"O(a^b)=o(a^b)=p-adic or power series zero with precision given by b",
"abs(x)=absolute value (or modulus) of x",
"acos(x)=inverse cosine of x",
"acosh(x)=inverse hyperbolic cosine of x",
"addell(e,z1,z2)=sum of the points z1 and z2 on elliptic curve e",
"adj(x)=adjoint matrix of x",
"agm(x,y)=arithmeticogeometric mean of x and y",
"akell(e,n)=computes the n-th Fourier coefficient of the L-function of the elliptic curve e",
"algdep(x,n)=algebraic relations up to degree n of x",
"algdep2(x,n,bit)=algebraic relations up to degree n of x where bit is as in \nlindep2",
"anell(e,n)=computes the first n Fourier coefficients of the L-function of\n the elliptic curve e (n<32768)",
"apell(e,p)=computes a_p for the elliptic curve e using Shanks-Mestre's method",
"apell2(e,p)=computes a_p for the elliptic curve e using Jacobi symbols",
"apprpadic(x,a)=p-adic roots of the polynomial x congruent to a mod p",
"arg(x)=argument of x,such that -pi<arg(x)<=pi",
"asin(x)=inverse sine of x",
"asinh(x)=inverse hyperbolic sine of x",
"assmat(x)=associated matrix to polynomial x",
"atan(x)=inverse tangent of x",
"atanh(x)=inverse hyperbolic tangent of x",
"base(x)=integral basis of the field Q[a], where a is a root of the polynomial x",
"bernreal(x)=Bernoulli number B_x, as a real number with the current precision",
"bernvec(x)=Vector of rational Bernoulli numbers B_0, B_2,... up to B_(2x)",
"bezout(x,y)= gives a 3-dimensional row vector [u,v,d] such that\nd=gcd(x,y) and u*x+v*y=d",
"bigomega(x)=number of repeated prime divisors of x",
"bin(x,y)=binomial coefficient x*(x-1)...*(x-y+1)/y! defined for y in Z and any x",
"binary(x)=gives the vector formed by the binary digits of x (x C-integer)",
"bittest(x,n)=gives bit number n (coefficient of 2^n) of the integer x",
"boundcf(x,lmax)=continued fraction expansion of x with at most lmax terms",
"boundfact(x,lim)=partial factorization of the integer x (using primes up to lim)",
"box(w,x1,y1,x2,y2)=draw a box with diagonal (x1,y1) and (x2,y2) in\nrectwindow w (cursor does not move)",
"buchimag(D,c,CO)=compute the structure of the class group of the complex\nquadratic field of discriminant D<0, where e.g. c=0.8 and CO=20 (see manual)",
"buchreal(D,n,c,CO)=compute the structure of the class group and the\nregulator of the real quadratic field of discriminant D>0, where n=0 (wide)\nn=1 (narrow) and e.g. c=0.8 and CO=20 (see manual)",
"ceil(x)=ceiling of x=smallest integer>=x",
"centerlift(x)=centered lift of x. Same as lift except for integermods",
"cf(x)=continued fraction expansion of x (x rational,real or rational function)",
"cf2(b,x)=continued fraction expansion of x (x rational,real or rational\nfunction), where b is the vector of numerators of the continued fraction",
"changevar(x,y)=change variables of x according to the vector y",
"char(x,y)=det(y*I-x)=characteristic polynomial of the matrix x using the\ncomatrix",
"char1(x,y)=det(y*I-x)=characteristic polynomial of the matrix x using Lagrange\ninterpolation",
"char2(x,y)=characteristic polynomial of the matrix x expressed with variable y,\nusing the Hessenberg form. Can be much faster or much slower than char, depending on the base ring",
"chell(x,y)=change data on elliptic curve according to y=[u,r,s,t]",
"chinese(x,y)=x,y being integers modulo mx and my,finds z such that\nz is congruent to x mod mx and y mod my",
"chptell(x,y)=change data on point or vector of points x on an elliptic curve\n according to y=[u,r,s,t]",
"classno(x)=class number of discriminant x",
