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32      <h3>Navigation</h3>
33      <ul>
34        <li class="right" style="margin-right: 10px">
35          <a href="genindex.html" title="General Index"
36             accesskey="I">index</a></li>
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41        <li class="nav-item nav-item-0"><a href="#">Sage Reference v9.2.beta9</a> &#187;</li>
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51           
52<h1>.. _reference-manual: (PDF version)</h1></p>
53<p>* <a href="repl/repl.pdf">Command Line Interface <img src="_static/pdf.png" /></a>
54* `Jupyter Notebook Interface <https://jupyter-notebook.readthedocs.io/en/latest/notebook.html>`_</p>
55<p>Graphics
56========</p>
57<p>* <a href="plotting/plotting.pdf">2D Graphics <img src="_static/pdf.png" /></a>
58* <a href="plot3d/plot3d.pdf">3D Graphics <img src="_static/pdf.png" /></a></p>
59<p>Mathematics
60===========</p>
61<p>Parents and Categories
62----------------------</p>
63<p>* <a href="structure/structure.pdf">Parents and Elements <img src="_static/pdf.png" /></a>
64* <a href="coercion/coercion.pdf">Coercion <img src="_static/pdf.png" /></a>
65* <a href="categories/categories.pdf">Categories <img src="_static/pdf.png" /></a></p>
66<p>Basic Rings and Fields
67----------------------</p>
68<p>* <a href="rings_standard/rings_standard.pdf">Integers and Rational Numbers <img src="_static/pdf.png" /></a>
69* <a href="rings_numerical/rings_numerical.pdf">Real and Complex Numbers <img src="_static/pdf.png" /></a>
70* <a href="polynomial_rings/polynomial_rings.pdf">Commutative Polynomials <img src="_static/pdf.png" /></a>
71* <a href="power_series/power_series.pdf">Power Series and Laurent Series <img src="_static/pdf.png" /></a>
72* <a href="finite_rings/finite_rings.pdf">Finite Rings and Fields <img src="_static/pdf.png" /></a>
73* <a href="padics/padics.pdf">\\(p\\)-adic Numbers <img src="_static/pdf.png" /></a>
74* <a href="noncommutative_polynomial_rings/noncommutative_polynomial_rings.pdf">Noncommutative Polynomials <img src="_static/pdf.png" /></a>
75* <a href="quat_algebras/quat_algebras.pdf">Quaternion Algebras <img src="_static/pdf.png" /></a></p>
76<p>Linear Algebra
77--------------</p>
78<p>* <a href="matrices/matrices.pdf">Matrices and Spaces of Matrices <img src="_static/pdf.png" /></a>
79* <a href="modules/modules.pdf">Vectors and Modules <img src="_static/pdf.png" /></a>
80* <a href="tensor_free_modules/tensor_free_modules.pdf">Tensors on Free Modules of Finite Rank <img src="_static/pdf.png" /></a></p>
81<p>Calculus and Analysis
82---------------------</p>
83<p>* <a href="calculus/calculus.pdf">Symbolic Calculus <img src="_static/pdf.png" /></a>
84* <a href="constants/constants.pdf">Mathematical Constants <img src="_static/pdf.png" /></a>
85* <a href="functions/functions.pdf">Elementary and Special Functions <img src="_static/pdf.png" /></a>
86* <a href="asymptotic/asymptotic.pdf">Asymptotic Expansions <img src="_static/pdf.png" /></a>
87* <a href="numerical/numerical.pdf">Numerical Optimization <img src="_static/pdf.png" /></a></p>
88<p>Probability and Statistics
89--------------------------</p>
90<p>* <a href="probability/probability.pdf">Probability <img src="_static/pdf.png" /></a>
91* <a href="stats/stats.pdf">Statistics <img src="_static/pdf.png" /></a>
92* <a href="finance/finance.pdf">Quantitative Finance <img src="_static/pdf.png" /></a></p>
93<p>Mathematical Structures
94-----------------------</p>
95<p>* <a href="sets/sets.pdf">Sets <img src="_static/pdf.png" /></a>
96* <a href="monoids/monoids.pdf">Monoids <img src="_static/pdf.png" /></a>
