SAGE
Reference Manual
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Front Matter
SAGE
Reference Manual
William A. Stein
wstein@gmail.com
Date:
May 25, 2006
Front Matter
Contents
1. Introduction
2. Miscellaneous
2.1 Miscellaneous functions
2.2 Installing shortcut scripts
2.3 Bug reporting1
2.4 Functional notation
2.5 Latex printing support
2.6 Logging of SAGE sessions
2.7 Object persistence
2.8 Support for persistent functions in .sage files
2.9 Miscellaneous arithmetic functions
2.10 Evaluating a string in
SAGE
3. Basic Structure
3.1 Abstract base class for
SAGE
objects
3.2 Base class for objects with generators
3.3 Formal sums
3.4 Factorizations
3.5 Elements
3.6 Mutability
3.7 Sequences
3.8 Sets
3.9 The set of prime numbers
4. Interpreter Interfaces
4.1 Common Interface Functionality
4.2 Interface to GAP
4.2.1 First Examples
4.2.2 GAP and Singular
4.2.3 Saving and loading objects
4.2.4 Long Input
4.2.5 Changing which GAP is used
4.3 Interface to GP/Pari
4.4 Interface to the Gnuplot interpreter
4.5 Interface to KASH
4.5.1 Issues
4.5.2 Tutorial
4.5.3 Long Input
4.6 Interface to Magma
4.6.1 Parameters
4.6.2 Multiple Return Values
4.6.3 Long Input
4.7 Interface to Maple
4.7.1 Tutorial
4.8 Interface to Maxima
4.8.1 Tutorial
4.8.2 Examples involving matrices
4.8.3 Laplace Transforms
4.8.4 Continued Fractions
4.8.5 Special examples
4.8.6 Miscellaneous
4.8.7 Interactivity
4.8.8 Latex Output
4.8.9 Long Input
4.9 Interface to Mathematica
4.9.1 Tutorial
4.9.2 Long Input
4.9.3 Loading and saving
4.10 Interface to mwrank
4.11 Interface to Octave
4.11.1 Computation of Special Functions
4.12 Interface to SAGE
4.13 Interface to Singular
4.13.1 Tutorial
4.13.2 Computing the Genus
4.13.3 An Important Concept
4.13.4 Long Input
5. C/C++ Library Interfaces
5.1 Cremona's mwrank C++ library
5.2 Victor Shoup's NTL C++ Library
5.3 PARI C-library interface
6. Constants and Mathematical Functions
6.1 Mathematical constants
6.2 Computation of transcendental functions
7. Plotting and Visualization
7.1 2D Plotting
8. Category Theory
8.1 Categories
8.2 Homsets
8.3 Morphisms
8.4 Functors
9. Monoids
9.1 Free Monoids
9.2 Monoid Elements
9.3 Free abelian monoids
9.4 Abelian monoid elements
10. Groups
10.1 Base class for all groups
10.2 Multiplicative Abelian Groups
10.3 Abelian group elements
10.4 Homomorphisms of abelian groups
10.5 Permutation groups
10.6 Permutation group elements
10.7 Permutation group homomorphisms
10.8 Matrix Groups
10.9 To contain general base classes for elements of the classical groups
10.10 Contains general base classes for homomorphisms between the matrix groups
10.11 Matrix Group Homsets
10.12 Contains general base classes for the classical groups
10.13 General Linear Groups
10.14 Special Linear Groups
10.15 Orthogonal Linear Groups
10.16 Symplectic Linear Groups
10.17 Unitary Groups
and
11. General Rings, Ideals, and Morphisms
11.1 Ideals
11.2 Monoid of Ring Ideals
11.3 Homomorphisms of rings
11.4 Space of homomorphisms between two rings
11.5 Infinity
11.6 Fraction Field of Integral Domains
11.7 Fraction Field Elements
11.8 Quotient Rings
11.9 Quotient Ring Elements
12. Standard Commutative Rings
12.1 Ring
of Integers
12.2 Elements of the ring
of integers
12.3 Ring
of integers modulo
12.4 Elements of
12.5 Field
of Rational Numbers
12.6 Rational Numbers
12.7 Finite Fields
12.8 Elements of Finite Fields
12.9 Real Numbers
12.10 Field
of Complex Numbers
12.11 Complex Numbers
13. Number Fields
13.1 Number Fields
13.2 Number Field Elements
14. Polynomial Rings
14.1 Univariate Polynomial Rings
14.2 Univariate Polynomials
14.3 Quotients of Univariate Polynomial Rings
