SAGE Reference Manual

William Stein
wstein@gmail.com

Date: October 28, 2007

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 In the following session, we do all of the examples in the Sympy
- 4.5 Further Examples
- 4.6 Further examples from Wester's paper
5. Plotting and Visualization
- 5.1 2D Plotting
- 5.2 3D Plotting Using Tachyon
6. Games
- 6.1 Sudoku Solver
7. Graph Theory
- 7.1 Graph Theory
- 7.2 A collection of constructors of common graphs
- 7.3 N.I.C.E. - Nice (as in open source) Isomorphism Check Engine
- 7.4 Graph Database Module
- 7.5 A module for dealing with lists of graphs
8. Constants
- 8.1 Mathematical constants
9. Functions
- 9.1 SAGE Functions Class
- 9.2 Transcendental Functions
- 9.3 Orthogonal Polynomials
- 9.4 Special Functions
10. Basic Structure
- 10.1 Abstract base class for SAGE objects
- 10.2 Base class for parent objects with generators
- 10.3 Formal sums
- 10.4 Factorizations
- 10.5 Elements
  - 10.5.1 The Abstract Element Class Heierarchy
  - 10.5.2 How to Define a New Element Class
- 10.6 Mutability Pyrex Implementation
- 10.7 Sequences
- 10.8 Sets
- 10.9 The set of prime numbers
11. Miscellaneous
- 11.1 Miscellaneous functions
- 11.2 SAGE package management commands
- 11.3 Get resource usage of process
- 11.4 Multidimensional enumeration
- 11.5 Installing shortcut scripts
- 11.6 SAGE Interface to the HG/Mercurial Revision Control System
- 11.7 Functional notation
- 11.8 Latex printing support
- 11.9 Logging of SAGE sessions
- 11.10 Object persistence
- 11.11 Support for persistent functions in .sage files
- 11.12 Evaluating a String in SAGE
- 11.13 Miscellaneous arithmetic functions
12. Databases
- 12.1 Cremona's tables of elliptic curves
- 12.2 The Stein-Watkins table of elliptic curves
- 12.3 John Jones's tables of number fields
- 12.4 Linear codes
- 12.5 Interface to Sloane On-Line Encyclopedia of Integer Sequences
- 12.6 Frank Luebeck's tables of Conway polynomials over finite fields
- 12.7 Tables of zeros of the Riemann-Zeta function
13. Interpreter Interfaces
- 13.1 Common Interface Functionality
- 13.2 Interface to Axiom
- 13.3 Interface to GAP
- 13.4 Interface to GP/Pari
- 13.5 Interface to the Gnuplot interpreter
- 13.6 Interface to KASH
- 13.7 Interface to Magma
- 13.8 Interface to Maple
  - 13.8.1 Tutorial
- 13.9 Interface to MATLAB
  - 13.9.1 Tutorial
- 13.10 Interface to Maxima
- 13.11 Interface to Mathematica
- 13.12 Interface to mwrank
- 13.13 Interface to Octave
  - 13.13.1 Computation of Special Functions
  - 13.13.2 Tutorial
- 13.14 Interface to SAGE
- 13.15 Interface to Singular
- 13.16 The Tachyon Ray Tracer
14. C/C++ Library Interfaces
- 14.1 PARI C-library interface
- 14.2 Victor Shoup's NTL C++ Library
- 14.3 Cremona's mwrank C++ library
15. Networking and Grid Computing
- 15.1 Wiki Interactive Web Page
- 15.2 Distributed SAGE
16. Cryptography
- 16.1 Cryptosystems
- 16.2 Ciphers
- 16.3 Classical Cryptosystems
- 16.4 Classical Ciphers
- 16.5 Stream Cryptosystems
- 16.6
- 16.7 Linear feedback shift register (LFSR) sequence commands
17. Combinatorics
- 17.1 Combinatorial Functions
- 17.2 Functions that compute some of the sequences in Sloane's tables
- 17.3 Compute Bell and Uppuluri-Carpenter numbers
18. Probability
- 18.1 Random variables and probability spaces
19. Category Theory
- 19.1 Categories
- 19.2 Homsets
- 19.3 Morphisms
- 19.4 Functors
20. Monoids
- 20.1 Free Monoids
- 20.2 Monoid Elements
- 20.3 Free abelian monoids
- 20.4 Abelian monoid elements
21. Groups
- 21.1 Base class for groups
- 21.2 Multiplicative Abelian Groups
- 21.3 Abelian group elements
- 21.4 Homomorphisms of abelian groups
- 21.5 Basic functionality for dual groups of finite multiplicative Abelian groups
- 21.6 Permutation groups
- 21.7 Permutation group elements
- 21.8 Permutation group homomorphisms
- 21.9 Rubik's cube group functions
- 21.10 Matrix Groups