97* <a href="groups/groups.pdf">Groups <img src="_static/pdf.png" /></a>
98* <a href="semirings/semirings.pdf">Semirings <img src="_static/pdf.png" /></a>
99* <a href="rings/rings.pdf">Rings <img src="_static/pdf.png" /></a>
100* <a href="algebras/algebras.pdf">Algebras <img src="_static/pdf.png" /></a></p>
101<p>Discrete Mathematics
102--------------------</p>
103<p>* <a href="combinat/combinat.pdf">Combinatorics <img src="_static/pdf.png" /></a>
104* <a href="graphs/graphs.pdf">Graph Theory <img src="_static/pdf.png" /></a>
105* <a href="quivers/quivers.pdf">Quivers <img src="_static/pdf.png" /></a>
106* <a href="matroids/matroids.pdf">Matroid Theory <img src="_static/pdf.png" /></a>
107* <a href="dynamics/dynamics.pdf">Discrete Dynamics <img src="_static/pdf.png" /></a>
108* <a href="coding/coding.pdf">Coding Theory <img src="_static/pdf.png" /></a>
109* <a href="cryptography/cryptography.pdf">Cryptography <img src="_static/pdf.png" /></a>
110* <a href="game_theory/game_theory.pdf">Game Theory <img src="_static/pdf.png" /></a>
111* <a href="logic/logic.pdf">Symbolic Logic <img src="_static/pdf.png" /></a>
112* <a href="sat/sat.pdf">SAT solvers <img src="_static/pdf.png" /></a></p>
113<p>Geometry and Topology
114---------------------</p>
115<p>* <a href="euclidean_spaces/euclidean_spaces.pdf">Euclidean Spaces and Vector Calculus <img src="_static/pdf.png" /></a>
116* <a href="discrete_geometry/discrete_geometry.pdf">Combinatorial and Discrete Geometry <img src="_static/pdf.png" /></a>
117* <a href="homology/homology.pdf">Cell Complexes and their Homology <img src="_static/pdf.png" /></a>
118* <a href="manifolds/manifolds.pdf">Manifolds and Differential Geometry <img src="_static/pdf.png" /></a>
119* <a href="hyperbolic_geometry/hyperbolic_geometry.pdf">Hyperbolic Geometry <img src="_static/pdf.png" /></a>
120* <a href="riemannian_geometry/riemannian_geometry.pdf">Parametrized Surfaces <img src="_static/pdf.png" /></a>
121* <a href="knots/knots.pdf">Knot Theory <img src="_static/pdf.png" /></a></p>
122<p>Number Fields, Function Fields, and Valuations
123----------------------------------------------</p>
124<p>* <a href="number_fields/number_fields.pdf">Number Fields <img src="_static/pdf.png" /></a>
125* <a href="function_fields/function_fields.pdf">Function Fields <img src="_static/pdf.png" /></a>
126* <a href="valuations/valuations.pdf">Discrete Valuations <img src="_static/pdf.png" /></a></p>
127<p>Number Theory
128-------------</p>
129<p>* <a href="diophantine_approximation/diophantine_approximation.pdf">Diophantine approximation <img src="_static/pdf.png" /></a>
130* <a href="quadratic_forms/quadratic_forms.pdf">Quadratic Forms <img src="_static/pdf.png" /></a>
131* <a href="lfunctions/lfunctions.pdf">\\(L\\)-Functions <img src="_static/pdf.png" /></a>
132* <a href="arithgroup/arithgroup.pdf">Arithmetic Subgroups of \\({\\rm SL}_2(\\ZZ)\\) <img src="_static/pdf.png" /></a>
133* <a href="hecke/hecke.pdf">General Hecke Algebras and Hecke Modules <img src="_static/pdf.png" /></a>
134* <a href="modsym/modsym.pdf">Modular Symbols <img src="_static/pdf.png" /></a>
135* <a href="modfrm/modfrm.pdf">Modular Forms <img src="_static/pdf.png" /></a>
136* <a href="modfrm_hecketriangle/modfrm_hecketriangle.pdf">Modular Forms for Hecke Triangle Groups <img src="_static/pdf.png" /></a>
137* <a href="modabvar/modabvar.pdf">Modular Abelian Varieties <img src="_static/pdf.png" /></a>
138* <a href="modmisc/modmisc.pdf">Miscellaneous Modular-Form-Related Modules <img src="_static/pdf.png" /></a></p>
139<p>Algebraic and Arithmetic Geometry
140---------------------------------
141* <a href="schemes/schemes.pdf">Schemes <img src="_static/pdf.png" /></a>
