PARI C-library interface¶

AUTHORS:

William Stein (2006-03-01): updated to work with PARI 2.2.12-beta (this involved changing almost every doc string, among other things; the precision behavior of PARI seems to change from any version to the next...).
William Stein (2006-03-06): added newtonpoly
Justin Walker: contributed some of the function definitions
Gonzalo Tornaria: improvements to conversions; much better error handling.

EXAMPLES:

sage: pari('5! + 10/x')
(120*x + 10)/x
sage: pari('intnum(x=0,13,sin(x)+sin(x^2) + x)')
85.1885681951527
sage: f = pari('x^3-1')
sage: v = f.factor(); v
[x - 1, 1; x^2 + x + 1, 1]
sage: v[0]   # indexing is 0-based unlike in GP.
[x - 1, x^2 + x + 1]~
sage: v[1]
[1, 1]~    

Arithmetic obeys the usual coercion rules.

sage: type(pari(1) + 1)
<type 'sage.libs.pari.gen.gen'>
sage: type(1 + pari(1))
<type 'sage.libs.pari.gen.gen'>    

GUIDE TO REAL PRECISION AND THE PARI LIBRARY

The default real precision in communicating with the Pari library is the same as the default Sage real precision, which is 53 bits. Inexact Pari objects are therefore printed by default to 15 decimal digits (even if they are actually more precise).

Default precision example (53 bits, 15 significant decimals):

sage: a = pari(1.23); a
1.23000000000000
sage: a.sin()
0.942488801931698

Example with custom precision of 200 bits (60 significant decimals):

sage: R = RealField(200)
sage: a = pari(R(1.23)); a   # only 15 significant digits printed
1.23000000000000
sage: R(a)         # but the number is known to precision of 200 bits
1.2300000000000000000000000000000000000000000000000000000000
sage: a.sin()      # only 15 significant digits printed
0.942488801931698  
sage: R(a.sin())   # but the number is known to precision of 200 bits
0.94248880193169751002382356538924454146128740562765030213504

It is possible to change the number of printed decimals:

sage: R = RealField(200)    # 200 bits of precision in computations
sage: old_prec = pari.set_real_precision(60)  # 60 decimals printed
sage: a = pari(R(1.23)); a
1.23000000000000000000000000000000000000000000000000000000000
sage: a.sin()
0.942488801931697510023823565389244541461287405627650302135038
sage: pari.set_real_precision(old_prec)  # restore the default printing behavior
60

Unless otherwise indicated in the docstring, most Pari functions that return inexact objects use the precision of their arguments to decide the precision of the computation. However, if some of these arguments happen to be exact numbers (integers, rationals, etc.), an optional parameter indicates the precision (in bits) to which these arguments should be converted before the computation. If this precision parameter is missing, the default precision of 53 bits is used. The following first converts 2 into a real with 53-bit precision:

sage: R = RealField()
sage: R(pari(2).sin())
0.909297426825682

We can ask for a better precision using the optional parameter:

sage: R = RealField(150)
sage: R(pari(2).sin(precision=150))
0.90929742682568169539601986591174484270225497

Warning regarding conversions Sage - Pari - Sage: Some care must be taken when juggling inexact types back and forth between Sage and Pari. In theory, calling p=pari(s) creates a Pari object p with the same precision as s; in practice, the Pari library’s precision is word-based, so it will go up to the next word. For example, a default 53-bit Sage real s will be bumped up to 64 bits by adding bogus 11 bits. The function p.python() returns a Sage object with exactly the same precision as the Pari object p. So pari(s).python() is definitely not equal to s, since it has 64 bits of precision, including the bogus 11 bits. The correct way of avoiding this is to coerce pari(s).python() back into a domain with the right precision. This has to be done by the user (or by Sage functions that use Pari library functions in gen.pyx). For instance, if we want to use the Pari library to compute sqrt(pi) with a precision of 100 bits:

sage: R = RealField(100)
sage: s = R(pi); s
3.1415926535897932384626433833
sage: p = pari(s).sqrt()
sage: x = p.python(); x  # wow, more digits than I expected!
1.7724538509055160272981674833410973484
sage: x.prec()           # has precision 'improved' from 100 to 128?
128
sage: x == RealField(128)(pi).sqrt()  # sadly, no!
False
sage: R(x)               # x should be brought back to precision 100
1.7724538509055160272981674833
sage: R(x) == s.sqrt()
True

