SAGE
Reference Manual
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Front Matter
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
7.1.1 Graph Format
7.1.2 Generators
7.1.3 Labels
7.1.4 Database
7.1.5 Visualization
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
13.3.1 First Examples
13.3.2 GAP and Singular
13.3.3 Saving and loading objects
13.3.4 Long Input
13.3.5 Changing which GAP is used
13.4 Interface to GP/Pari
13.5 Interface to the Gnuplot interpreter
13.6 Interface to KASH
13.6.1 Issues
13.6.2 Tutorial
13.6.3 Long Input
13.7 Interface to Magma
13.7.1 Parameters
13.7.2 Multiple Return Values
13.7.3 Long Input
13.7.4 Other Examples
13.8 Interface to Maple
13.8.1 Tutorial
13.9 Interface to MATLAB
13.9.1 Tutorial
13.10 Interface to Maxima
13.10.1 Tutorial
13.10.2 Examples involving matrices
13.10.3 Laplace Transforms
13.10.4 Continued Fractions
13.10.5 Special examples
13.10.6 Miscellaneous
13.10.7 Interactivity
13.10.8 Latex Output
13.10.9 Long Input
13.11 Interface to Mathematica
13.11.1 Tutorial
13.11.2 Long Input
13.11.3 Loading and saving
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
13.15.1 Introduction
13.15.2 Tutorial
13.15.3 Computing the Genus
13.15.4 An Important Concept
13.15.5 Long Input
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
of Integers
23.2 Elements of the ring
of integers
23.3 Ring
of integers modulo
23.4 Elements of
23.5 Field
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
24.2.1 Fixed Modulus Rings
24.2.2 Capped Absolute Rings
24.2.3 Capped Relative Rings and Fields
24.2.4 Lazy Rings and Fields
24.2.5 Unramified Extensions
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
for
small
31.15 Sparse matrices over
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
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
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
over
41.6 Modular Forms for
over
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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SAGE
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Release 2007.10.28, documentation updated on October 28, 2007.
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