This is the manual for the Sage mathematical software system. Sage is free open source math software that supports research and teaching in algebra, geometry, number theory, cryptography, and related areas. Both the Sage development model and the technology in Sage itself are distinguished by an extremely strong emphasis on openness, community, cooperation, and collaboration: we are building the car, not reinventing the wheel.

This reference manual contains many examples that illustrate the usage of Sage. The examples are all tested with each release of Sage, and should produce exactly the same output as in this manual, except for line breaks.

The Sage command line is briefly described in The Sage Command Line , which lists the command line options. For more details about the command line, see the Sage tutorial.

The Sage graphical user interface is described in The Sage Notebook . This graphical user interface is unusual in that it operates via your web browser. It provides you with Sage worksheets that you can edit and evaluate, which contain scalable typeset mathematics and beautiful antialiased images.

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

Enjoy Sage!

- General Rings, Ideals, and Morphisms
- Standard Commutative Rings
- Fixed and Arbitrary Precision Numerical Fields
- Finite Rings
- Algebraic Number Fields
- Function Fields
- p-Adics
- Polynomial Rings
- Power Series Rings
- Standard Semirings
- Algebras
- Quaternion Algebras

- Quadratic Forms
- L-Functions
- Schemes
- Elliptic, Plane, and Hyperelliptic Curves
- Arithmetic Subgroups of SL_2(Z)
- General Hecke Algebras and Hecke Modules
- Modular Symbols
- Modular Forms
- Modular Abelian Varieties
- Miscellaneous Modular-Form-Related Modules

- Games
- Symbolic Logic
- SAT solvers
- Cryptography
- Numerical Optimization
- Probability
- Statistics
- Quantitative Finance
- Coding Theory

- Doctesting
- Development Scripts
- Interpreter Interfaces
- C/C++ Library Interfaces
- Databases
- Parallel Computing
- Miscellaneous