Elliptic curves and precision: If you are working with elliptic curves and want to compute with a precision other than the default 53 bits, you should use the precision parameter of ellinit():

sage: R = RealField(150)
sage: e = pari([0,0,0,-82,0]).ellinit(precision=150)
sage: eta1 = e.elleta()[0]
sage: R(eta1)
3.6054636014326520863839536934492002728802618

Number fields and precision: TODO

exception sage.libs.pari.gen.PariError¶

Bases: exceptions.RuntimeError

errmessage(d)¶

class sage.libs.pari.gen.PariInstance¶

Bases: sage.structure.parent_base.ParentWithBase

allocatemem(s=0, silent=False)¶: Double the PARI stack.

complex(re, im)¶: Create a new complex number, initialized from re and im.

default(variable, value=None)¶

double_to_gen(x)¶

euler(precision=0)¶

Return Euler’s constant to the requested real precision (in bits).

EXAMPLES:

sage: pari.euler()
0.577215664901533
sage: pari.euler(precision=100).python()
0.577215664901532860606512090082...

factorial(n)¶

Return the factorial of the integer n as a PARI gen.

EXAMPLES:

sage: pari.factorial(0)
1
sage: pari.factorial(1)
1
sage: pari.factorial(5)
120
sage: pari.factorial(25)
15511210043330985984000000

get_debug_level()¶: Set the debug PARI C library variable.

get_real_precision()¶

Returns the current PARI default real precision.

This is used both for creation of new objects from strings and for printing. It is the number of digits IN DECIMAL in which real numbers are printed. It also determines the precision of objects created by parsing strings (e.g. pari(‘1.2’)), which is not the normal way of creating new pari objects in Sage. It has no effect on the precision of computations within the pari library.

get_series_precision()¶

getrand()¶

Returns Pari’s current random number seed.

EXAMPLES:

sage: pari.setrand(50)
sage: pari.getrand()
50
sage: pari.pari_rand31()
621715893
sage: pari.getrand()
621715893

init_primes(_M)¶

Recompute the primes table including at least all primes up to M (but possibly more).

EXAMPLES:

sage: pari.init_primes(200000)

listcreate(n)¶

listcreate(n): return an empty pari list of maximal length n.

EXAMPLES:

sage: pari.listcreate(20)
List([])

matrix(m, n, entries=None)¶: matrix(long m, long n, entries=None): Create and return the m x n PARI matrix with given list of entries.

new_with_bits_prec(s, precision)¶: pari.new_with_bits_prec(self, s, precision) creates s as a PARI gen with (at most) precision bits of precision.

nth_prime(n)¶

pari_rand31()¶: Returns a random number from Pari’s random number generator.

Warning

You probably don’t want to use this; it’s a very poor random number generator. Sage exposes it only as a way to test getrand() and setrand().

pari_version()¶

pi(precision=0)¶

Return the value of the constant pi to the requested real precision (in bits).

EXAMPLES:

sage: pari.pi()
3.14159265358979
sage: pari.pi(precision=100).python()
3.1415926535897932384626433832...

polcyclo(n, v=-1)¶

polcyclo(n, v=x): cyclotomic polynomial of degree n, in variable v.

EXAMPLES:

sage: pari.polcyclo(8)
x^4 + 1
sage: pari.polcyclo(7, 'z')
z^6 + z^5 + z^4 + z^3 + z^2 + z + 1
sage: pari.polcyclo(1)
x - 1

pollegendre(n, v=-1)¶

pollegendre(n, v=x): Legendre polynomial of degree n (n C-integer), in variable v.

EXAMPLES:

sage: pari.pollegendre(7)
429/16*x^7 - 693/16*x^5 + 315/16*x^3 - 35/16*x
sage: pari.pollegendre(7, 'z')
429/16*z^7 - 693/16*z^5 + 315/16*z^3 - 35/16*z
sage: pari.pollegendre(0)
1

polsubcyclo(n, d, v=-1)¶

polsubcyclo(n, d, v=x): return the pari list of polynomial(s) defining the sub-abelian extensions of degree $d$ of the cyclotomic field $\QQ(\zeta_n)$ , where $d$ divides $\phi(n)$ .

EXAMPLES:

sage: pari.polsubcyclo(8, 4)
[x^4 + 1]
sage: pari.polsubcyclo(8, 2, 'z')
[z^2 - 2, z^2 + 1, z^2 + 2]
sage: pari.polsubcyclo(8, 1)
[x - 1]
sage: pari.polsubcyclo(8, 3)
[]

poltchebi(n, v=-1)¶

poltchebi(n, v=x): Chebyshev polynomial of the first kind of degree n, in variable v.

