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<!DOCTYPE html>
<!--
DO NOT EDIT THIS PAGE -- AUTOGENERATED VIA index.py
-->
<head>
<title>SageMath Documentation </title>
<link rel="stylesheet" href="/html/en/website/_static/sage.css" type="text/css" />
<style>
html {margin: 20px; font-family: sans-serif; font-size: 90%; }
body { background-color: white; max-width: 1200px; }
h2 { margin: 0;}
a {text-decoration: none;}
a:hover {text-decoration: underline; }
div.table { display: table; }
div.table.ov { margin-bottom: 1rem; }
div.row { display: table-row; }
div.row:hover {background: #eef; }
div.cell { display: table-cell; padding: 0 2em 0.5em 0; line-height: 1.5em;}
div.lang {font-weight: bold; }
div.row.en {border: 1px solid #ccf;}
div.row.last {height: 3em;}
div.entry {margin: 1em 1em 1em 0; display: inline; white-space: nowrap;}
div.entry.lang-en {font-weight: bold;}
</style>
<link rel="shortcut icon" href="/html/en/website/_static/favicon.ico"/>
</head>
<body>
<a href="https://www.sagemath.org/">
<img width="350" alt="SageMath Logo"
style="border-radius: 10px;"
src="https://www.sagemath.org/pix/sage_logo_new_l_hc_edgy-nq8.png"
title="SageMath Mathematical Software">
</a>
<h1>Documentation</h1>
</div></div></div>
<div class="table">
<div class="row first">
<div class='cell'><h2>HTML</h2></div>
<div class="cell lang"><a href="html/ca">Catalan</a></div>
<div class="cell doc">
<div class='entry lang-ca'><a href='html/ca/intro/index.html'>Intro</a></div>
</div></div>
<div class="row el"><div class="cell"></div>
<div class="cell lang"><a href="html/el">el</a></div>
<div class="cell doc">
<div class='entry lang-el'><a href='html/el/a_tour_of_sage/index.html'>A Tour Of Sage</a></div>
</div></div>
<div class="row en"><div class="cell"></div>
<div class="cell lang"><a href="html/en">English</a></div>
<div class="cell doc">
<div class='entry lang-en'><a href='html/en/a_tour_of_sage/index.html'>A Tour Of Sage</a></div>
<div class='entry lang-en'><a href='html/en/constructions/index.html'>Constructions</a></div>
<div class='entry lang-en'><a href='html/en/developer/index.html'>Developer</a></div>
<div class='entry lang-en'><a href='html/en/faq/index.html'>Faq</a></div>
<div class='entry lang-en'><a href='html/en/installation/index.html'>Installation</a></div>
<div class='entry lang-en'><a href='html/en/jupyter_execute/index.html'>Jupyter Execute</a></div>
<div class='entry lang-en'><a href='html/en/prep/index.html'>Prep</a></div>
<div class='entry lang-en'><a href='html/en/reference/index.html'>Reference</a></div>
<div class='entry lang-en'><a href='html/en/thematic_tutorials/index.html'>Thematic Tutorials</a></div>
<div class='entry lang-en'><a href='html/en/tutorial/index.html'>Tutorial</a></div>
</div></div>
<div class="row fr"><div class="cell"></div>
<div class="cell lang"><a href="html/fr">French</a></div>
<div class="cell doc">
<div class='entry lang-fr'><a href='html/fr/a_tour_of_sage/index.html'>A Tour Of Sage</a></div>
<div class='entry lang-fr'><a href='html/fr/tutorial/index.html'>Tutorial</a></div>
</div></div>
<div class="row de"><div class="cell"></div>
<div class="cell lang"><a href="html/de">German</a></div>
<div class="cell doc">
<div class='entry lang-de'><a href='html/de/a_tour_of_sage/index.html'>A Tour Of Sage</a></div>
<div class='entry lang-de'><a href='html/de/thematische_anleitungen/index.html'>Thematische Anleitungen</a></div>
<div class='entry lang-de'><a href='html/de/tutorial/index.html'>Tutorial</a></div>
</div></div>
<div class="row hu"><div class="cell"></div>
<div class="cell lang"><a href="html/hu">Hungarian</a></div>
<div class="cell doc">
<div class='entry lang-hu'><a href='html/hu/a_tour_of_sage/index.html'>A Tour Of Sage</a></div>
</div></div>
<div class="row it"><div class="cell"></div>
<div class="cell lang"><a href="html/it">Italian</a></div>
<div class="cell doc">
<div class='entry lang-it'><a href='html/it/a_tour_of_sage/index.html'>A Tour Of Sage</a></div>
<div class='entry lang-it'><a href='html/it/faq/index.html'>Faq</a></div>
<div class='entry lang-it'><a href='html/it/tutorial/index.html'>Tutorial</a></div>
</div></div>
<div class="row ja"><div class="cell"></div>
<div class="cell lang"><a href="html/ja">Japanese</a></div>
<div class="cell doc">
<div class='entry lang-ja'><a href='html/ja/a_tour_of_sage/index.html'>A Tour Of Sage</a></div>