- 21.11 Matrix Group Elements
- 21.12 Homomorphisms Between Matrix Groups
- 21.13 Matrix Group Homsets
- 21.14 Linear Groups
- 21.15 General Linear Groups
- 21.16 Special Linear Groups
- 21.17 Orthogonal Linear Groups
- 21.18 Symplectic Linear Groups
- 21.19 Unitary Groups and
22. General Rings, Ideals, and Morphisms
- 22.1 Ideals
- 22.2 Monoid of Ring Ideals
- 22.3 Homomorphisms of rings
- 22.4 Space of homomorphisms between two rings
- 22.5 Infinity Rings
- 22.6 Fraction Field of Integral Domains
- 22.7 Fraction Field Elements
- 22.8 Quotient Rings
- 22.9 Quotient Ring Elements
23. Standard Commutative Rings
- 23.1 Ring $\mathbf{Z}$ of Integers
- 23.2 Elements of the ring $\mathbf{Z}$ of integers
- 23.3 Ring $\mathbf{Z}/n\mathbf{Z}$ of integers modulo
- 23.4 Elements of $\mathbf{Z}/n\mathbf{Z}$
- 23.5 Field $\mathbf{Q}$ of Rational Numbers
- 23.6 Rational Numbers
- 23.7 Finite Fields
- 23.8 Elements of Finite Fields
24. p-adic Rings
- 24.1 Introduction to the -adics
- 24.2 Terminology and types of -adics
25. Fixed and Arbitrary Precision Numerical Fields
- 25.1 Double Precision Real Numbers
- 25.2 Double Precision Complex Numbers
- 25.3 Field of Arbitrary Precision Real Numbers
- 25.4 Field of Arbitrary Precision Complex Numbers
- 25.5 Arbitrary Precision Complex Numbers
- 25.6 Field of Arbitrary Precision Real Intervals
26. Number Fields
- 26.1 Number Fields
- 26.2 Number Field Elements
27. Polynomial Rings
- 27.1 Univariate Polynomial Rings
- 27.2 Univariate Polynomial Base Class
- 27.3 Quotients of Univariate Polynomial Rings
- 27.4 Elements of Quotients of Univariate Polynomial Rings
- 27.5 Term Orderings
- 27.6 Multivariate Polynomial Rings
- 27.7 Multivariate Polynomials
- 27.8 Ideals in multivariate polynomial rings
- 27.9 Generic Convolution
28. Power Series Rings
- 28.1 Univariate Power Series Rings
- 28.2 Power Series
- 28.3 Laurent Series Rings
- 28.4 Laurent Series
29. Algebras
- 29.1 Free algebras
- 29.2 Free algebra elements
- 29.3 Free algebra quotients
- 29.4 Free algebra quotient elements
30. Quaternion Algebras
- 30.1 Quaternion algebras
- 30.2 Quaternion algebra elements
- 30.3 Quaternion orders
- 30.4 Quaternion order elements
- 30.5 Quaternion ideal
- 30.6 Quaternion ideal elements
31. Matrices and Spaces of Matrices
- Introduction
- 31.1 Matrix Spaces
- 31.2 Matrix Constructor
- 31.3 Matrices over an arbitrary ring
  - 31.3.1 Implementation and Design
- 31.4 Abstract base class for matrices
- 31.5 Base class for matrices, part 0
- 31.6 Base class for matrices, part 1
- 31.7 Base class for matrices, part 2
- 31.8 Generic Asymptotically Fast Strassen Algorithms
- 31.9 Minimal Polynomials of Linear Recurrence Sequences
- 31.10 Base class for dense matrices
- 31.11 Base class for sparse matrices
- 31.12 Dense Matrices over a general ring
- 31.13 Sparse Matrices over a general ring
- 31.14 Dense matrices over $\mathbf{Z}/n\mathbf{Z}$ for small
- 31.15 Sparse matrices over $\mathbf{Z}/n\mathbf{Z}$ for small
- 31.16 Dense matrices over the integer ring
- 31.17 Dense matrices over the rational field
- 31.18 Dense matrices over the real double field
- 31.19 Dense matrices over the Complex Double Field
32. Modules
- 32.1 Abstract base class for modules
- 32.2 Free modules
- 32.3 Elements of free modules
- 32.4 Complex double vectors
- 32.5 Real double vectors
- 32.6 Homspaces between free modules
- 32.7 Morphisms of free modules
- 32.8 Morphisms defined by a matrix
33. Combinatorial Geometry
- 33.1 Lattice and reflexive polytopes
- 33.2 Groebner Fans
- 33.3 Polytopes
34. L-functions
- 34.1 Rubinstein's -function Calculator
- 34.2 Watkins Symmetric Power -function Calculator
- 34.3 Dokchitser's L-functions Calculator
35. Schemes
- 35.1 Scheme implementation overview
  - 35.1.1 TODO List
- 35.2 Schemes
- 35.3 Spec of a ring
- 35.4 Scheme obtained by glueing two other schemes