EXAMPLES:

sage: pari.poltchebi(7)
64*x^7 - 112*x^5 + 56*x^3 - 7*x
sage: pari.poltchebi(7, 'z')
64*z^7 - 112*z^5 + 56*z^3 - 7*z
sage: pari.poltchebi(0)
1

polzagier(n, m)¶

prime_list(n)¶

prime_list(n): returns list of the first n primes

To extend the table of primes use pari.init_primes(M).

INPUT:

n - C long

OUTPUT:

gen - PARI list of first n primes

EXAMPLES:

sage: pari.prime_list(0)
[]
sage: pari.prime_list(-1)
[]
sage: pari.prime_list(3)
[2, 3, 5]
sage: pari.prime_list(10)
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
sage: pari.prime_list(20)
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71]
sage: len(pari.prime_list(1000))
1000

primes_up_to_n(n)¶

Return the primes <= n as a pari list.

EXAMPLES:

sage: pari.primes_up_to_n(1)
[]
sage: pari.primes_up_to_n(20)
[2, 3, 5, 7, 11, 13, 17, 19]

read(filename)¶

Read a script from the named filename into the interpreter, where s is a string. The functions defined in the script are then available for use from Sage/PARI.

EXAMPLE:

If foo.gp is a script that contains

{foo(n) =
    n^2
}

and you type read("foo.gp"), then the command pari("foo(12)") will create the Python/PARI gen which is the integer 144.

CONSTRAINTS: The PARI script must not contain the following function calls:

print, default, ??? (please report any others that cause trouble)

set_debug_level(level)¶: Set the debug PARI C library variable.

set_real_precision(n)¶

Sets the PARI default real precision.

This is used both for creation of new objects from strings and for printing. It is the number of digits IN DECIMAL in which real numbers are printed. It also determines the precision of objects created by parsing strings (e.g. pari(‘1.2’)), which is not the normal way of creating new pari objects in Sage. It has no effect on the precision of computations within the pari library.

Returns the previous PARI real precision.

set_series_precision(n)¶

setrand(seed)¶

Sets Pari’s current random number seed.

This should not be called directly; instead, use Sage’s global random number seed handling in sage.misc.randstate and call current_randstate().set_seed_pari().

EXAMPLES:

sage: pari.setrand(12345)
sage: pari.getrand()
12345

vector(n, entries=None)¶

vector(long n, entries=None): Create and return the length n PARI vector with given list of entries.

EXAMPLES:

sage: pari.vector(5, [1, 2, 5, 4, 3])
[1, 2, 5, 4, 3]
sage: pari.vector(2, [x, 1])
[x, 1]
sage: pari.vector(2, [x, 1, 5])
...
IndexError: length of entries (=3) must equal n (=2)

class sage.libs.pari.gen.gen¶

Bases: sage.structure.element.RingElement

Python extension class that models the PARI GEN type.

Col(x)¶

Col(x): Transforms the object x into a column vector.

The vector will have only one component, except in the following cases:

When x is a vector or a quadratic form, the resulting vector is the initial object considered as a column vector.
When x is a matrix, the resulting vector is the column of row vectors comprising the matrix.
When x is a character string, the result is a column of individual characters.
When x is a polynomial, the coefficients of the vector start with the leading coefficient of the polynomial.
When x is a power series, only the significant coefficients are taken into account, but this time by increasing order of degree.

INPUT:

x - gen

OUTPUT: gen

EXAMPLES:

sage: pari(1.5).Col()
[1.50000000000000]~
sage: pari([1,2,3,4]).Col()
[1, 2, 3, 4]~
sage: pari('[1,2; 3,4]').Col()
[[1, 2], [3, 4]]~
sage: pari('"Sage"').Col()
["S", "a", "g", "e"]~
sage: pari('3*x^3 + x').Col()
[3, 0, 1, 0]~
sage: pari('x + 3*x^3 + O(x^5)').Col()
[1, 0, 3, 0]~

List(x)¶

List(x): transforms the PARI vector or list x into a list.

EXAMPLES:

sage: v = pari([1,2,3])
sage: v
[1, 2, 3]
sage: v.type()
't_VEC'
sage: w = v.List()
sage: w
List([1, 2, 3])
sage: w.type()
't_LIST'

Mat(x)¶

Mat(x): Returns the matrix defined by x.