<div class='entry lang-ja'><a href='html/ja/tutorial/index.html'>Tutorial</a></div>
</div></div>
<div class="row pt"><div class="cell"></div>
<div class="cell lang"><a href="html/pt">Portuguese</a></div>
<div class="cell doc">
<div class='entry lang-pt'><a href='html/pt/a_tour_of_sage/index.html'>A Tour Of Sage</a></div>
<div class='entry lang-pt'><a href='html/pt/tutorial/index.html'>Tutorial</a></div>
</div></div>
<div class="row ru"><div class="cell"></div>
<div class="cell lang"><a href="html/ru">Russian</a></div>
<div class="cell doc">
<div class='entry lang-ru'><a href='html/ru/tutorial/index.html'>Tutorial</a></div>
</div></div>
<div class="row es"><div class="cell"></div>
<div class="cell lang"><a href="html/es">Spanish</a></div>
<div class="cell doc">
<div class='entry lang-es'><a href='html/es/a_tour_of_sage/index.html'>A Tour Of Sage</a></div>
<div class='entry lang-es'><a href='html/es/tutorial/index.html'>Tutorial</a></div>
</div></div>
<div class="row tr"><div class="cell"></div>
<div class="cell lang"><a href="html/tr">Turkish</a></div>
<div class="cell doc">
<div class='entry lang-tr'><a href='html/tr/a_tour_of_sage/index.html'>A Tour Of Sage</a></div>
</div></div>
<div class="row last"></div>
<div class="row first">
<div class='cell'><h2>PDF</h2></div>
<div class="cell lang"><a href="pdf/ca">Catalan</a></div>
<div class="cell doc">
<div class='entry lang-ca'><a href='pdf/ca/intro/TutorialdeSage.pdf'>Tutorialdesage</a></div>
</div></div>
<div class="row el"><div class="cell"></div>
<div class="cell lang"><a href="pdf/el">el</a></div>
<div class="cell doc">
<div class='entry lang-el'><a href='pdf/el/a_tour_of_sage/a_tour_of_sage.pdf'>A Tour Of Sage</a></div>
</div></div>
<div class="row en"><div class="cell"></div>
<div class="cell lang"><a href="pdf/en">English</a></div>
<div class="cell doc">
<div class='entry lang-en'><a href='pdf/en/a_tour_of_sage/a_tour_of_sage.pdf'>A Tour Of Sage</a></div>
<div class='entry lang-en'><a href='pdf/en/constructions/constructions.pdf'>Constructions</a></div>
<div class='entry lang-en'><a href='pdf/en/developer/developer.pdf'>Developer</a></div>
<div class='entry lang-en'><a href='pdf/en/faq/faq.pdf'>Faq</a></div>
<div class='entry lang-en'><a href='pdf/en/installation/installation.pdf'>Installation</a></div>
<div class='entry lang-en'><a href='pdf/en/prep/prep_tutorials.pdf'>Prep Tutorials</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/reference.pdf'>Reference</a></div>
<div class='entry lang-en'><a href='pdf/en/thematic_tutorials/thematic_tutorials.pdf'>Thematic Tutorials</a></div>
<div class='entry lang-en'><a href='pdf/en/tutorial/sage_tutorial.pdf'>Sage Tutorial</a></div>
<div class='entry lang-en'><a href='pdf/en/website/sage_documentation.pdf'>Sage Documentation</a></div>
</div></div>
<div class="row en"><div class="cell"></div>
<div class="cell lang"><a href="pdf/en">English/Reference</a></div>
<div class="cell doc">
<div class='entry lang-en'><a href='pdf/en/reference/algebras/algebras.pdf'>Algebras</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/arithgroup/arithgroup.pdf'>Arithgroup</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/arithmetic_curves/arithmetic_curves.pdf'>Arithmetic Curves</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/asymptotic/asymptotic.pdf'>Asymptotic</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/calculus/calculus.pdf'>Calculus</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/categories/categories.pdf'>Categories</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/coding/coding.pdf'>Coding</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/coercion/coercion.pdf'>Coercion</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/combinat/combinat.pdf'>Combinat</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/constants/constants.pdf'>Constants</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/cpython/cpython.pdf'>Cpython</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/cryptography/cryptography.pdf'>Cryptography</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/curves/curves.pdf'>Curves</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/data_structures/data_structures.pdf'>Data Structures</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/databases/databases.pdf'>Databases</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/diophantine_approximation/diophantine_approximation.pdf'>Diophantine