An Introduction to Sage - College of Natural Sciences (CNS ...jli/talks/introduction...l Uses the Python programming language Jennifer Li(Louisiana State University) An Introduction

An Introduction to Sage

Jennifer Li

Department of Mathematics

Louisiana State University

Baton Rouge

What is Sage?

l Mathematical softwarel Originated by William Stein (University of Washington)l System for Algebra and Geometry Experimentationl Uses the Python programming language

Jennifer Li (Louisiana State University) An Introduction to Sage May 2, 2015 2 / 83

What is Sage?

l Mathematical software

l Originated by William Stein (University of Washington)l System for Algebra and Geometry Experimentationl Uses the Python programming language


What is Sage?

l Mathematical softwarel Originated by William Stein (University of Washington)

l System for Algebra and Geometry Experimentationl Uses the Python programming language


What is Sage?

l Mathematical softwarel Originated by William Stein (University of Washington)l System for Algebra and Geometry Experimentation

l Uses the Python programming language


What is Sage?

l Mathematical softwarel Originated by William Stein (University of Washington)l System for Algebra and Geometry Experimentationl Uses the Python programming language


Why use Sage?

l Free and open source!l Initial goal:

"open source alternative to Magma, Maple, Mathematica, and MATLAB"

lAllows review, reuse, and editing of previous computationsl Covers many areas of mathematics:

abstract algebracalculuscombinatoricslinear algebranumerical mathematicsnumber theory


Why use Sage?

l Free and open source!

l Initial goal:





Why use Sage?






Why use Sage?



lAllows review, reuse, and editing of previous computations

l Covers many areas of mathematics:abstract algebracalculuscombinatoricslinear algebranumerical mathematicsnumber theory


Why use Sage?






Why use Sage?




abstract algebra

calculuscombinatoricslinear algebranumerical mathematicsnumber theory


Why use Sage?




abstract algebracalculus

combinatoricslinear algebranumerical mathematicsnumber theory


Why use Sage?




abstract algebracalculuscombinatorics

linear algebranumerical mathematicsnumber theory


Why use Sage?




abstract algebracalculuscombinatoricslinear algebra

numerical mathematicsnumber theory


Why use Sage?




abstract algebracalculuscombinatoricslinear algebranumerical mathematics

number theory


Why use Sage?






Downloading: Basic Idea

l www.sagemath.org



l www.sagemath.org



l Need virtual machine: Oracle VM Virtual Box Managerl Computer inside of a computer.



l Need virtual machine: Oracle VM Virtual Box Manager

l Computer inside of a computer.



l Need virtual machine: Oracle VM Virtual Box Managerl Computer inside of a computer.


Getting Started

l Choose New Worksheet


Getting Started

l Name the worksheet


Getting Started

l New worksheet


Getting Started

l Top left corner:


Getting Started

l Option to choose a di�erent software package


Getting Started


Software Packages in Sage


Evaluating Sage Commands

l Rectangular box = celll Click on cell to �activate"



l Rectangular box = cell

l Click on cell to �activate"



l Rectangular box = celll Click on cell to �activate"



l Type command(s)



l Hit Evaluate



l Sage returns result directly below cell with command


Functions

l De�ning functions is straightforward


Functions

l We can check the de�nition of f (x) by asking Sage what f (x) is


Functions

l Sage returns the correct de�nition


Functions

l Other variables may also be used

B


Functions

l Other variables may also be used B


Functions

l Click in the green box.


Functions

l More details about error are given.


Functions

l To use a new variable:


Comments

l Comments can be inserted above each cell.


New Cells

l Click icon for new cell.


Cell Memory


Plotting Graphs in 2D


A Cool Trick

l Command? Shift+Enter


A Cool Trick



l Right click to save image.



l Click and drag on image to rotate!


Graphs


Modular Arithmetic

l Ring of integers modulo n is available in Sage

l What are the elements?


Modular Arithmetic

l Addition in Z/nZ:


Modular Arithmetic

l For multiple commands on one line, separate with ';'


Modular Arithmetic

l Hitting Evaluate returns results for all commands


Groups

l Many groups are prede�ned.l Example: Symmetric group of n elements.