142* <a href="curves/curves.pdf">Plane and Space Curves <img src="_static/pdf.png" /></a>
143* <a href="arithmetic_curves/arithmetic_curves.pdf">Elliptic and Hyperelliptic Curves <img src="_static/pdf.png" /></a></p>
144<p>Miscellaneous
145-------------</p>
146<p>* <a href="databases/databases.pdf">Databases <img src="_static/pdf.png" /></a>
147* <a href="games/games.pdf">Games <img src="_static/pdf.png" /></a></p>
148<p>Programming
149===========</p>
150<p>Facilities
151----------</p>
152<p>* <a href="data_structures/data_structures.pdf">Data Structures <img src="_static/pdf.png" /></a>
153* <a href="misc/misc.pdf">Utilities <img src="_static/pdf.png" /></a>
154* <a href="doctest/doctest.pdf">Test Framework <img src="_static/pdf.png" /></a>
155* <a href="parallel/parallel.pdf">Parallel Computing <img src="_static/pdf.png" /></a></p>
156<p>Interfaces
157----------</p>
158<p>* <a href="interfaces/interfaces.pdf">Interpreter Interfaces <img src="_static/pdf.png" /></a>
159* <a href="libs/libs.pdf">C/C++ Library Interfaces <img src="_static/pdf.png" /></a>
160* <a href="cpython/cpython.pdf">Python Technicalities <img src="_static/pdf.png" /></a></p>
161<p>General Information
162===================</p>
163<p>* <a href="spkg/spkg.pdf">External Packages <img src="_static/pdf.png" /></a>
164* <a href="references/references.pdf">Bibliographic References <img src="_static/pdf.png" /></a>
165* <a href="history_and_license/history_and_license.pdf">History and License <img src="_static/pdf.png" /></a></p>
166<p>Indices and Tables
167==================</p>
168<p>* :ref:`genindex`
169* :ref:`modindex`
170* :ref:`search`
171<h2>Table of Contents</h2>
172
173<ul><li><li><li><li><li><li><li><li><li><li>
174Sage Reference Manual
175<li><li><li><li><li><li><li><li><li><li><li><li><li><li><li><li><li><li><li><li><li>
176
177Welcome to the Sage reference manual. Here you find documentation for all of
178`Sage <http://www.sagemath.org/>`_'s features, illustrated with lots of examples.
179A thematic index follows.
180
181This documentation is licensed under the `Creative Commons Attribution-Share Alike 3.0 License`__:math:`.`
182
183__ http://creativecommons.org/licenses/by-sa/3.0/
184
185User Interfaces
186===============
187
188<li> <a href="repl/repl.pdf">Command Line Interface <img src="_static/pdf.png" /></a>
189<li> `Jupyter Notebook Interface <https://jupyter-notebook.readthedocs.io/en/latest/notebook.html>`_
190
191Graphics
192========
193
194<li> <a href="plotting/plotting.pdf">2D Graphics <img src="_static/pdf.png" /></a>
195<li> <a href="plot3d/plot3d.pdf">3D Graphics <img src="_static/pdf.png" /></a>
196
197Mathematics
198===========
199</ul>
200
201
202<h2>Parents and Categories</h2>
203
204<ul>
205
206<li> <a href="structure/structure.pdf">Parents and Elements <img src="_static/pdf.png" /></a>
207<li> <a href="coercion/coercion.pdf">Coercion <img src="_static/pdf.png" /></a>
208<li> <a href="categories/categories.pdf">Categories <img src="_static/pdf.png" /></a>
209</ul>
210
211
212<h2>Basic Rings and Fields</h2>
213
214<ul>
215
216<li> <a href="rings_standard/rings_standard.pdf">Integers and Rational Numbers <img src="_static/pdf.png" /></a>
217<li> <a href="rings_numerical/rings_numerical.pdf">Real and Complex Numbers <img src="_static/pdf.png" /></a>
218<li> <a href="polynomial_rings/polynomial_rings.pdf">Commutative Polynomials <img src="_static/pdf.png" /></a>
219<li> <a href="power_series/power_series.pdf">Power Series and Laurent Series <img src="_static/pdf.png" /></a>
220<li> <a href="finite_rings/finite_rings.pdf">Finite Rings and Fields <img src="_static/pdf.png" /></a>