Approximation</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/discrete_geometry/discrete_geometry.pdf'>Discrete Geometry</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/doctest/doctest.pdf'>Doctest</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/documentation/documentation.pdf'>Documentation</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/drinfeld_modules/drinfeld_modules.pdf'>Drinfeld Modules</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/dynamics/dynamics.pdf'>Dynamics</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/euclidean_spaces/euclidean_spaces.pdf'>Euclidean Spaces</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/finite_rings/finite_rings.pdf'>Finite Rings</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/function_fields/function_fields.pdf'>Function Fields</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/functions/functions.pdf'>Functions</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/game_theory/game_theory.pdf'>Game Theory</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/games/games.pdf'>Games</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/graphs/graphs.pdf'>Graphs</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/groups/groups.pdf'>Groups</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/hecke/hecke.pdf'>Hecke</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/history_and_license/history_and_license.pdf'>History And License</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/homology/homology.pdf'>Homology</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/hyperbolic_geometry/hyperbolic_geometry.pdf'>Hyperbolic Geometry</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/interfaces/interfaces.pdf'>Interfaces</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/knots/knots.pdf'>Knots</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/lfunctions/lfunctions.pdf'>Lfunctions</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/libs/libs.pdf'>Libs</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/logic/logic.pdf'>Logic</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/manifolds/manifolds.pdf'>Manifolds</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/matrices/matrices.pdf'>Matrices</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/matroids/matroids.pdf'>Matroids</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/misc/misc.pdf'>Misc</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/modabvar/modabvar.pdf'>Modabvar</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/modfrm/modfrm.pdf'>Modfrm</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/modsym/modsym.pdf'>Modsym</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/modules/modules.pdf'>Modules</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/monoids/monoids.pdf'>Monoids</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/noncommutative_polynomial_rings/noncommutative_polynomial_rings.pdf'>Noncommutative Polynomial Rings</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/number_fields/number_fields.pdf'>Number Fields</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/numerical/numerical.pdf'>Numerical</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/padics/padics.pdf'>Padics</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/parallel/parallel.pdf'>Parallel</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/plot3d/plot3d.pdf'>Plot3D</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/plotting/plotting.pdf'>Plotting</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/polynomial_rings/polynomial_rings.pdf'>Polynomial Rings</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/power_series/power_series.pdf'>Power Series</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/probability/probability.pdf'>Probability</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/quadratic_forms/quadratic_forms.pdf'>Quadratic Forms</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/quat_algebras/quat_algebras.pdf'>Quat Algebras</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/quivers/quivers.pdf'>Quivers</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/references/references.pdf'>References</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/repl/repl.pdf'>Repl</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/resolutions/resolutions.pdf'>Resolutions</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/riemannian_geometry/riemannian_geometry.pdf'>Riemannian