Sage Quick Reference
William Stein (based on work of P. Jipsen)
GNU Free Document License, extend for your own use
Notebook
Evaluate cell: shift-enter
Evaluate cell creating new cell: alt-enter
Split cell: control-;
Join cells: control-backspace
Insert math cell: click blue line between cells
Insert text/HTML cell: shift-click blue line between cells
Delete cell: delete content then backspace
Command line
com tab complete command
*bar*? list command names containing “bar”
command? tab shows documentation
command?? tab shows source code
a. tab shows methods for object a (more: dir(a))
a._ tab shows hidden methods for object a
search_doc("string or regexp") fulltext search of docs
search_src("string or regexp") search source code
_ is previous output
Numbers
Integers: Z = ZZ e.g. -2 -1 0 1 10^100
Rationals: Q = QQ e.g. 1/2 1/1000 314/100 -2/1
Reals: R ≈ RR e.g. .5 0.001 3.14 1.23e10000
Complex: C ≈ CC e.g. CC(1,1) CC(2.5,-3)
Double precision: RDF and CDF e.g. CDF(2.1,3)
Mod n: Z/nZ = Zmod e.g. Mod(2,3) Zmod(3)(2)
Finite fields: Fq = GF e.g. GF(3)(2) GF(9,"a").0
Polynomials: R[x, y] e.g. S.<x,y>=QQ[] x+2*y^3
Series: R[[t]] e.g. S.<t>=QQ[[]] 1/2+2*t+O(t^2)
p-adic numbers: Zp ≈Zp, Qp ≈Qp e.g. 2+3*5+O(5^2)
Algebraic closure: Q = QQbar e.g. QQbar(2^(1/5))
Interval arithmetic: RIF e.g. sage: RIF((1,1.00001))
Number field: R.<x>=QQ[];K.<a>=NumberField(x^3+x+1)
Arithmetic
ab = a*b a
b = a/b ab
= a^b
√
x = sqrt(x)
n
√
x = x^(1/n) |x| = abs(x) logb(x) = log(x,b)
Sums:
n
i=k
f(i) = sum(f(i) for i in (k..n))
Products:
n
i=k
f(i) = prod(f(i) for i in (k..n))
Constants and functions
Constants: π = pi e = e i = i ∞ = oo
φ = golden_ratio γ = euler_gamma
Approximate: pi.n(digits=18) = 3.14159265358979324
Functions: sin cos tan sec csc cot sinh cosh tanh
sech csch coth log ln exp ...
Python function: def f(x): return x^2
Interactive functions
Put @interact before function (vars determine controls)
@interact
def f(n=[0..4], s=(1..5), c=Color("red")):
var("x");show(plot(sin(n+x^s),-pi,pi,color=c))
Symbolic expressions
Define new symbolic variables: var("t u v y z")
Symbolic function: e.g. f(x) = x2
f(x)=x^2
Relations: f==g f<=g f>=g f<g f>g
Solve f = g: solve(f(x)==g(x), x)
solve([f(x,y)==0, g(x,y)==0], x,y)
factor(...) expand(...) (...).simplify_...
find_root(f(x), a, b) find x ∈ [a, b] s.t. f(x) ≈ 0
Calculus
lim
x→a
f(x) = limit(f(x), x=a)
d
dx (f(x)) = diff(f(x),x)
∂
∂x (f(x, y)) = diff(f(x,y),x)
diff = differentiate = derivative
f(x)dx = integral(f(x),x)
b
a
f(x)dx = integral(f(x),x,a,b)
b
a
f(x)dx ≈ numerical_integral(f(x),a,b)
Taylor polynomial, deg n about a: taylor(f(x),x,a,n)
2D graphics
-6 -4 -2 2 4 6
-1
-0.75
-0.5
-0.25
0.25
0.5
0.75
1
line([(x1,y1),. . .,(xn,yn)],options)
polygon([(x1,y1),. . .,(xn,yn)],options)
circle((x,y),r,options)
text("txt",(x,y),options)
options as in plot.options, e.g. thickness=pixel,
rgbcolor=(r,g,b), hue=h where 0 ≤ r, b, g, h ≤ 1
show(graphic, options)
use figsize=[w,h] to adjust size
use aspect_ratio=number to adjust aspect ratio
plot(f(x),(x, xmin, xmax),options)
parametric plot((f(t),g(t)),(t, tmin, tmax),options)
polar plot(f(t),(t, tmin, tmax),options)
combine: circle((1,1),1)+line([(0,0),(2,2)])
animate(list of graphics, options).show(delay=20)
3D graphics
line3d([(x1,y1,z1),. . .,(xn,yn,zn)],options)
sphere((x,y,z),r,options)
text3d("txt", (x,y,z), options)
tetrahedron((x,y,z),size,options)
cube((x,y,z),size,options)
octahedron((x,y,z),size,options)
dodecahedron((x,y,z),size,options)
icosahedron((x,y,z),size,options)
plot3d(f(x, y),(x, xb, xe), (y, yb, ye),options)
parametric plot3d((f,g,h),(t, tb, te),options)
parametric plot3d((f(u, v),g(u, v),h(u, v)),
(u, ub, ue),(v, vb, ve),options)
options: aspect ratio=[1, 1, 1], color="red"
opacity=0.5, figsize=6, viewer="tachyon"
Discrete math
x = floor(x) x = ceil(x)
Remainder of n divided by k = n%k k|n iff n%k==0
n! = factorial(n) x
m = binomial(x,m)
φ(n) = euler phi(n)
Strings: e.g. s = "Hello" = "He"+’llo’
s[0]="H" s[-1]="o" s[1:3]="el" s[3:]="lo"
Lists: e.g. [1,"Hello",x] = []+[1,"Hello"]+[x]
Tuples: e.g. (1,"Hello",x) (immutable)
Sets: e.g. {1, 2, 1, a} = Set([1,2,1,"a"]) (= {1, 2, a})
List comprehension ≈ set builder notation, e.g.
