Introduction to SAGE for Number Theory

Pure and Applied Number Theory School2015.8.11

Cheolmin ParkNIMS

SAGE (System for Algebra and Geometry Experimenta-tion):

open-source mathematics software

Sage is a free open-source mathematics software system licensed under the GPL. It combines the power of many existing open-source packages into a common Python-based interface. ◦ Mission: Creating a viable free open source alter-

native to Magma, Maple, Mathematica and Mat-lab.

http://www.sagemath.org/ Current version: 6.8 Free, open source

Sage: what is it?

Sage provides ◦ Basic Algebra and Calculus

Solving Equations, Differentiation, … ◦ Plotting 2,3-dimensional plots ◦ Linear Algebra, Polynomials, Groups ◦ p-adic numbers ◦ Elliptic Curve, …

Easy to use

Sage: introduction

SAGE is Linux based software.

If you want to use SAGE on Window OS, you need to install VirtualBox for Window

which can run linux in Window OS.

Refer to “install guide” in www.sagemath.org/

We use SAGE Online!!!◦ https://cloud.sagemath.com/

Installing SAGE

https://cloud.sagemath.com/

IE9 cannot connect to this site. In this case, use higher version of IE or chrome

browser.

If you already have account, sign in

If not, you need to create account.◦ It is for free.◦ Click item “First, agree to the Terms of Service”◦ Write Name/Email/ Choose a password◦ Click item “Create account for free”

https://cloud.sagemath.com/

Create New Project…

Title…and click “create project”

Click item “⨁New”

https://cloud.sagemath.com/projects

Click “Sage Worksheet”

A very brief Tour of Sage

Run code: click or shift+Enter◦ e.g. 2+3 shift+Enter

Sage uses = for assignment Comparison check: ==, <=, >=, <, > Basic arithmetic

◦ **, ^: exponentiation ( 2^3 == 2**3)◦ %: remainder (10%3)◦ //: quotient◦ sqrt(10), sin(5), sin(pi)

Run, Assignment, Comparison, Arithmetic

Sage use for calling function ◦ variable.function() or function(variable)◦ (eg1) n=2015 n.factor() factor(n)◦ (eg2) M=matrix(2, 2, [1,pi, e,5]) M.det() det(M)

For defining function,

Call and define function

(eg1)def f(a,b): return a+b

(eg2)def fm(a,b): if a >=b: c=a else: c=b return c

Note: indentation is important.

var('x y') f = x^3 * e^(y*x) * sin(y*x); f.diff(x) latex(f.diff(x)) show(f.diff(x)) f(x,3): You can evaluate f at y=3 plot(f(x,3))

Calculus

CC: complex field RR: real field QQ: rational field ZZ: integer ring Integers(n): ring Z/nZ of integers modulo n GF(p): finite field of order p GF(p^n): finite field of order p^n QQ[x]: Univariate Polynomial Ring in x over Ra-

tional Field QQ[x,y]: Multivariate Polynomial Ring in x, y

over Rational Field

Rings and Fields

k.<x> = CC[] (k is univariate polynomial ring in x over complex field)

f=x^3+x factor(f) f.roots()

Change field and try factor(f), f.roots()

Example of Rings and Fields

EllipticCurve(fields(rings),[a1,a2,a3,a4,a5]):Elliptic Curve defined by y^2 + a1*x*y +

a3*y = x^3 + a2*x^2 + a4*x + a5 over Fields

Example ◦ EllipticCurve(QQ,[10,20,30,40,50])◦ k=EllipticCurve(GF(127),[10,20,30,40,50])◦ k.order(), P=k.random_point(), Q=k(75,4)◦ 123*P, P+Q

Elliptic curves

mod(8+5, 7) == (8+5)%7 : (8+5) mod 7 pow(3,6,11) : 3^6 mod 11 inverse_mod(2,19): 2^-1 mod 19 gcd(12, 36); xgcd(3,5) functions related to prime

◦ prime_factor(n), divisors(n), next_prime(n), nth_prime(i), is_prime(n), prime_pi(n) (# of primes less than n), euler_phi(n), primitive_root(p)…

x=crt(y1,y2,p1,p2): x mod p1 =y1, x mod p2 =y2

Basic number functions

P.<x>=QQ[] K.<a>=NumberField(x^2+1) R.<y> = K[]

f = y^2 + y + 1 L.<b> = K.extension(f)

S.<c>=CyclotomicField(n)

Number field

P.<x>=QQ[] K.<a>=NumberField(x^2+1)

OK=K.ring_of_integers() O3 = K.order(3*a); O3 #Z+3aZ O3.gens()

Order

K.<a> = NumberField(x^2 + 23) I = K.fractional_ideal(2, 1/2*a - 1/2) J = I^2 I*J factor(I) factor(J) is_prime(I) I = K.fractional_ideal(2) I.factor() K.class_group()

Fractional ideal of Number field

You can use reference manual or quick search in www.sage.org

use help() in Sage

function??: can see source code of function◦ factor??

Tab completion: Type obj followed by tab to see all completions of obj.◦ fac+tab

Help in SAGE

Thank you!