"classno2(x)=class number of discriminant x",
"coeff(x,s)=coefficient of degree s of x, or the s-th component for vectors or matrices\n(for which it is simpler to use x[])",
"compo(x,s)=the s'th component of the internal representation of x.\nFor vectors or matrices, it is simpler to use x[]",
"compose(x,y)=Gaussian composition of the binary quadratic forms x and y of negative\ndiscriminant",
"comprealraw(x,y)=Gaussian composition without reduction of the binary quadratic\nforms x and y of positive discriminant",
"concat(x,y)=concatenation of x and y",
"conj(x)=the algebraic conjugate of x",
"content(x)=gcd of all the components of x, when this makes sense",
"convol(x,y)=convolution (or Hadamard product) of two power series",
"cos(x)=cosine of x",
"cosh(x)=hyperbolic cosine of x",
"cursor(w)=current position of cursor in rectwindow w",
"cvtoi(x)=truncation of x, without taking into account loss of integer part precision",
"cyclo(n)=n-th cyclotomic polynomial",
"denom(x)=denominator of x (or lowest common denominator in case of an array).",
"deriv(x,y)=derivative of x with respect to the main variable of y",
"det(x)=determinant of the matrix x",
"det2(x)=determinant of the matrix x (better for integer entries)",
"detr(x)=determinant of the matrix x having real entries",
"dilog(x)=dilogarithm of x",
"disc(x)=discriminant of the polynomial x",
"discf(x)=discriminant of the number field defined by the polynomial x",
"divisors(x)=gives a vector formed by the divisors of x in increasing order",
"divres(x,y)=euclidean division of x by y giving in a 2-dimensional column vector\nthe quotient and the remainder",
"divsum(n,X,expr)=sum of expression expr, X running over the divisors of n",
"draw(list)=draw vector of rectwindows list at indicated x,y positions;\nlist is a vector w1,x1,y1,w2,x2,y2,etc...",
"eigen(x)=eigenvectors of the matrix x",
"eint1(x)=exponential integral E1(x)",
"erfc(x)=complementary error function",
"eta(x)=eta function without the q^(1/24)",
"euler=euler()=euler's constant with current precision",
"eval(x)=evaluation of x, replacing variables by their value",
"exp(x)=exponential of x",
"extract(x,y)=extraction of the components of the vector x according to the vector\n or mask y, from left to right (1,2,4,8,...for the first,second,third,fourth,...component)",
"fact(x)=factorial of x (x C-integer), the result being given as a real number",
"factfq(x,p,a)=factorization of the polynomial x in the finite field\nF_p[X]/a(X)F_p[X]",
"factmod(x,p)=factorization mod p of the polynomial x",
"factor(x)=factorization of x",
"factoredbase(x,p)=integral basis of the maximal order defined by the polynomial x,\nwhere p is the matrix of the factorization of the discriminant of x",
"factoreddiscf(x,p)=discriminant of the maximal order defined by the polynomial x,\nwhere p is the matrix of the factorization of the discriminant of x",
"factoredpolred(x,p)=reduction of the polynomial x, where p is the matrix of the\nfactorization of the discriminant of x (gives minimal polynomials only)",
"factoredpolred2(x,p)=reduction of the polynomial x, where p is the matrix of the\nfactorization of the discriminant of x (gives elements and minimal polynomials)",
"factornf(x,t)=factorization of the polynomial x over the number field defined by the polynomial\nt, which must have X as main variable",
"factorpadic(x,p,r)=p-adic factorization of the polynomial x to precision r",
"factpol(x,l)=factorization over Z of the polynomial x up to degree l (complete if l=0)\nusing Hensel lift",
