Welcome to the Sage Reference Manual

This manual contains documentation for (almost) all of Sage’s features, each illustrated with examples that are systematically tested with each release. A thematic index is available below.

User Interface

• Command Line Interface (REPL)
• For the Jupyter notebook interface, visit its documentation.
• For the legacy notebook interface, which is no longer actively maintained, visit the source repository.

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
• Polynomials
• Power Series and Laurent Series
• Finite Rings and Fields
• p-Adic Numbers
• 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 and their Homology
• Manifolds and Differential Geometry
• Hyperbolic Geometry
• Parametrized Surfaces
• Knot Theory

Number Fields, Function Fields, and Valuations

• Number Fields
• Function Fields
• Discrete Valuations

Number Theory

• Diophantine approximation
• Quadratic Forms
• L-Functions
• Arithmetic Subgroups of SL_2(Z)
• General Hecke Algebras and Hecke Modules
• Modular Symbols
• Modular Forms
• Modular Forms for Hecke Triangle Groups
• Modular Abelian Varieties
• Miscellaneous Modular-Form-Related Modules

Algebraic and Arithmetic Geometry

• Schemes
• Plane, Elliptic and Hyperelliptic Curves

Miscellaneous

• Databases
• Games

Programming

• Data Structures
• Utilities
• Test Framework
• Parallel Computing

Interfaces

• Interpreter Interfaces
• C/C++ Library Interfaces
• Python technicalities

General Information

• References
• History and License
• Index
• Module Index
• Search Page

This work is licensed under a Creative Commons Attribution-Share Alike 3.0 License.