SAGE Reference Manual

William Stein
wstein@gmail.com

Date: October 18, 2006

Front Matter
Contents
1. Introduction
2. The SAGE Command Line
- 2.1 sage optional arguments
3. The SAGE Notebook
- 3.1 SAGE Notebook Interface
  - 3.1.1 Supported Browsers
  - 3.1.2 Tutorial
- 3.2 Web Server Component of SAGE Notebook
- 3.3 A Cell
- 3.4 A Worksheet
- 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
4. Miscellaneous
- 4.1 Miscellaneous functions
- 4.2 Get resource usage of process
- 4.3 Multidimensional enumeration
- 4.4 Installing shortcut scripts
- 4.5 HG from SAGE
- 4.6 Bug reporting1
- 4.7 Functional notation
- 4.8 Latex printing support
- 4.9 Logging of SAGE sessions
- 4.10 Object persistence
- 4.11 Support for persistent functions in .sage files
- 4.12 Miscellaneous arithmetic functions
- 4.13 Evaluating a string in SAGE
5. Basic Structure
- 5.1 Abstract base class for SAGE objects
- 5.2 Base class for objects with generators
- 5.3 Formal sums
- 5.4 Factorizations
- 5.5 Elements
- 5.6 Mutability
- 5.7 Sequences
- 5.8 Sets
- 5.9 The set of prime numbers
6. Interpreter Interfaces
- 6.1 Common Interface Functionality
- 6.2 Interface to Axiom
- 6.3 Interface to GAP
- 6.4 Interface to GP/Pari
- 6.5 Interface to the Gnuplot interpreter
- 6.6 Interface to KASH
- 6.7 Interface to Magma
- 6.8 Interface to Maple
  - 6.8.1 Tutorial
- 6.9 Interface to MATLAB
  - 6.9.1 Tutorial
- 6.10 Interface to Maxima
- 6.11 Interface to Mathematica
- 6.12 Interface to mwrank
- 6.13 Interface to Octave
  - 6.13.1 Computation of Special Functions
- 6.14 Interface to SAGE
- 6.15 Interface to Singular
- 6.16 The Tachyon Ray Tracer
7. C/C++ Library Interfaces
- 7.1 Cremona's mwrank C++ library
- 7.2 Victor Shoup's NTL C++ Library
- 7.3 PARI C-library interface
8. Plotting and Visualization
- 8.1 2D Plotting
- 8.2 Interface to the Tachyon Ray Tracer
9. Constants
- 9.1 Mathematical constants
10. Functions
- 10.1 SAGE Functions Class
- 10.2 Elementary Functions
- 10.3 Transcendental Functions
- 10.4 Piecewise-defined Functions
- 10.5 Orthogonal Polynomials
- 10.6 Special Functions
11. Category Theory
- 11.1 Categories
- 11.2 Homsets
- 11.3 Morphisms
- 11.4 Functors
12. Numerical Computation
- 12.1 Overview
- 12.2 Field of Double-Precision Real Numbers
- 12.3 Double Precision Complex Numbers
13. Monoids
- 13.1 Free Monoids
- 13.2 Monoid Elements
- 13.3 Free abelian monoids
- 13.4 Abelian monoid elements
14. Groups
- 14.1 Base class for all groups
- 14.2 Multiplicative Abelian Groups
- 14.3 Abelian group elements
- 14.4 Homomorphisms of abelian groups
- 14.5 Permutation groups
- 14.6 Permutation group elements
- 14.7 Permutation group homomorphisms
- 14.8 Matrix Groups
- 14.9 To contain general base classes for elements of the classical groups
- 14.10 Contains general base classes for homomorphisms between the matrix groups
- 14.11 Matrix Group Homsets
- 14.12 Contains general base classes for the classical groups
- 14.13 General Linear Groups
- 14.14 Special Linear Groups
- 14.15 Orthogonal Linear Groups
- 14.16 Symplectic Linear Groups
- 14.17 Unitary Groups and
15. General Rings, Ideals, and Morphisms
- 15.1 Ideals
- 15.2 Monoid of Ring Ideals
- 15.3 Homomorphisms of rings
- 15.4 Space of homomorphisms between two rings
- 15.5 Infinity
- 15.6 Fraction Field of Integral Domains
- 15.7 Fraction Field Elements
- 15.8 Quotient Rings
- 15.9 Quotient Ring Elements
16. Standard Commutative Rings
- 16.1 Ring $\mathbf {Z}$ of Integers
- 16.2 Elements of the ring $\mathbf {Z}$ of integers
- 16.3 Ring $\mathbf {Z}/n\mathbf {Z}$ of integers modulo
- 16.4 Elements of $\mathbf {Z}/n\mathbf {Z}$
- 16.5 Field $\mathbf {Q}$ of Rational Numbers
- 16.6 Rational Numbers
- 16.7 Finite Fields
- 16.8 Elements of Finite Fields
- 16.9 Field of Arbitrary Precision Real Numbers
- 16.10 Field $\mathbf{C}$ of Complex Numbers
- 16.11 Complex Numbers
17. Number Fields
- 17.1 Number Fields
- 17.2 Number Field Elements
18. Polynomial Rings
- 18.1 Univariate Polynomial Rings
- 18.2 Univariate Polynomials
- 18.3 Quotients of Univariate Polynomial Rings
- 18.4 Elements of Quotients of Univariate Polynomial Rings
- 18.5 Multivariate Polynomial Rings
- 18.6 Multivariate Polynomials