"factpol2(x,l)=factorization over Z of the polynomial x up to degree l (complete if l=0)\nusing root finding",
"fibo(x)=fibonacci number of index x (x C-integer)",
"floor(x)=floor of x=largest integer<=x",
"for(X=a,b,seq)=the sequence is evaluated, X going from a up to b",
"fordiv(n,X,seq)=the sequence is evaluated, X running over the divisors of n",
"forprime(X=a,b,seq)=the sequence is evaluated, X running over the primes between a and b",
"forstep(X=a,b,s,seq)=the sequence is evaluated, X going from a to b in steps of s",
"frac(x)=fractional part of x=x-floor(x)",
"galois(x)=Galois group of the polynomial x",
"galoisconj(x)=list of conjugates of a root of the polynomial x in the same\nnumber field (not always complete)",
"gamh(x)=gamma of x+1/2 (x integer)",
"gamma(x)=gamma function at x",
"gauss(a,b)=gaussian solution of ax=b (a matrix,b vector)",
"gcd(x,y)=greatest common divisor of x and y",
"globalred(e)=e being an elliptic curve, returns [N, [u, r, s, t]], where N\nis the conductor of e and [u, r, s, t] leads to the standard model for e",
"goto(n)=go to label number n",
"hclassno(x)=Hurwitz-Kronecker class number of x>0",
"hell(e,x)=archimedean height of point x on elliptic curve E defined by datavector e\ncomputed using theta-functions",
"hell2(e,x)=archimedean height of point x on elliptic curve E defined by datavector e\ncomputed using Tate's method",
"hell3(e,x)=archimedean height of point x on elliptic curve E defined by datavector e\ncomputed using theta-functions (for debugging only)",
"hermite(x)=(upper triangular) Hermite normal form of x, basis for the lattice formed\nby the columns of x",
"hess(x)=Hessenberg form of x",
"hilb(x,y,p)=Hilbert symbol at p of x,y (integers or fractions)",
"hilbert(n)=Hilbert matrix of order n (n C-integer)",
"hilbp(x,y)=Hilbert symbol of x,y (where x or y is integermod or p-adic)",
"hvector(n,X,expr)=horizontal vector with n components of expression expr,the variable X goes from 1 to n",
"incgam4(s,x,y)=incomplete gamma function where y=gamma(s) is precomputed",
"indexrank(x)=gives two extraction vectors (rows and columns) for the matrix x such\nthat the exracted matrix is square of maximal rank",
"indsort(x)=indirect sorting of the vector x",
"initalg(x)=x being a nonconstant irreducible polynomial, gives the vector:\n[pol,[r1,r2],discf,index,l2norm of roots,roots,integral basis]",
"initell(x)=x being the vector [a1,a2,a3,a4,a6], gives the vector:\n[a1,a2,a3,a4,a6,b2,b4,b6,b8,c4,c6,delta,j,[e1,e2,e3],w1,w2,eta1,eta2,q,area]",
"initell2(x)=x being the vector [a1,a2,a3,a4,a6], gives the vector:\n[a1,a2,a3,a4,a6,b2,b4,b6,b8,c4,c6,delta,j,[e1,e2,e3],w1,w2,eta1,eta2,q,area]",
"initrect(w,x,y)=initialize rectwindow w to size x,y",
"integ(x,y)=formal integration of x with respect to the main variable of y",
"intersect(x,y)=intersection of the vector spaces whose bases are the columns of x and y",
"intgen(X=a,b,s)=general numerical integration of s from a to b with respect to X, to be used after removing singularities",
"intinf(X=a,b,s)=numerical integration of s from a to b with respect to X, where a or b can be plus or minus infinity (1.0e4000), but of same sign",
"intnum(X=a,b,s)=numerical integration of s from a to b with respect to X",
"intopen(X=a,b,s)=numerical integration of s from a to b with respect to X, where s has only limits at a or b",
"inverseimage(x,y)=an element of the inverse image of the vector y by the\nmatrix x if one exists, the empty vector otherwise",
