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# Welcome to the Sage Constructions documentation!

This document collects the answers to some questions along the line “How do I construct ... in Sage?” Though much of this material can be found in the manual or tutorial or string documentation of the Python code, it is hoped that this will provide the casual user with some basic examples on how to start using Sage in a useful way.

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

Please send suggestions, additions, corrections to the `sage-devel` Google group!

The Sage wiki http://wiki.sagemath.org/ contains a wealth of information. Moreover, all these can be tested in the Sage notebook interface http://www.sagenb.org/ on the web.

- Calculus
  - Differentiation
  - Integration
  - Ordinary differential equations
  - Fourier series of periodic functions
- Plotting
  - Plotting functions in 2D
  - Plotting curves
  - openmath
  - Tachyon 3D plotting
  - gnuplot
  - Plotting surfaces
- Groups
  - Permutation groups
  - Conjugacy classes
  - Normal subgroups
  - Centers
  - The group id database
  - Construction instructions for every group of order less than 32
  - Construction instructions for every finitely presented group of order 15 or less
- Linear algebra
  - Vector spaces
  - Matrix powers
  - Kernels
  - Eigenvectors and eigenvalues
  - Row reduction
  - Characteristic polynomial
  - Solving systems of linear equations
- Linear codes and ciphers
  - Codes
  - Ciphers
- Representation theory
  - Ordinary characters
  - Brauer characters
- Rings
  - Matrix rings
  - Polynomial rings
  - \(p\)-adic numbers
  - Quotient rings of polynomials
- Polynomials
  - Polynomial powers
  - Factorization
  - Polynomial GCD’s
  - Roots of polynomials
  - Evaluation of multivariate functions
  - Roots of multivariate polynomials
  - Gröbner bases
- Elementary number theory
  - Taking modular powers
  - Discrete logs
  - Prime numbers
  - Divisors
  - Quadratic residues
- Modular forms
  - Cusp forms
  - Coset representatives
  - Modular symbols and Hecke operators
  - Genus formulas
- Elliptic curves
  - Conductor
  - \(j\)-invariant
  - The \(GF(q)\)-rational points on E
  - Modular form associated to an elliptic curve over \(\QQ\)
- Number fields
  - Ramification
  - Class numbers
  - Integral basis
- Algebraic Geometry
  - Point counting on curves
  - Riemann-Roch spaces using Singular
- Interface Issues
  - Background jobs
  - Referencing Sage
  - Logging your Sage session
  - LaTeX conversion
  - Sage and other computer algebra systems
  - Command-line Sage help
  - Reading and importing files into Sage
  - Installation for the impatient
  - Python language program code for Sage commands
  - “Special functions” in Sage
  - What is Sage?
- Contributions to this document

# Indices and tables

- Index
- Module Index
- Search Page

### Navigation

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- Sage Constructions v6.4.1 »