221<li> <a href="padics/padics.pdf">\\(p\\)-adic Numbers <img src="_static/pdf.png" /></a>
222<li> <a href="noncommutative_polynomial_rings/noncommutative_polynomial_rings.pdf">Noncommutative Polynomials <img src="_static/pdf.png" /></a>
223<li> <a href="quat_algebras/quat_algebras.pdf">Quaternion Algebras <img src="_static/pdf.png" /></a>
224</ul>
225
226
227<h2>Linear Algebra</h2>
228
229<ul>
230
231<li> <a href="matrices/matrices.pdf">Matrices and Spaces of Matrices <img src="_static/pdf.png" /></a>
232<li> <a href="modules/modules.pdf">Vectors and Modules <img src="_static/pdf.png" /></a>
233<li> <a href="tensor_free_modules/tensor_free_modules.pdf">Tensors on Free Modules of Finite Rank <img src="_static/pdf.png" /></a>
234</ul>
235
236
237<h2>Calculus and Analysis</h2>
238
239<ul>
240
241<li> <a href="calculus/calculus.pdf">Symbolic Calculus <img src="_static/pdf.png" /></a>
242<li> <a href="constants/constants.pdf">Mathematical Constants <img src="_static/pdf.png" /></a>
243<li> <a href="functions/functions.pdf">Elementary and Special Functions <img src="_static/pdf.png" /></a>
244<li> <a href="asymptotic/asymptotic.pdf">Asymptotic Expansions <img src="_static/pdf.png" /></a>
245<li> <a href="numerical/numerical.pdf">Numerical Optimization <img src="_static/pdf.png" /></a>
246</ul>
247
248
249<h2>Probability and Statistics</h2>
250
251<ul>
252
253<li> <a href="probability/probability.pdf">Probability <img src="_static/pdf.png" /></a>
254<li> <a href="stats/stats.pdf">Statistics <img src="_static/pdf.png" /></a>
255<li> <a href="finance/finance.pdf">Quantitative Finance <img src="_static/pdf.png" /></a>
256</ul>
257
258
259<h2>Mathematical Structures</h2>
260
261<ul>
262
263<li> <a href="sets/sets.pdf">Sets <img src="_static/pdf.png" /></a>
264<li> <a href="monoids/monoids.pdf">Monoids <img src="_static/pdf.png" /></a>
265<li> <a href="groups/groups.pdf">Groups <img src="_static/pdf.png" /></a>
266<li> <a href="semirings/semirings.pdf">Semirings <img src="_static/pdf.png" /></a>
267<li> <a href="rings/rings.pdf">Rings <img src="_static/pdf.png" /></a>
268<li> <a href="algebras/algebras.pdf">Algebras <img src="_static/pdf.png" /></a>
269</ul>
270
271
272<h2>Discrete Mathematics</h2>
273
274<ul>
275
276<li> <a href="combinat/combinat.pdf">Combinatorics <img src="_static/pdf.png" /></a>
277<li> <a href="graphs/graphs.pdf">Graph Theory <img src="_static/pdf.png" /></a>
278<li> <a href="quivers/quivers.pdf">Quivers <img src="_static/pdf.png" /></a>
279<li> <a href="matroids/matroids.pdf">Matroid Theory <img src="_static/pdf.png" /></a>
280<li> <a href="dynamics/dynamics.pdf">Discrete Dynamics <img src="_static/pdf.png" /></a>
281<li> <a href="coding/coding.pdf">Coding Theory <img src="_static/pdf.png" /></a>
282<li> <a href="cryptography/cryptography.pdf">Cryptography <img src="_static/pdf.png" /></a>
283<li> <a href="game_theory/game_theory.pdf">Game Theory <img src="_static/pdf.png" /></a>
284<li> <a href="logic/logic.pdf">Symbolic Logic <img src="_static/pdf.png" /></a>
285<li> <a href="sat/sat.pdf">SAT solvers <img src="_static/pdf.png" /></a>
286</ul>
287
288
289<h2>Geometry and Topology</h2>
290
291<ul>
292
293<li> <a href="euclidean_spaces/euclidean_spaces.pdf">Euclidean Spaces and Vector Calculus <img src="_static/pdf.png" /></a>
294<li> <a href="discrete_geometry/discrete_geometry.pdf">Combinatorial and Discrete Geometry <img src="_static/pdf.png" /></a>
295<li> <a href="homology/homology.pdf">Cell Complexes and their Homology <img src="_static/pdf.png" /></a>
296<li> <a href="manifolds/manifolds.pdf">Manifolds and Differential Geometry <img src="_static/pdf.png" /></a>
