Welcome to the Sage Reference Manual!#

Here you find documentation for all of Sage’s features, illustrated with lots of examples. A thematic index follows.

This documentation is licensed under the Creative Commons Attribution-Share Alike 3.0 License.

User Interfaces#

Graphics#

  • 2D Graphics

  • 3D Graphics

Mathematics#

Parents and Categories#

  • Parents and Elements

  • Coercion

  • Categories

Basic Rings and Fields#

  • Integers and Rational Numbers

  • Real and Complex Numbers

  • Commutative Polynomials

  • Power Series and Laurent Series

  • Finite Rings and Fields

  • \(p\)-adic Numbers

  • Noncommutative Polynomials

  • Quaternion Algebras

Linear Algebra#

  • Matrices and Spaces of Matrices

  • Vectors and Modules

  • Tensors on Free Modules of Finite Rank

Calculus and Analysis#

  • Symbolic Calculus

  • Mathematical Constants

  • Elementary and Special Functions

  • Asymptotic Expansions

  • Numerical Optimization

Probability and Statistics#

  • Probability

  • Statistics

  • Quantitative Finance

Mathematical Structures#

  • Sets

  • Monoids

  • Groups

  • Semirings

  • Rings

  • Algebras

Discrete Mathematics#

  • Combinatorics

  • Graph Theory

  • Quivers

  • Matroid Theory

  • Discrete Dynamics

  • Coding Theory

  • Cryptography

  • Game Theory

  • Symbolic Logic

  • SAT solvers

Geometry and Topology#

  • Euclidean Spaces and Vector Calculus

  • Combinatorial and Discrete Geometry

  • Cell Complexes, Simplicial Complexes, and Simplicial Sets

  • Manifolds and Differential Geometry

  • Hyperbolic Geometry

  • Parametrized Surfaces

  • Knot Theory

Homological Algebra#

  • Chain Complexes and their Homology

  • Resolutions

Number Fields, Function Fields, and Valuations#

  • Number Fields

  • Function Fields

  • Discrete Valuations

Number Theory#

  • Diophantine approximation

  • Quadratic Forms

  • \(L\)-Functions

  • Arithmetic Subgroups of \({\rm SL}_2(\ZZ)\)

  • General Hecke Algebras and Hecke Modules

  • Modular Symbols

  • Modular Forms

  • Quasimodular Forms

  • Modular Forms for Hecke Triangle Groups

  • Modular Abelian Varieties

  • Miscellaneous Modular-Form-Related Modules

Algebraic and Arithmetic Geometry#

  • Schemes

  • Plane and Space Curves

  • Elliptic and Hyperelliptic Curves

Miscellaneous#

  • Databases

  • Games

Programming#

Facilities#

  • Data Structures

  • Utilities

  • Test Framework

  • Parallel Computing

Interfaces#

  • Interpreter Interfaces

  • C/C++ Library Interfaces

  • Python Technicalities

Documentation System#

  • Documentation System

General Information#

Indices and Tables#