"isfund(x)=true(1) if x is a fundamental discriminant (including 1), false(0) if not",
"isincl(x,y)=tests whether the number field defined by the polynomial x is\nisomorphic to a subfield of the one defined by y; 0 if not, otherwise all\nthe isomorphisms",
"isisom(x,y)=tests whether the number field defined by the polynomial x is\nisomorphic to the one defined by y; 0 if not, otherwise all the isomorphisms",
"isoncurve(e,x)=true(1) if x is on elliptic curve e, false(0) if not",
"isprime(x)=true(1) if x is a strong pseudoprime for 10 random bases, false(0) if not",
"ispsp(x)=true(1) if x is a strong pseudoprime, false(0) if not",
"isqrt(x)=integer square root of x (x integer)",
"issqfree(x)=true(1) if x is squarefree, false(0) if not",
"issquare(x)=true(1) if x is a square, false(0) if not",
"jacobi(x)=eigenvalues and orthogonal matrix of eigenvectors of the real symmetric matrix x",
"jbesselh(n,x)=J-bessel function of index n+1/2 and argument x, where n is a non\n negative integer",
"jell(x)=elliptic j invariant of x",
"karamul(x,y,k)=karatsuba multiplication of polynomials x and y, recursively called\nk times",
"kbessel(nu,x)=K-bessel function of index nu and argument x (x positive real\nof type real,nu of any scalar type)",
"kbessel2(nu,x)=K-bessel function of index nu and argument x (x positive real\nof type real,nu of any scalar type)",
"ker(x)=basis of the kernel of the matrix x",
"keri(x)=basis of the kernel of the matrix x with integer entries",
"kerint(x)=LLL-reduced Z-basis of the kernel of the matrix x with integral\nentries using a modified LLL",
"kerint1(x)=LLL-reduced Z-basis of the kernel of the matrix x with rational\nentries using matrixqz3 and the HNF",
"kerint2(x)=LLL-reduced Z-basis of the kernel of the matrix x with integral\nentries using a modified LLL",
"kerr(x)=basis of the kernel of the matrix x with real or nonexact complex entries",
"kill(x)= kills the present value of the variable or function x. Returns new value or 0",
"killrect(w)=erase the rectwindow w",
"kro(x,y)=kronecker symbol (x/y)",
"label(n)=place at this point label number n",
"laplace(x)=replaces the power series sum of a_n*x^n/n! by sum of a_n*x^n",
"lcm(x,y)=least common multiple of x and y=x*y/gcd(x,y).",
"legendre(n)=legendre polynomial of degree n (n C-integer)",
"length(x)=number of non code words in x",
"lex(x,y)=compare x and y lexicographically (1 if x>y, 0 if x=y, -1 if x<y)",
"lexsort(x)=sort the elements of the vector x in ascending lexicographic order",
"lift(x)=lifts every element of Z/nZ to Z or Z[x]/PZ[x] to Z[x]",
"lindep(x)=Z-linear dependencies between components of x (Hastad et al)",
"lindep2(x,bit)=Z-linear dependencies between components of x using LLL, where\nbit should be about one half the number of bits of precision",
"line(w,x1,y1,x2,y2)=draw a line from x1,y1 to x2,y2 (and move the cursor)\nin the rectwindow w",
"lines(w,listx,listy)=draws an open polygon in rectwindow w where listx and\nlisty contain the x (resp. y) coordinates of the vertices",
"lll(x)=lll reduction of the vectors forming the matrix x",
"lll1(x)=old version of lll reduction of the vectors forming the matrix x",
"lllgen(x)=lll reduction of the vectors forming the matrix x with\npolynomial coefficients",
"lllgram(x)=lll reduction of the lattice whose gram matrix is x",
"lllgram1(x)=old version of lll reduction of the lattice whose gram matrix is x",
"lllgramgen(x)=lll reduction of the lattice whose gram matrix is x with\npolynomial coefficients",
"lllgramint(x)=lll reduction of the lattice whose gram matrix is the integral matrix x",