297<li> <a href="hyperbolic_geometry/hyperbolic_geometry.pdf">Hyperbolic Geometry <img src="_static/pdf.png" /></a>
298<li> <a href="riemannian_geometry/riemannian_geometry.pdf">Parametrized Surfaces <img src="_static/pdf.png" /></a>
299<li> <a href="knots/knots.pdf">Knot Theory <img src="_static/pdf.png" /></a>
300</ul>
301
302
303<h2>Number Fields, Function Fields, and Valuations</h2>
304
305<ul>
306
307<li> <a href="number_fields/number_fields.pdf">Number Fields <img src="_static/pdf.png" /></a>
308<li> <a href="function_fields/function_fields.pdf">Function Fields <img src="_static/pdf.png" /></a>
309<li> <a href="valuations/valuations.pdf">Discrete Valuations <img src="_static/pdf.png" /></a>
310</ul>
311
312
313<h2>Number Theory</h2>
314
315<ul>
316
317<li> <a href="diophantine_approximation/diophantine_approximation.pdf">Diophantine approximation <img src="_static/pdf.png" /></a>
318<li> <a href="quadratic_forms/quadratic_forms.pdf">Quadratic Forms <img src="_static/pdf.png" /></a>
319<li> <a href="lfunctions/lfunctions.pdf">\\(L\\)-Functions <img src="_static/pdf.png" /></a>
320<li> <a href="arithgroup/arithgroup.pdf">Arithmetic Subgroups of \\({\\rm SL}_2(\\ZZ)\\) <img src="_static/pdf.png" /></a>
321<li> <a href="hecke/hecke.pdf">General Hecke Algebras and Hecke Modules <img src="_static/pdf.png" /></a>
322<li> <a href="modsym/modsym.pdf">Modular Symbols <img src="_static/pdf.png" /></a>
323<li> <a href="modfrm/modfrm.pdf">Modular Forms <img src="_static/pdf.png" /></a>
324<li> <a href="modfrm_hecketriangle/modfrm_hecketriangle.pdf">Modular Forms for Hecke Triangle Groups <img src="_static/pdf.png" /></a>
325<li> <a href="modabvar/modabvar.pdf">Modular Abelian Varieties <img src="_static/pdf.png" /></a>
326<li> <a href="modmisc/modmisc.pdf">Miscellaneous Modular-Form-Related Modules <img src="_static/pdf.png" /></a>
327</ul>
328
329
330<h2>Algebraic and Arithmetic Geometry</h2>
331
332<ul>
333<li> <a href="schemes/schemes.pdf">Schemes <img src="_static/pdf.png" /></a>
334<li> <a href="curves/curves.pdf">Plane and Space Curves <img src="_static/pdf.png" /></a>
335<li> <a href="arithmetic_curves/arithmetic_curves.pdf">Elliptic and Hyperelliptic Curves <img src="_static/pdf.png" /></a>
336</ul>
337
338
339<h2>Miscellaneous</h2>
340
341<ul>
342
343<li> <a href="databases/databases.pdf">Databases <img src="_static/pdf.png" /></a>
344<li> <a href="games/games.pdf">Games <img src="_static/pdf.png" /></a>
345
346Programming
347===========
348</ul>
349
350
351<h2>Facilities</h2>
352
353<ul>
354
355<li> <a href="data_structures/data_structures.pdf">Data Structures <img src="_static/pdf.png" /></a>
356<li> <a href="misc/misc.pdf">Utilities <img src="_static/pdf.png" /></a>
357<li> <a href="doctest/doctest.pdf">Test Framework <img src="_static/pdf.png" /></a>
358<li> <a href="parallel/parallel.pdf">Parallel Computing <img src="_static/pdf.png" /></a>
359</ul>
360
361
362<h2>Interfaces</h2>
363
364<ul>
365
366<li> <a href="interfaces/interfaces.pdf">Interpreter Interfaces <img src="_static/pdf.png" /></a>
367<li> <a href="libs/libs.pdf">C/C++ Library Interfaces <img src="_static/pdf.png" /></a>
368<li> <a href="cpython/cpython.pdf">Python Technicalities <img src="_static/pdf.png" /></a>
369
370General Information
371===================
372
373<li> <a href="spkg/spkg.pdf">External Packages <img src="_static/pdf.png" /></a>
374<li> <a href="references/references.pdf">Bibliographic References <img src="_static/pdf.png" /></a>
375<li> <a href="history_and_license/history_and_license.pdf">History and License <img src="_static/pdf.png" /></a>
376
377</ul>
378
379
380
381
382
383
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