- 35.5 Points on schemes
- 35.6 Ambient Spaces
- 35.7 Affine space over a ring
- 35.8 Projective space over a ring
- 35.9 Algebraic schemes
- 35.10 Set of homomorphisms between two schemes
- 35.11 Scheme morphism
- 35.12 Divisors on schemes
36. Elliptic and Plane Curves
- 36.1 Plane curve constructors
- 36.2 Affine plane curves over a general ring
- 36.3 Plane curves over a general ring
- 36.4 Elliptic curve constructor
- 36.5 Elliptic curves over a general ring
- 36.6 Elliptic curves over a general field
- 36.7 Elliptic curves over the rational numbers
- 36.8 Elliptic curves over finite fields
- 36.9 Formal groups of elliptic curves
- 36.10 Computation of Frobenius matrix on Monsky-Washnitzer cohomology
37. Hyperelliptic Curves
- 37.1 Hyperelliptic curve constructor
- 37.2 Hyperelliptic curves over a finite field
- 37.3 Hyperelliptic curves over a general ring
- 37.4 Constructor for Jacobian of a hyperelliptic curve
- 37.5 Jacobian of a Hyperelliptic curve of Genus 2
- 37.6 Jacobian of a General Hyperelliptic Curve
- 37.7 Rational point sets on a Jacobian
- 37.8 Jacobian ``morphism'' as a class in the Picard group
- 37.9 Conductor and Reduction Types for Genus 2 Curves
38. Coding Theory
- 38.1 Linear Codes
- 38.2 Bounds for Parameters of Codes
39. Modular Forms: General Hecke Algebras and Hecke Modules
- 39.1 Congruence subgroups of SL2(Z)
- 39.2 Dirichlet characters
- 39.3 The set $\mathbf{P}^1(\mathbf{Q})$ of cusps
- 39.4 Dimensions of spaces of modular forms
- 39.5 Conjectural Slopes of Hecke Polynomial
- 39.6 Hecke modules
- 39.7 Submodule of a Hecke module
- 39.8 Ambient Hecke modules
- 39.9 Elements of Hecke modules
- 39.10 Hom spaces between objects of the category of hecke modules over a given base ring
- 39.11 Morphism of Hecke modules
- 39.12 Hecke algebras and modules
- 39.13 Hecke operators
40. Modular Symbols
- 40.1 Creation of modular symbols spaces
- 40.2 Space of modular symbols (base class)
- 40.3 Ambient spaces of modular symbols
- 40.4 Subspace of ambient spaces of modular symbols
- 40.5 A single element of an ambient space of modular symbols
- 40.6 Manin symbols
- 40.7 Space of boundary modular symbols
- 40.8 Heilbronn matrix computation
- 40.9 List of Elements of $P^1(\mathbf{Z}/N\mathbf{Z})$
- 40.10 Relation matrices for ambient modular symbols spaces
41. Modular Forms
- 41.1 Creating Spaces of Modular Forms
- 41.2 Generic spaces of modular forms
- 41.3 Ambient Spaces of Modular Forms
- 41.4 Modular Forms with Character
- 41.5 Modular Forms for $\Gamma_0(N)$ over $\mathbf{Q}$
- 41.6 Modular Forms for $\Gamma_1(N)$ over $\mathbf{Q}$
- 41.7 Modular Forms over a Non-minimal Base Ring
- 41.8 Submodules of spaces of modular forms
- 41.9 The Cuspidal Subspace
- 41.10 The Eisenstein Subspace
- 41.11 Eisenstein Series
- 41.12 Elements of modular forms spaces
- 41.13 Hecke Operators on -expansions
- 41.14 Numerical computation of newforms
- 41.15 The Victor Miller Basis
- 41.16 Ambient Spaces of Modular Forms
- 41.17 Compute spaces of half-integral weight modular forms
42. Modular Abelian Varieties
- 42.1 Constructors for certain modular abelian varieties
- 42.2 Base class for modular abelian varieties
- 42.3 Ambient Jacobian Abelian Variety
- 42.4 Optimal abelian variety quotients of modular jacobians
- 42.5 Finite subgroups of modular abelian varieties
- 42.6 Torsion points on modular abelan varieties.
- 42.7 Torsion subgroups of modular abelian varieties
- 42.8 Cuspidal subgroups of modular abelian varieties
- 42.9 Hecke operators on modular abelian varieties
- 42.10 Homology of modular abelian varieties
- 42.11 Spaces of homomorphisms between modular abelian varieties
- 42.12 Morphisms between modular abelian varieties
A. History and License
- A.1 License
  - A.1.1 The GNU General Public License
Module Index
Index
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