"lllgramkerim(x)=kernel and lll reduction of the lattice whose gram matrix is\nthe integral matrix x",
"lllgramkerimgen(x)=kernel and lll reduction of the lattice whose gram matrix is\nthe matrix x with polynomial coefficients",
"lllint(x)=lll reduction of the vectors forming the matrix x when the gram matrix\nis integral",
"lllkerim(x)=kernel and lll reduction of the vectors forming the integral matrix x",
"lllkerimgen(x)=kernel and lll reduction of the vectors forming the matrix x\nwith polynomial coefficients",
"lllrat(x)=lll reduction of the vectors forming the matrix x,computations done with rational numbers",
"ln(x)=log(x)=natural logarithm of x",
"lngamma(x)=logarithm of the gamma function of x",
"localred(e, p)= e being an ellliptic curve, returns [f, kod, [u, r, s, t]],\nwhere f is the conductor's exponent and kod is the kodaira type for e at p",
"log(x)=ln(x)=natural logarithm of x",
"logagm(x)=natural logarithm of x, computed using agm (faster than log for more\nthan a few hundred decimal digits)",
"lseriesell(e,s,N,A)=L-series at s of the elliptic curve e, where |N| is the conductor, sign(N) the sign\nof the functional equation, and A a cut-off point close to 1",
"mat(x)=transforms any GEN x into a matrix",
"matell(e,x)=gives the height matrix for vector of points x on elliptic curve e\nusing theta functions",
"matextract(x,y,z)=extraction of the components of the matrix x according to the\nvector or masks y (for the rows) and z (for the columns) from left to right (1,2,4,8,...for the\nfirst, second,third,fourth,...rows or columns)",
"matinvr(x)=inverse of the real matrix x",
"matlength(x)=number of rows and columns of the vector/matrix x as a 2-vector",
"matrix(m,n,X,Y,expr)=mXn matrix of expression expr, the row variable X going \nfrom 1 to m and the column variable Y going from 1 to n",
"matrixqz(x,p)=transforms the rational or integral mxn (m>=n) matrix x into an\nintegral matrix with gcd of maximal determinants equal to 1 if p is equal to 0, not \ndivisible by p otherwise",
"matrixqz2(x)=finds a basis of the intersection with Z^n of the lattice spanned by\nthe columns of x",
"matrixqz3(x)=finds a basis of the intersection with Z^n of the Q-vector space spanned by\nthe columns of x",
"max(x,y)=maximum of x and y",
"min(x,y)=minimum of x and y",
"minim(x,bound,maxnum)=number of vectors of square norm <= bound, maximum\nnorm and list of vectors for the integral and definite quadratic form x;\nminimal non-zero vectors if bound=0",
"mod(x,y)=creates the integer x modulo y on the PARI stack",
"modp(x,y)=creates the integer x modulo y as a permanent object (on the heap)",
"modreverse(x)=reverse polymod of the polymod x, if it exists",
"moveto(w,x,y)=move cursor to position x,y in rectwindow w",
"mu(x)=Moebius function of x",
"newtonpoly(x,p)=Newton polygon of polynomial x with respect to the prime p",
"nextprime(x)=smallest prime number>=x",
"norm(x)=norm of x",
"norml2(x)=square of the L2-norm of the vector x",
"nucomp(x,y,l)=composite of primitive positive definite quadratic forms x and y\nusing nucomp and nudupl, where l=[|D/4|^(1/4)] is precomputed",
"numdiv(x)=number of divisors of x",
"numer(x)=numerator of x.",
"nupow(x,n)=n-th power of primitive positive definite quadratic form x using\nnucomp and nudupl",
"o(a^b)=O(a^b)=p-adic or power series zero with precision given by b",
"omega(x)=number of unrepeated prime divisors of x",
"ordell(e,x)=y-coordinates corresponding to x-ordinate x on elliptic curve e",