If x is already a matrix, a copy of x is created and returned.
If x is not a vector or a matrix, this function returns a 1x1 matrix.
If x is a row (resp. column) vector, this functions returns a 1-row (resp. 1-column) matrix, unless all elements are column (resp. row) vectors of the same length, in which case the vectors are concatenated sideways and the associated big matrix is returned.

INPUT:

x - gen

OUTPUT:

gen - a PARI matrix

EXAMPLES:

sage: x = pari(5)
sage: x.type()
't_INT'
sage: y = x.Mat()
sage: y
Mat(5)
sage: y.type()
't_MAT'
sage: x = pari('[1,2;3,4]')
sage: x.type()
't_MAT'
sage: x = pari('[1,2,3,4]')
sage: x.type()
't_VEC'
sage: y = x.Mat()
sage: y
Mat([1, 2, 3, 4])
sage: y.type()
't_MAT'

sage: v = pari('[1,2;3,4]').Vec(); v
[[1, 3]~, [2, 4]~]
sage: v.Mat()
[1, 2; 3, 4]
sage: v = pari('[1,2;3,4]').Col(); v
[[1, 2], [3, 4]]~
sage: v.Mat()
[1, 2; 3, 4]

Mod(x, y)¶

Mod(x, y): Returns the object x modulo y, denoted Mod(x, y).

The input y must be a an integer or a polynomial:

If y is an INTEGER, x must also be an integer, a rational number, or a p-adic number compatible with the modulus y.
If y is a POLYNOMIAL, x must be a scalar (which is not a polmod), a polynomial, a rational function, or a power series.

Warning

This function is not the same as x % y which is an integer or a polynomial.

INPUT:

x - gen
y - integer or polynomial

OUTPUT:

gen - intmod or polmod

EXAMPLES:

sage: z = pari(3)
sage: x = z.Mod(pari(7))
sage: x
Mod(3, 7)
sage: x^2
Mod(2, 7)
sage: x^100
Mod(4, 7)
sage: x.type()
't_INTMOD'

sage: f = pari("x^2 + x + 1")
sage: g = pari("x")
sage: a = g.Mod(f)
sage: a
Mod(x, x^2 + x + 1)
sage: a*a
Mod(-x - 1, x^2 + x + 1)
sage: a.type()
't_POLMOD'

Pol(v=-1)¶

Pol(x, v): convert x into a polynomial with main variable v and return the result.

If x is a scalar, returns a constant polynomial.
If x is a power series, the effect is identical to truncate, i.e. it chops off the $O(X^k)$ .
If x is a vector, this function creates the polynomial whose coefficients are given in x, with x[0] being the leading coefficient (which can be zero).

Warning

This is not a substitution function. It will not transform an object containing variables of higher priority than v:

sage: pari('x+y').Pol('y')
...
PariError:  (8)

INPUT:

x - gen
v - (optional) which variable, defaults to ‘x’

OUTPUT:

gen - a polynomial

EXAMPLES:

sage: v = pari("[1,2,3,4]")
sage: f = v.Pol()
sage: f
x^3 + 2*x^2 + 3*x + 4
sage: f*f
x^6 + 4*x^5 + 10*x^4 + 20*x^3 + 25*x^2 + 24*x + 16

sage: v = pari("[1,2;3,4]")
sage: v.Pol()
[1, 3]~*x + [2, 4]~

Polrev(v=-1)¶

Polrev(x, v): Convert x into a polynomial with main variable v and return the result. This is the reverse of Pol if x is a vector, otherwise it is identical to Pol. By “reverse” we mean that the coefficients are reversed.

INPUT:

x - gen

OUTPUT:

gen - a polynomial

EXAMPLES:

sage: v = pari("[1,2,3,4]")
sage: f = v.Polrev()
sage: f
4*x^3 + 3*x^2 + 2*x + 1
sage: v.Pol()
x^3 + 2*x^2 + 3*x + 4
sage: v.Polrev('y')
4*y^3 + 3*y^2 + 2*y + 1

Note that Polrev does not reverse the coefficients of a polynomial!

sage: f
4*x^3 + 3*x^2 + 2*x + 1
sage: f.Polrev()
4*x^3 + 3*x^2 + 2*x + 1
sage: v = pari("[1,2;3,4]")
sage: v.Polrev()
[2, 4]~*x + [1, 3]~

Qfb(a, b, c, D=0)¶

Qfb(a,b,c,D=0.): Returns the binary quadratic form

$ax^2 + bxy + cy^2.$

The optional D is 0 by default and initializes Shank’s distance if $b^2 - 4ac > 0$ .

