptosystems
- 17.6 Stream Ciphers
- 17.7 Linear feedback shift register (LFSR) sequence commands
- 17.8 Small Scale Variants of the AES (SR) Polynomial System Generator
- 17.9 Multivariate Polynomial Systems
18. Combinatorics
- 18.1 Combinatorial Functions
- 18.2 Functions that compute some of the sequences in Sloane's tables
- 18.3 Compute Bell and Uppuluri-Carpenter numbers
- 18.4 Alternating Sign Matrices
- 18.5 Cartesian Products
- 18.6 Combinations
- 18.7 Combinatorial Algebras
- 18.8 Signed Compositions
- 18.9 Compositions
- 18.10 Exact Cover Problem via Dancing Links
- 18.11 Dyck Words
- 18.12 Finite combinatorial classes
- 18.13 Paths in Directed Acyclic Graphs
- 18.14 Kostka-Foulkes Polynomials
- 18.15 Lyndon words
- 18.16 Miscellaneous
- 18.17 Necklaces
- 18.18 Partition/Diagram Algebras
- 18.19 Partitions
- 18.20 Permutations
- 18.21 q-Analogues
- 18.22 Ribbons
- 18.23 Schubert Polynomials
- 18.24 Ordered Set Partitions
- 18.25 Set Partitions
- 18.26 Skew Partitions
- 18.27 Skew Tableaux
- 18.28 Subsets
- 18.29 Subwords
- 18.30 Symmetric Functions
- 18.31 Hall-Littlewood Polynomials
- 18.32 Jack Polynomials
- 18.33 Macdonald Polynomials - under development
- 18.34 Symmetric Group Algebra
- 18.35 Tableaux
- 18.36 Tools
- 18.37 Tuples
- 18.38 Words
19. Probability
- 19.1 Random variables and probability spaces
20. Category Theory
- 20.1 Categories
- 20.2 Homsets
- 20.3 Morphisms
- 20.4 Functors
21. Monoids
- 21.1 Free Monoids
- 21.2 Monoid Elements
- 21.3 Free abelian monoids
- 21.4 Abelian monoid elements
22. Groups
- 22.1 Base class for groups
- 22.2 Multiplicative Abelian Groups
- 22.3 Abelian group elements
- 22.4 Homomorphisms of abelian groups
- 22.5 Basic functionality for dual groups of finite multiplicative Abelian groups
- 22.6 Permutation groups
- 22.7 Permutation group elements
- 22.8 Permutation group homomorphisms
- 22.9 Rubik's cube group functions
- 22.10 Matrix Groups
- 22.11 Matrix Group Elements
- 22.12 Homomorphisms Between Matrix Groups
- 22.13 Matrix Group Homsets
- 22.14 Linear Groups
- 22.15 General Linear Groups
- 22.16 Special Linear Groups
- 22.17 Orthogonal Linear Groups
- 22.18 Symplectic Linear Groups
- 22.19 Unitary Groups and
23. General Rings, Ideals, and Morphisms
- 23.1 Ideals
- 23.2 Monoid of Ring Ideals
- 23.3 Homomorphisms of rings
- 23.4 Space of homomorphisms between two rings
- 23.5 Infinity Rings
- 23.6 Fraction Field of Integral Domains
- 23.7 Fraction Field Elements
- 23.8 Quotient Rings
- 23.9 Quotient Ring Elements
24. Standard Commutative Rings
- 24.1 Ring $\mathbf{Z}$ of Integers
- 24.2 Elements of the ring $\mathbf{Z}$ of integers
- 24.3 Ring $\mathbf{Z}/n\mathbf{Z}$ of integers modulo
- 24.4 Elements of $\mathbf{Z}/n\mathbf{Z}$
- 24.5 Field $\mathbf{Q}$ of Rational Numbers
- 24.6 Rational Numbers
- 24.7 Finite Fields
- 24.8 Elements of Finite Fields
25. p-adic Rings
- 25.1 Introduction to the -adics
- 25.2 Terminology and types of -adics
26. Fixed and Arbitrary Precision Numerical Fields
- 26.1 Double Precision Real Numbers
- 26.2 Double Precision Complex Numbers
- 26.3 Field of Arbitrary Precision Real Numbers
- 26.4 Field of Arbitrary Precision Complex Numbers
- 26.5 Arbitrary Precision Complex Numbers
- 26.6 Field of Arbitrary Precision Real Intervals