"order(x)=order of the integermod x in (Z/nZ)*",
"ordred(x)=reduction of the polynomial x, staying in the same order",
"pascal(n)=pascal triangle of order n (n C-integer)",
"permutation(n,k)=permutation number k (mod n!) of n letters (n C-integer)",
"pf(x,p)=returns the prime form whose first coefficient is p, of discriminant x",
"phi(x)=Euler's totient function of x",
"pi=pi()=the constant pi, with current precision",
"plot(X=a,b,expr)=crude plot of expression expr, X goes from a to b",
"ploth(X=a,b,expr)=plot of expression expr, X goes from a to b in high resolution",
"ploth2(X=a,b,[expr1,expr2])=plot of points [expr1,expr2], X goes from a to b in high resolution",
"plothraw(listx,listy)=plot in high resolution points whose x (resp. y)\ncoordinates are in listx (resp. listy)",
"pnqn(x)=[p_n,p_{n-1};q_n,q_{n-1}] corresponding to the continued fraction x",
"point(w,x,y)=draw a point (and move cursor) at position x,y in rectwindow w",
"pointell(e,z)=coordinates of point on the curve e corresponding to the complex number z",
"points(w,listx,listy)=draws in rectwindow w the points whose x (resp y)\ncoordinates are in listx (resp listy)",
"polint(xa,ya,x)=polynomial interpolation at x according to data vectors xa, ya",
"polred(x)=reduction of the polynomial x (gives minimal polynomials only)",
"polred2(x)=reduction of the polynomial x (gives elements and minimal polynomials)",
"polsym(x,n)=vector of symmetric powers of the roots of x up to n",
"poly(x,v)=convert x (usually a vector or a power series) into a polynomial with variable v,\nstarting with the leading coefficient",
"polylog(m,x)=m-th polylogarithm of x",
"polylogd(m,x)=D_m~-modified m-th polylog of x",
"polylogdold(m,x)=D_m-modified m-th polylog of x",
"polylogp(m,x)=P_m-modified m-th polylog of x",
"polyrev(x,v)=convert x (usually a vector or a power series) into a polynomial with variable v,\nstarting with the constant term",
"postdraw(list)=same as draw, except that the output is a postscript\nprogram in the file pari.ps",
"postploth(X=a,b,expr)=same as ploth, except that the output is a postscript\nprogram in the file pari.ps",
"postploth2(X=a,b,[expr1,expr2])=same as ploth2, except that the output is a\npostscript program in the file pari.ps",
"postplothraw(listx,listy)=same as plothraw, except that the output is a\npostscript program in the file pari.ps",
"powell(e,n,x)=n times the point x on elliptic curve e (n in Z)",
"powrealraw(x,n)=n-th power without reduction of the binary quadratic form x of\npositive discriminant",
"pprint(a)=outputs a in beautified format ending with newline",
"pprint1(a)=outputs a in beautified format without ending with newline",
"prec(x,n)=change the precision of x to be n (n C-integer)",
"prime(n)=returns the n-th prime (n C-integer)",
"primes(n)=returns the vector of the first n primes (n C-integer)",
"primroot(n)=returns a primitive root of n when it exists",
"print(a)=outputs a in raw format ending with newline",
"print1(a)=outputs a in raw format without ending with newline",
"prod(x,X=a,b,expr)=x times the product (X runs from a to b) of expression",
"prodeuler(X=a,b,expr)=Euler product (X runs over the primes between a and b) of real or complex expression",
"prodinf(X=a,expr)=infinite product (X goes from a to infinity) of real or complex expression",
"prodinf1(X=a,expr)=infinite product (X goes from a to infinity) of real or complex 1+expression",
"psi(x)=psi-function at x",
"qfi(a,b,c)=binary quadratic form a*x^2+b*x*y+c*y^2 with b^2-4*a*c<0",