Groups

l Many groups are prede�ned.

l Example: Symmetric group of n elements.


Groups

l Many groups are prede�ned.l Example: Symmetric group of n elements.


Groups

l What are the elements in this group?


Groups

l Example: Symmetric Group of �ve elements.


Groups

l What is the size of this group?


Groups

l What are the elements in this group?


Groups

l Right click on .txt �le


Groups

l Open link in new tab or new window


Groups

l Save or copy and paste text


Other Group Functions

l Is a group abelian?


Another Cool Trick

l Object. Tab


Another Cool Trick

l All commands for a group!


Other Prede�ned Groups

l Alternating Groups (all even permutations of n elements)l Dihedral Groups (symmetries of n-gon)l Klein Four Group



l Alternating Groups (all even permutations of n elements)

l Dihedral Groups (symmetries of n-gon)l Klein Four Group



l Alternating Groups (all even permutations of n elements)l Dihedral Groups (symmetries of n-gon)

l Klein Four Group



l Alternating Groups (all even permutations of n elements)l Dihedral Groups (symmetries of n-gon)l Klein Four Group


Even More Group Functions

l Sage can construct the multiplication table, or Cayley table, of a group


Finite Fields

l Sage knows about �nite �elds

l Another way to de�ne a �nite �eld:


Finite Fields

l All �nite �elds must have prime power order.


Elliptic Curves

l All elliptic curves can be written in Weierstrass form:

y2+a1xy +a3y = x3+a2x2+a4x +a6


Elliptic Curves

l There are many ways to construct an elliptic curve in Sage

l One way: simply enter the coe�cients a1,a2,a3,a4,a6


Elliptic Curves

l There are many ways to construct an elliptic curve in Sagel One way: simply enter the coe�cients a1,a2,a3,a4,a6


Elliptic Curves

l Where do the coe�cients live?

l If ai are all integers: Ql Elliptic Curves over �nite �elds:


Elliptic Curves

l Where do the coe�cients live?l If ai are all integers: Q

l Elliptic Curves over �nite �elds:


Elliptic Curves

l Where do the coe�cients live?l If ai are all integers: Ql Elliptic Curves over �nite �elds:


Plotting Elliptic Curves


Databases in Sage

l There are many mathematical databases in Sage.

l An example: Cremona Tables

l Cremona number −→ More information about curve


Databases in Sage

l There are many mathematical databases in Sage.l An example: Cremona Tables

l Cremona number −→ More information about curve


Saving

l Save options are at the top right corner.


Closing Sage


SageMathCloud


Conclusion

l Sage is great!Quick calculationsExtensive projectsBeautiful plotsSpecialized topicsE�cient collaborationNo costYou can contribute!


Conclusion

l Sage is great!

Quick calculationsExtensive projectsBeautiful plotsSpecialized topicsE�cient collaborationNo costYou can contribute!


Conclusion

l Sage is great!Quick calculations

Extensive projectsBeautiful plotsSpecialized topicsE�cient collaborationNo costYou can contribute!


Conclusion

l Sage is great!Quick calculationsExtensive projects

Beautiful plotsSpecialized topicsE�cient collaborationNo costYou can contribute!


Conclusion

l Sage is great!Quick calculationsExtensive projectsBeautiful plots

Specialized topicsE�cient collaborationNo costYou can contribute!


Conclusion

l Sage is great!Quick calculationsExtensive projectsBeautiful plotsSpecialized topics

E�cient collaborationNo costYou can contribute!


Conclusion

l Sage is great!Quick calculationsExtensive projectsBeautiful plotsSpecialized topicsE�cient collaboration

No costYou can contribute!


Conclusion

l Sage is great!Quick calculationsExtensive projectsBeautiful plotsSpecialized topicsE�cient collaborationNo cost

You can contribute!


Conclusion

l Sage is great!Quick calculationsExtensive projectsBeautiful plotsSpecialized topicsE�cient collaborationNo costYou can contribute!


Documents

An Introduction to Sage - College of Natural Sciences (CNS ...jli/talks/introduction...l Uses the Python programming language Jennifer Li(Louisiana State University) An Introduction