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

• Command Line Interface

• Jupyter Notebook Interface

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, Topology, and Homological Algebra

• 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

• Chain Complexes and their Homology

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

General Information

• External Packages

• Bibliographic References

• History and License

Indices and Tables

• Index

• Module Index

• Search Page