- 18.7 Ideals in multivariate polynomial rings
19. Power Series Rings
- 19.1 Univariate Power Series Rings
- 19.2 Power Series
- 19.3 Laurent Series Rings
- 19.4 Laurent Series
- 19.5 Field $\mathbf{Q}_p$ of -adic Numbers
- 19.6 -adic Numbers
20. Algebras
- 20.1 Free algebras
- 20.2 Free algebra elements
- 20.3 Free algebra quotients
- 20.4 Free algebra quotient elements
21. Quaternion Algebras
- 21.1 Quaternion algebras
- 21.2 Quaternion algebra elements
- 21.3 Quaternion orders
- 21.4 Quaternion order elements
- 21.5 Quaternion ideal
- 21.6 Quaternion ideal elements
22. Matrices and Spaces of Matrices
- 22.1 Spaces of matrices over a ring or field
- 22.2 Matrices over an arbitrary ring
- 22.3 Matrices over specific rings
- 22.4 Constructor for making matrices
- 22.5 Sparse matrices
- 22.6 Minimal Polynomials of Linear Recurrence Sequences
- 22.7 Dense matrices over $\mathbf{F}_p$ and $\mathbf {Q}$
- 22.8 Sparse matrices over $\mathbf{F}_p$ and $\mathbf {Q}$
23. Free Modules
- 23.1 Free modules
- 23.2 Elements of free modules
- 23.3 Homspaces between free modules
- 23.4 Morphisms of free modules
- 23.5 Morphisms defined by a matrix
24. Geometry
- 24.1 Groebner Fans
25. L-functions
- 25.1 Rubinstein's -function Calculator
- 25.2 Watkins Symmetric Power -function Calculator
- 25.3 Dokchitser's L-functions Calculator
26. Schemes
- 26.1 Scheme implementation overview
  - 26.1.1 TODO List
- 26.2 Schemes
- 26.3 Spec of a ring
- 26.4 Scheme obtained by glueing two other schemes
- 26.5 Points on schemes
- 26.6 Ambient Spaces
- 26.7 Affine space over a ring
- 26.8 Projective space over a ring
- 26.9 Algebraic schemes
- 26.10 Set of homomorphisms between two schemes
- 26.11 Scheme morphism
- 26.12 Divisors on schemes
27. Elliptic and Plane Curves
- 27.1 Plane curve constructors
- 27.2 Affine plane curves over a general ring
- 27.3 Plane curves over a general ring
- 27.4 Elliptic curve constructor
- 27.5 Elliptic curves over a general ring
- 27.6 Elliptic curves over the rational numbers
- 27.7 Elliptic curves over finite fields
28. Hyperelliptic Curves
- 28.1 Hyperelliptic curve constructor
- 28.2 Hyperelliptic curves over a finite field
- 28.3 Hyperelliptic curves over a general ring
- 28.4 Constructor for Jacobian of a hyperelliptic curve
- 28.5 Jacobian of a Hyperelliptic curve of Genus 2
- 28.6 Jacobian of a General Hyperelliptic Curve
- 28.7 Rational point sets on a Jacobian
- 28.8 Jacobian "morphism" as class in Picard group
- 28.9 Conductor and Reduction Types for Genus 2 Curves
29. Combinatorics
- 29.1 Combinatorial Functions
30. Coding Theory
- 30.1 Linear Codes
- 30.2 Bounds for Parameters of Codes
31. General Hecke Algebras and Hecke Modules
- 31.1 Hecke modules
- 31.2 Submodule of a Hecke module
- 31.3 Ambient Hecke modules
- 31.4 Elements of Hecke modules
- 31.5 Hom spaces between objects of the category of hecke modules over a given base ring
- 31.6 Morphism of Hecke modules
- 31.7 Hecke algebras and modules
- 31.8 Hecke algebras and modules
32. Modular Symbols
- 32.1 Creation of modular symbols spaces
- 32.2 Space of modular symbols (base class)
- 32.3 Ambient spaces of modular symbols
- 32.4 Subspace of ambient spaces of modular symbols
- 32.5 A single element of an ambient space of modular symbols
- 32.6 Manin symbols
- 32.7 Space of boundary modular symbols
- 32.8 Heilbronn matrix computation
- 32.9 List of Elements of $P^1(\mathbf {Z}/N\mathbf {Z})$
- 32.10 Relation matrices for ambient modular symbols spaces
33. Modular Forms
- 33.1 Congruence subgroups of $\SL _2(\mathbf{Z})$
- 33.2 Dirichlet characters
- 33.3 The set $\mathbf {P}^1(\mathbf {Q})$ of cusps
- 33.4 Dimensions of spaces of modular forms
- 33.5 Conjectural Slopes of Hecke Polynomial
34. Databases
- 34.1 Cremona's tables of elliptic curves
- 34.2 The Stein-Watkins table of elliptic curves
- 34.3 John Jones's tables of number fields
- 34.4 Linear codes
- 34.5 Interface to Sloane On-Line Encyclopedia of Integer Sequences
- 34.6 Frank Luebeck's tables of Conway polynomials over finite fields
A. History and License
- A.1 License
  - A.1.1 The GNU General Public License
Module Index
Index
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