Geometry</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/rings/rings.pdf'>Rings</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/rings_numerical/rings_numerical.pdf'>Rings Numerical</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/rings_standard/rings_standard.pdf'>Rings Standard</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/sat/sat.pdf'>Sat</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/schemes/schemes.pdf'>Schemes</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/semirings/semirings.pdf'>Semirings</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/sets/sets.pdf'>Sets</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/spkg/spkg.pdf'>Spkg</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/stats/stats.pdf'>Stats</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/structure/structure.pdf'>Structure</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/tensor_free_modules/tensor_free_modules.pdf'>Tensor Free Modules</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/topology/topology.pdf'>Topology</a></div>
<div class='entry lang-en'><a href='pdf/en/reference/valuations/valuations.pdf'>Valuations</a></div>
</div></div>
<div class="row fr"><div class="cell"></div>
<div class="cell lang"><a href="pdf/fr">French</a></div>
<div class="cell doc">
<div class='entry lang-fr'><a href='pdf/fr/a_tour_of_sage/a_tour_of_sage.pdf'>A Tour Of Sage</a></div>
<div class='entry lang-fr'><a href='pdf/fr/tutorial/tutorial-fr.pdf'>Tutorial-Fr</a></div>
</div></div>
<div class="row de"><div class="cell"></div>
<div class="cell lang"><a href="pdf/de">German</a></div>
<div class="cell doc">
<div class='entry lang-de'><a href='pdf/de/a_tour_of_sage/a_tour_of_sage.pdf'>A Tour Of Sage</a></div>
<div class='entry lang-de'><a href='pdf/de/thematische_anleitungen/ThematischeAnleitungen-de.pdf'>Thematischeanleitungen-De</a></div>
<div class='entry lang-de'><a href='pdf/de/tutorial/SageTutorial-de.pdf'>Sagetutorial-De</a></div>
</div></div>
<div class="row hu"><div class="cell"></div>
<div class="cell lang"><a href="pdf/hu">Hungarian</a></div>
<div class="cell doc">
<div class='entry lang-hu'><a href='pdf/hu/a_tour_of_sage/a_tour_of_sage.pdf'>A Tour Of Sage</a></div>
</div></div>
<div class="row it"><div class="cell"></div>
<div class="cell lang"><a href="pdf/it">Italian</a></div>
<div class="cell doc">
<div class='entry lang-it'><a href='pdf/it/a_tour_of_sage/a_tour_of_sage.pdf'>A Tour Of Sage</a></div>
<div class='entry lang-it'><a href='pdf/it/faq/faq.pdf'>Faq</a></div>
<div class='entry lang-it'><a href='pdf/it/tutorial/tutorial-it.pdf'>Tutorial-It</a></div>
</div></div>
<div class="row ja"><div class="cell"></div>
<div class="cell lang"><a href="pdf/ja">Japanese</a></div>
<div class="cell doc">
<div class='entry lang-ja'><a href='pdf/ja/a_tour_of_sage/a_tour_of_sage.pdf'>A Tour Of Sage</a></div>
<div class='entry lang-ja'><a href='pdf/ja/tutorial/tutorial-jp.pdf'>Tutorial-Jp</a></div>
</div></div>
<div class="row pt"><div class="cell"></div>
<div class="cell lang"><a href="pdf/pt">Portuguese</a></div>
<div class="cell doc">
<div class='entry lang-pt'><a href='pdf/pt/a_tour_of_sage/a_tour_of_sage.pdf'>A Tour Of Sage</a></div>
<div class='entry lang-pt'><a href='pdf/pt/tutorial/tutorial-pt.pdf'>Tutorial-Pt</a></div>
</div></div>
<div class="row ru"><div class="cell"></div>
<div class="cell lang"><a href="pdf/ru">Russian</a></div>
<div class="cell doc">
<div class='entry lang-ru'><a href='pdf/ru/tutorial/SageTutorial_ru.pdf'>Sagetutorial Ru</a></div>
</div></div>
<div class="row es"><div class="cell"></div>
<div class="cell lang"><a href="pdf/es">Spanish</a></div>
<div class="cell doc">
<div class='entry lang-es'><a href='pdf/es/a_tour_of_sage/a_tour_of_sage.pdf'>A Tour Of Sage</a></div>
<div class='entry lang-es'><a href='pdf/es/tutorial/SageTutorial.pdf'>Sagetutorial</a></div>
</div></div>
<div class="row tr"><div class="cell"></div>
<div class="cell lang"><a href="pdf/tr">Turkish</a></div>
<div class="cell doc">
<div class='entry lang-tr'><a href='pdf/tr/a_tour_of_sage/a_tour_of_sage.pdf'>A Tour Of Sage</a></div>
</div></div>
<div class="row last"></div>
</div>
</body></html>