"qfr(a,b,c,d)=binary quadratic form a*x^2+b*x*y+c*y^2 with b^2-4*a*c>0 and distance d",
"quadgen(x)=standard generator of quadratic order of discriminant x",
"quadpoly(x)=quadratic polynomial corresponding to the discriminant x",
"random()=random integer between 0 and 2^31-1",
"rank(x)=rank of the matrix x",
"rbox(w,x,y)=draw a box with current position of cursor and x,y as diagonal\nin rectwindow w (cursor does not move)",
"read()=read an expression from the input file or standard input",
"real(x)=real part of x",
"recip(x)=reciprocal polynomial of x",
"redcomp(x)=reduction of the binary quadratic form x with D<0",
"redreal(x)=reduction of the binary quadratic form x with D>0",
"redrealnod(x,sq)=reduction of the binary quadratic form x with D>0 without\ndistance function where sq=[sqrt D]",
"regula(x)=regulator of the real quadratic field of discriminant x",
"reorder(x)=reorder the variables for output according to the vector x",
"resultant(x,y)=resultant of the polynomials x and y with exact entries",
"resultant2(x,y)=resultant of the polynomials x and y",
"reverse(x)=reversion of the power series x",
"rhoreal(x)=single reduction step of the binary quadratic form x of positive discriminant",
"rhorealnod(x,sq)=single reduction step of the binary quadratic form x with D>0\nwithout distance function where sq=[sqrt D]",
"rline(w,x,y)=draw a line from the present position of the cursor to x,y\n(and move the cursor) in the rectwindow w",
"rmoveto(w,x,y)=move cursor to position x,y relative to present position in\nrectwindow w",
"rndtoi(x)=take the nearest integer to all the coefficients of x,without taking into account loss of integer part precision",
"rootmod(x,p)=roots mod p of the polynomial x",
"rootmod2(x,p)=roots mod p of the polynomial x, when p is small",
"rootpadic(x,p,r)=p-adic roots of the polynomial x to precision r",
"roots(x)=roots of the polynomial x",
"rootslong(x)=roots of the polynomial x (takes more time, but more sturdy\nthan roots)",
"rootsof1(x)=number of roots of unity in the number field defined by x",
"round(x)=take the nearest integer to all the coefficients of x",
"rounderror(x)=maximum error found in rounding x",
"rpoint(w,x,y)=draw a point (and move cursor) at position x,y relative to\npresent position of the cursor in rectwindow w",
"series(x,v)=convert x (usually a vector) into a power series with variable v, starting with\nthe constant coefficient",
"setprecision(n)=set the current precision to n decimal digits if n>0, or\nreturn the current precision if n<=0",
"setserieslength(n)=set the default length of power series to n if n>0, or\nreturn the current default length if n<=0",
"shift(x,n)=shift x left n bits if n>=0, right -n bits if n<0",
"shiftmul(x,n)=multiply x by 2^n (n>=0 or n<0)",
"sigma(x)=sum of the divisors of x",
"sigmak(k,x)=sum of the k-th powers of the divisors of x (k C-integer)",
"sign(x)=sign of x, of type integer, real or fraction",
"signat(x)=signature of the symmetric matrix x",
"simplify(x)=simplify the object x as much as possible",
"sin(x)=sine of x",
"sinh(x)=hyperbolic sine of x",
"size(x)=maximum number of decimal digits minus one of (the coefficients of) x",
"smallbase(x)=integral basis of the field Q[a], where a is a root of the polynomial x where one\nassumes that no square of a prime>500000 divides the discriminant of x",
"smalldiscf(x)=discriminant of the number field defined by the polynomial x where one assumes that\n no square of a prime>500000 divides the discriminant of x",