27. Number Fields
- 27.1 Number Fields
- 27.2 Number Field Elements
28. Polynomial Rings
- 28.1 Univariate Polynomial Rings
- 28.2 Univariate Polynomial Base Class
- 28.3 Quotients of Univariate Polynomial Rings
- 28.4 Elements of Quotients of Univariate Polynomial Rings
- 28.5 Term Orderings
- 28.6 Multivariate Polynomial Rings
- 28.7 Multivariate Polynomials
- 28.8 Ideals in multivariate polynomial rings
- 28.9 Boolean Polynomials
  - 28.9.1 Implementation specific notes
  - 28.9.2 Access to the original POLYBORI interface
- 28.10 Generic Convolution
29. Power Series Rings
- 29.1 Univariate Power Series Rings
- 29.2 Power Series
- 29.3 Laurent Series Rings
- 29.4 Laurent Series
30. Algebras
- 30.1 Free algebras
- 30.2 Free algebra elements
- 30.3 Free algebra quotients
- 30.4 Free algebra quotient elements
31. Quaternion Algebras
- 31.1 Quaternion algebras
- 31.2 Quaternion algebra elements
- 31.3 Quaternion orders
- 31.4 Quaternion order elements
- 31.5 Quaternion ideal
- 31.6 Quaternion ideal elements
32. Matrices and Spaces of Matrices
- Introduction
- 32.1 Matrix Spaces
- 32.2 Matrix Constructor
- 32.3 Matrices over an arbitrary ring
  - 32.3.1 Implementation and Design
- 32.4 Abstract base class for matrices
- 32.5 Base class for matrices, part 0
- 32.6 Base class for matrices, part 1
- 32.7 Base class for matrices, part 2
- 32.8 Generic Asymptotically Fast Strassen Algorithms
- 32.9 Minimal Polynomials of Linear Recurrence Sequences
- 32.10 Base class for dense matrices
- 32.11 Base class for sparse matrices
- 32.12 Dense Matrices over a general ring
- 32.13 Sparse Matrices over a general ring
- 32.14 Dense matrices over $\mathbf{Z}/n\mathbf{Z}$ for small
- 32.15 Sparse matrices over $\mathbf{Z}/n\mathbf{Z}$ for small
- 32.16 Dense matrices over the integer ring
- 32.17 Dense matrices over the rational field
- 32.18 Dense matrices over the real double field
- 32.19 Dense matrices over the Complex Double Field
33. Modules
- 33.1 Abstract base class for modules
- 33.2 Free modules
- 33.3 Elements of free modules
- 33.4 Complex double vectors
- 33.5 Real double vectors
- 33.6 Homspaces between free modules
- 33.7 Morphisms of free modules
- 33.8 Morphisms defined by a matrix
34. Combinatorial Geometry
- 34.1 Lattice and reflexive polytopes
- 34.2 Groebner Fans
- 34.3 Polytopes
35. L-functions
- 35.1 Rubinstein's -function Calculator
- 35.2 Watkins Symmetric Power -function Calculator
- 35.3 Dokchitser's L-functions Calculator
36. Schemes
- 36.1 Scheme implementation overview
  - 36.1.1 TODO List
- 36.2 Schemes
- 36.3 Spec of a ring
- 36.4 Scheme obtained by glueing two other schemes
- 36.5 Points on schemes
- 36.6 Ambient Spaces
- 36.7 Affine space over a ring
- 36.8 Projective space over a ring
- 36.9 Algebraic schemes
- 36.10 Set of homomorphisms between two schemes
- 36.11 Scheme morphism
- 36.12 Divisors on schemes
37. Elliptic and Plane Curves
- 37.1 Plane curve constructors
- 37.2 Affine plane curves over a general ring
- 37.3 Plane curves over a general ring
- 37.4 Elliptic curve constructor
- 37.5 Elliptic curves over a general ring
- 37.6 Elliptic curves over a general field
- 37.7 Elliptic curves over the rational numbers
- 37.8 Elliptic curves over finite fields
- 37.9 Formal groups of elliptic curves