Note

Negative definite forms are not implemented, so use their positive definite counterparts instead. (I.e., if f is a negative definite quadratic form, then -f is positive definite.)

INPUT:

a - gen
b - gen
c - gen
D - gen (optional, defaults to 0)

OUTPUT:

gen - binary quadratic form

EXAMPLES:

sage: pari(3).Qfb(7, 2)
Qfb(3, 7, 2, 0.E-19)

Ser(x, v=-1)¶

Ser(x,v=x): Create a power series from x with main variable v and return the result.

If x is a scalar, this gives a constant power series with precision given by the default series precision, as returned by get_series_precision().
If x is a polynomial, the precision is the greatest of get_series_precision() and the degree of the polynomial.
If x is a vector, the precision is similarly given, and the coefficients of the vector are understood to be the coefficients of the power series starting from the constant term (i.e. the reverse of the function Pol).

Warning

This is not a substitution function. It will not transform an object containing variables of higher priority than v.

INPUT:

x - gen
v - PARI variable (default: x)

OUTPUT:

gen - PARI object of PARI type t_SER

EXAMPLES:

sage: pari(2).Ser()
2 + O(x^16)
sage: x = pari([1,2,3,4,5])
sage: x.Ser()
1 + 2*x + 3*x^2 + 4*x^3 + 5*x^4 + O(x^5)
sage: f = x.Ser('v'); print f
1 + 2*v + 3*v^2 + 4*v^3 + 5*v^4 + O(v^5)
sage: pari(1)/f
1 - 2*v + v^2 + O(v^5)
sage: pari(1).Ser()
1 + O(x^16)

Set(x)¶

Set(x): convert x into a set, i.e. a row vector of strings in increasing lexicographic order.

INPUT:

x - gen

OUTPUT:

gen - a vector of strings in increasing lexicographic order.

EXAMPLES:

sage: pari([1,5,2]).Set()
["1", "2", "5"]
sage: pari([]).Set()     # the empty set
[]
sage: pari([1,1,-1,-1,3,3]).Set()
["-1", "1", "3"]
sage: pari(1).Set()
["1"]
sage: pari('1/(x*y)').Set()
["1/(y*x)"]
sage: pari('["bc","ab","bc"]').Set()
["ab", "bc"]

Str()¶

Str(self): Return the print representation of self as a PARI object.

INPUT:

self - gen

OUTPUT:

gen - a PARI gen of type t_STR, i.e., a PARI string

EXAMPLES:

sage: pari([1,2,['abc',1]]).Str()
[1, 2, [abc, 1]]
sage: pari([1,1, 1.54]).Str()
[1, 1, 1.54000000000000]
sage: pari(1).Str()       # 1 is automatically converted to string rep
1
sage: x = pari('x')       # PARI variable "x"
sage: x.Str()             # is converted to string rep.
x
sage: x.Str().type()
't_STR'

Strchr(x)¶

Strchr(x): converts x to a string, translating each integer into a character (in ASCII).

Note

Vecsmall() is (essentially) the inverse to Strchr().

INPUT:

x - PARI vector of integers

OUTPUT:

gen - a PARI string

EXAMPLES:

sage: pari([65,66,123]).Strchr()
AB{
sage: pari('"Sage"').Vecsmall()   # pari('"Sage"') --> PARI t_STR
Vecsmall([83, 97, 103, 101])
sage: _.Strchr()
Sage
sage: pari([83, 97, 103, 101]).Strchr()
Sage

Strexpand(x)¶: Strexpand(x): Concatenate the entries of the vector x into a single string, performing tilde expansion.

Note

I have no clue what the point of this function is. - William

Strtex(x)¶

Strtex(x): Translates the vector x of PARI gens to TeX format and returns the resulting concatenated strings as a PARI t_STR.

INPUT:

x - gen

OUTPUT:

gen - PARI t_STR (string)

EXAMPLES:

sage: v=pari('x^2')
sage: v.Strtex()
x^2
sage: v=pari(['1/x^2','x'])
sage: v.Strtex()
\frac{1}{x^2}x
sage: v=pari(['1 + 1/x + 1/(y+1)','x-1'])
sage: v.Strtex()
\frac{ \left(y
 + 2\right)  x
 + \left(y
 + 1\right) }{ \left(y
 + 1\right)  x}x
 - 1

Vec(x)¶

Vec(x): Transforms the object x into a vector.