"smallfact(x)=partial factorization of the integer x (using only the stored primes)",
"smallinitell(x)=x being the vector [a1,a2,a3,a4,a6], gives the vector:\n[a1,a2,a3,a4,a6,b2,b4,b6,b8,c4,c6,delta,j]",
"smallpolred(x)=partial reduction of the polynomial x (gives minimal polynomials only)",
"smallpolred2(x)=partial reduction of the polynomial x (gives elements and minimal polynomials)",
"smith(x)=Smith normal form (i.e. elementary divisors) of the matrix x, expressed as a vector",
"smith2(x)=Smith normal form (i.e. elementary divisors) of the matrix x, expressed as a vector",
"solve(X=a,b,expr)=real root of expression expr (X between a and b), where expr(a)*expr(b)<=0",
"sort(x)=sort in ascending order of the vector x",
"sqr(x)=square of x. NOT identical to x*x",
"sqred(x)=square reduction of the (symmetric) matrix x ( returns a square matrix whose i-th\ndiagonal term is the coefficient of the i-th square in which coeff of i-th variable is 1)",
"sqrt(x)=square root of x",
"srgcd(x,y)=polynomial gcd of x and y using the subresultant algorithm",
"string(w,x)=draw in rectwindow w the string corresponding to x, where x is\neither a string, or a number in R, written in format 9.3",
"sturm(x)=number of real roots of the polynomial x",
"sturmpart(x,a,b)=number of real roots of the polynomial x in the interval (a,b]",
"subell(e,z1,z2)=difference of the points z1 and z2 on elliptic curve e",
"subst(x,y,z)=substitute z for y in x",
"sum(x,X=a,b,expr)=x plus the sum (X goes from a to b) of expression expr",
"sumalt(X=a,expr)=euler's acceleration of alternating series expr, X starting at a",
"suminf(X=a,expr)=infinite sum (X goes from a to infinity) of real or complex expression expr",
"sumpos(X=a,expr)=sum of positive series expr, the formal variable\n X starting at a",
"supplement(x)=supplement the columns of the matrix x to an invertible matrix",
"tan(x)=tangent of x",
"tanh(x)=hyperbolic tangent of x",
"taylor(x,y)=taylor expansion of x with respect to the main variable of y",
"tchebi(n)=tchebitcheff polynomial of degree n (n C-integer)",
"tchirnhausen(x)=random Tchirnhausen transformation of the polynomial x",
"teich(x)=teichmuller character of p-adic number x",
"texprint(a)=outputs a in TeX format",
"theta(q,z)=Jacobi sine theta-function",
"thetanullk(q,k)=k'th derivative at z=0 of theta(q,z)",
"trace(x)=trace of x",
"trans(x)=x~=transpose of x",
"trunc(x)=truncation of x;when x is a power series,take away the O(X^)",
"type(x)=internal type number of the GEN x",
"unit(x)=fundamental unit of the quadratic field of discriminant x;\n x must be positive",
"until(a,seq)=evaluate the expression sequence seq until a is nonzero",
"valuation(x,p)=valuation of x with respect to p",
"vec(x)=transforms the object x into a vector. Used mainly if x is\na polynomial or a power series",
"vecsort(x,k)=sorts the vector of vector (or matrix) x according to the\nvalue of its k-th component",
"vector(n,X,expr)=horizontal vector with n components of expression expr (X goes from 1 to n)",
"vvector(n,X,expr)=vertical vector with n components of expression expr (X goes from 1 to n)",
"wf(x)=weber's f function of x (j=(f^24-16)^3/f^24)",
"wf2(x)=weber's f2 function of x (j=(f2^24+16)^3/f2^24)",
"while(a,seq)= while a is nonzero evaluate the expression sequence seq. Otherwise 0",
"zell(e,z)=In the complex case, lattice point corresponding to the point z on\nthe elliptic curve e",
"zeta(s)=Riemann zeta function at s",
"zzzz(...)=can be any function which is being tested. For developing purposes only."