{f(x) : x ∈ X, x > 0} = Set([f(x) for x in X if x>0])
Graph theory
Graph: G = Graph({0:[1,2,3], 2:[4]})
Directed Graph: DiGraph(dictionary)
Graph families: graphs. tab
Invariants: G.chromatic polynomial(), G.is planar()
Paths: G.shortest path()
Visualize: G.plot(), G.plot3d()
Automorphisms: G.automorphism group(),
G1.is isomorphic(G2), G1.is subgraph(G2)
Combinatorics
Integer sequences: sloane find(list), sloane. tab
Partitions: P=Partitions(n) P.count()
Combinations: C=Combinations(list) C.list()
Cartesian product: CartesianProduct(P,C)
Tableau: Tableau([[1,2,3],[4,5]])
Words: W=Words("abc"); W("aabca")
Posets: Poset([[1,2],[4],[3],[4],[]])
Root systems: RootSystem(["A",3])
Crystals: CrystalOfTableaux(["A",3], shape=[3,2])
Lattice Polytopes: A=random_matrix(ZZ,3,6,x=7)
L=LatticePolytope(A) L.npoints() L.plot3d()
Matrix algebra
1
2
= vector([1,2])
1 2
3 4
= matrix(QQ,[[1,2],[3,4]], sparse=False)
1 2 3
4 5 6
= matrix(QQ,2,3,[1,2,3, 4,5,6])
1 2
3 4
= det(matrix(QQ,[[1,2],[3,4]]))
Av = A*v A−1
= A^-1 At
= A.transpose()
Solve Ax = v: A\v or A.solve_right(v)
Solve xA = v: A.solve_left(v)
Reduced row echelon form: A.echelon_form()
Rank and nullity: A.rank() A.nullity()
Hessenberg form: A.hessenberg_form()
Characteristic polynomial: A.charpoly()
Eigenvalues: A.eigenvalues()
Eigenvectors: A.eigenvectors_right() (also left)
Gram-Schmidt: A.gram_schmidt()
Visualize: A.plot()
LLL reduction: matrix(ZZ,...).LLL()
Hermite form: matrix(ZZ,...).hermite_form()
Linear algebra
Vector space Kn
= K^n e.g. QQ^3 RR^2 CC^4
Subspace: span(vectors, field )
E.g., span([[1,2,3], [2,3,5]], QQ)
Kernel: A.right_kernel() (also left)
Sum and intersection: V + W and V.intersection(W)
Basis: V.basis()
Basis matrix: V.basis_matrix()
Restrict matrix to subspace: A.restrict(V)
Vector in terms of basis: V.coordinates(vector)
Numerical mathematics
Packages: import numpy, scipy, cvxopt
Minimization: var("x y z")
minimize(x^2+x*y^3+(1-z)^2-1, [1,1,1])
Number theory
Primes: prime_range(n,m), is_prime, next_prime
Factor: factor(n), qsieve(n), ecm.factor(n)
Kronecker symbol: a
b = kronecker symbol(a,b)
Continued fractions: continued_fraction(x)
Bernoulli numbers: bernoulli(n), bernoulli mod p(p)
Elliptic curves: EllipticCurve([a1, a2, a3, a4, a6])
Dirichlet characters: DirichletGroup(N)
Modular forms: ModularForms(level, weight)
Modular symbols: ModularSymbols(level, weight, sign)
Brandt modules: BrandtModule(level, weight)
Modular abelian varieties: J0(N), J1(N)
Group theory
G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]])
SymmetricGroup(n), AlternatingGroup(n)
Abelian groups: AbelianGroup([3,15])
Matrix groups: GL, SL, Sp, SU, GU, SO, GO
Functions: G.sylow subgroup(p), G.character table(),
G.normal subgroups(), G.cayley graph()
Noncommutative rings
Quaternions: Q.<i,j,k> = QuaternionAlgebra(a,b)
Free algebra: R.<a,b,c> = FreeAlgebra(QQ, 3)
Python modules
import module name
module_name. tab and help(module_name)
Profiling and debugging
time command: show timing information
timeit("command"): accurately time command
t = cputime(); cputime(t): elapsed CPU time
t = walltime(); walltime(t): elapsed wall time
%pdb: turn on interactive debugger (command line only)
%prun command: profile command (command line only)