14.4 Elements of Quotients of Univariate Polynomial Rings
14.5 Multivariate Polynomial Rings
14.6 Multivariate Polynomials
14.7 Ideals in multivariate polynomial rings
15. Power Series Rings
15.1 Univariate Power Series Rings
15.2 Power Series
15.3 Laurent Series Rings
15.4 Laurent Series
15.5 Field
of
-adic Numbers
15.6
-adic Numbers
16. Algebras
16.1 Free algebras
16.2 Free algebra elements
16.3 Free algebra quotients
16.4 Free algebra quotient elements
17. Quaternion Algebras
17.1 Quaternion algebras
17.2 Quaternion algebra elements
17.3 Quaternion orders
17.4 Quaternion order elements
17.5 Quaternion ideal
17.6 Quaternion ideal elements
18. Matrices
18.1 Space of matrices over a ring
18.2 Matrices
19. Free Modules
19.1 Free modules
19.2 Elements of free modules
19.3 Homspaces between free modules
19.4 Morphisms of free modules
19.5 Morphisms defined by a matrix
20. Functions
20.1 Piecewise-defined functions
21. Geometry
21.1 Groebner Fans
22. L-functions
22.1 Rubinstein's
-function Calculator
22.2 Watkins Symmetric Power
-function Calculator
22.3 Dokchitser's L-functions Calculator
23. Schemes
23.1 Scheme implementation overview
23.1.1 TODO List
23.2 Schemes
23.3 Spec of a ring
23.4 Scheme obtained by glueing two other schemes
23.5 Points on schemes
23.6 Ambient Spaces
23.7 Affine
space over a ring
23.8 Projective
space over a ring
23.9 Algebraic schemes
23.10 Set of homomorphisms between two schemes
23.11 Scheme morphism
23.12 Divisors on schemes
24. Elliptic and Plane Curves
24.1 Plane curve constructors
24.2 Affine plane curves over a general ring
24.3 Plane curves over a general ring
24.4 Elliptic curve constructor
24.5 Elliptic curves over a general ring
24.6 Elliptic curves over the rational numbers
24.7 Elliptic curves over finite fields
25. Hyperelliptic Curves
25.1 Hyperelliptic curve constructor
25.2 Hyperelliptic curves over a finite field
25.3 Hyperelliptic curves over a general ring
25.4 Constructor for Jacobian of a hyperelliptic curve
25.5 Jacobian of a hyperelliptic curve
25.6 Jacobian of a hyperelliptic curve
25.7 Rational point sets on a Jacobian
25.8 Jacobian "morphism" as class in Picard group
25.9 Conductor and Reduction Types for Genus 2 Curves
26. General Hecke Algebras and Hecke Modules
26.1 Hecke modules
26.2 Submodule of a Hecke module
26.3 Ambient Hecke modules
26.4 Elements of Hecke modules
26.5 Hom spaces between objects of the category of hecke modules over a given base ring
26.6 Morphism of Hecke modules
26.7 Hecke algebras and modules
26.8 Hecke algebras and modules
27. Modular Symbols
27.1 Creation of modular symbols spaces
27.2 Space of modular symbols (base class)
27.3 Ambient spaces of modular symbols.
27.4 Subspace of ambient spaces of modular symbols
27.5 A single element of an ambient space of modular symbols
27.6 Manin symbols
27.7 Space of boundary modular symbols
27.8 Heilbronn matrix computation
27.9 List of Elements of
27.10 Relation matrices for ambient modular symbols spaces
28. Modular Forms
28.1 Congruence subgroups of
28.2 Dirichlet characters
28.3 The set
of cusps
28.4 Dimensions of spaces of modular forms
28.5 Conjectural Slopes of Hecke Polynomial
29. Databases
29.1 Cremona's tables of elliptic curves
29.2 The Stein-Watkins table of elliptic curves
29.3 John Jones's tables of number fields
29.4 Linear codes
29.5 Interface to Sloane On-Line Encyclopedia of Integer Sequences
29.6 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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SAGE
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Release 2006.05.25, documentation updated on May 25, 2006.
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