- 37.10 Computation of Frobenius matrix on Monsky-Washnitzer cohomology
38. Hyperelliptic Curves
- 38.1 Hyperelliptic curve constructor
- 38.2 Hyperelliptic curves over a finite field
- 38.3 Hyperelliptic curves over a general ring
- 38.4 Constructor for Jacobian of a hyperelliptic curve
- 38.5 Jacobian of a Hyperelliptic curve of Genus 2
- 38.6 Jacobian of a General Hyperelliptic Curve
- 38.7 Rational point sets on a Jacobian
- 38.8 Jacobian `morphism' as a class in the Picard group
- 38.9 Conductor and Reduction Types for Genus 2 Curves
39. Coding Theory
- 39.1 Linear Codes
- 39.2 AUTHOR:
- 39.3 This module implements functions useful for studying binary self-dual codes
- 39.4 Bounds for Parameters of Codes
40. Modular Forms: General Hecke Algebras and Hecke Modules
- 40.1 Congruence subgroups of SL2(Z)
- 40.2 Dirichlet characters
- 40.3 The set $\mathbf{P}^1(\mathbf{Q})$ of cusps
- 40.4 Dimensions of spaces of modular forms
- 40.5 Conjectural Slopes of Hecke Polynomial
- 40.6 Hecke modules
- 40.7 Submodule of a Hecke module
- 40.8 Ambient Hecke modules
- 40.9 Elements of Hecke modules
- 40.10 Hom spaces between objects of the category of hecke modules over a given base ring
- 40.11 Morphism of Hecke modules
- 40.12 Hecke algebras and modules
- 40.13 Hecke operators
41. Modular Symbols
- 41.1 Creation of modular symbols spaces
- 41.2 Space of modular symbols (base class)
- 41.3 Ambient spaces of modular symbols
- 41.4 Subspace of ambient spaces of modular symbols
- 41.5 A single element of an ambient space of modular symbols
- 41.6 Manin symbols
- 41.7 Space of boundary modular symbols
- 41.8 Heilbronn matrix computation
- 41.9 List of Elements of $P^1(\mathbf{Z}/N\mathbf{Z})$
- 41.10 Relation matrices for ambient modular symbols spaces
42. Modular Forms
- 42.1 Creating Spaces of Modular Forms
- 42.2 Generic spaces of modular forms
- 42.3 Ambient Spaces of Modular Forms
- 42.4 Modular Forms with Character
- 42.5 Modular Forms for $\Gamma_0(N)$ over $\mathbf{Q}$
- 42.6 Modular Forms for $\Gamma_1(N)$ over $\mathbf{Q}$
- 42.7 Modular Forms over a Non-minimal Base Ring
- 42.8 Submodules of spaces of modular forms
- 42.9 The Cuspidal Subspace
- 42.10 The Eisenstein Subspace
- 42.11 Eisenstein Series
- 42.12 Elements of modular forms spaces
- 42.13 Hecke Operators on -expansions
- 42.14 Numerical computation of newforms
- 42.15 The Victor Miller Basis
- 42.16 Ambient Spaces of Modular Forms
- 42.17 Compute spaces of half-integral weight modular forms
43. Modular Abelian Varieties
- 43.1 Constructors for certain modular abelian varieties
- 43.2 Base class for modular abelian varieties
- 43.3 Ambient Jacobian Abelian Variety
- 43.4 Finite subgroups of modular abelian varieties
- 43.5 Torsion points on modular abelan varieties.
- 43.6 Torsion subgroups of modular abelian varieties
- 43.7 Cuspidal subgroups of modular abelian varieties
- 43.8 Hecke operators on modular abelian varieties
- 43.9 Homology of modular abelian varieties
- 43.10 Spaces of homomorphisms between modular abelian varieties
- 43.11 Morphisms between modular abelian varieties, including Hecke operators
A. History and License
- A.1 License
  - A.1.1 The GNU General Public License
Module Index
Index
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