INPUT:

x - gen

OUTPUT:

gen - of PARI type t_VEC

EXAMPLES:

sage: pari(1).Vec()
[1]
sage: pari('x^3').Vec()
[1, 0, 0, 0]
sage: pari('x^3 + 3*x - 2').Vec()
[1, 0, 3, -2]
sage: pari([1,2,3]).Vec()
[1, 2, 3]
sage: pari('ab').Vec()
[1, 0]

Vecrev(x)¶

Vecrev(x): Transforms the object x into a vector. Identical to Vec(x) except when x is - a polynomial, this is the reverse of Vec. - a power series, this includes low-order zero coefficients. - a Laurent series, raises an exception

INPUT:

x - gen

OUTPUT:

gen - of PARI type t_VEC

EXAMPLES:

sage: pari(1).Vecrev()
[1]
sage: pari('x^3').Vecrev()
[0, 0, 0, 1]
sage: pari('x^3 + 3*x - 2').Vecrev()
[-2, 3, 0, 1]
sage: pari([1, 2, 3]).Vecrev()
[1, 2, 3]
sage: pari('Col([1, 2, 3])').Vecrev()
[1, 2, 3]
sage: pari('[1, 2; 3, 4]').Vecrev()
[[1, 3]~, [2, 4]~]
sage: pari('ab').Vecrev()
[0, 1]
sage: pari('x^2 + 3*x^3 + O(x^5)').Vecrev()
[0, 0, 1, 3, 0]
sage: pari('x^-2 + 3*x^3 + O(x^5)').Vecrev()
...
ValueError: Vecrev() is not defined for Laurent series

Vecsmall(x)¶

Vecsmall(x): transforms the object x into a t_VECSMALL.

INPUT:

x - gen

OUTPUT:

gen - PARI t_VECSMALL

EXAMPLES:

sage: pari([1,2,3]).Vecsmall()
Vecsmall([1, 2, 3])
sage: pari('"Sage"').Vecsmall()
Vecsmall([83, 97, 103, 101])
sage: pari(1234).Vecsmall()
Vecsmall([1234])

abs(x)¶

Returns the absolute value of x (its modulus, if x is complex). Rational functions are not allowed. Contrary to most transcendental functions, an exact argument is not converted to a real number before applying abs and an exact result is returned if possible.

EXAMPLES:

sage: x = pari("-27.1")
sage: x.abs()
27.1000000000000

If x is a polynomial, returns -x if the leading coefficient is real and negative else returns x. For a power series, the constant coefficient is considered instead.

EXAMPLES:

sage: pari('x-1.2*x^2').abs()
1.20000000000000*x^2 - x

acos(x, precision=0)¶

The principal branch of $\cos^{-1}(x)$ , so that $\RR e(\mathrm{acos}(x))$ belongs to $[0,Pi]$ . If $x$ is real and $|x| > 1$ , then $\mathrm{acos}(x)$ is complex.

If $x$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If $x$ is inexact (e.g. real), its own precision is used in the computation, and the parameter precision is ignored.

EXAMPLES:

sage: pari(0.5).acos()
1.04719755119660
sage: pari(1/2).acos()
1.04719755119660
sage: pari(1.1).acos()
-0.443568254385115*I
sage: C.<i> = ComplexField()
sage: pari(1.1+i).acos()
0.849343054245252 - 1.09770986682533*I

acosh(x, precision=0)¶

The principal branch of $\cosh^{-1}(x)$ , so that $\Im(\mathrm{acosh}(x))$ belongs to $[0,Pi]$ . If $x$ is real and $x < 1$ , then $\mathrm{acosh}(x)$ is complex.

If $x$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If $x$ is inexact (e.g. real), its own precision is used in the computation, and the parameter precision is ignored.

EXAMPLES:

sage: pari(2).acosh()
1.31695789692482
sage: pari(0).acosh()
1.57079632679490*I
sage: C.<i> = ComplexField()
sage: pari(i).acosh()
0.881373587019543 + 1.57079632679490*I

agm(x, y, precision=0)¶

The arithmetic-geometric mean of x and y. In the case of complex or negative numbers, the principal square root is always chosen. p-adic or power series arguments are also allowed. Note that a p-adic AGM exists only if x/y is congruent to 1 modulo p (modulo 16 for p=2). x and y cannot both be vectors or matrices.

If any of $x$ or $y$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If the arguments are inexact (e.g. real), the smallest of their two precisions is used in the computation, and the parameter precision is ignored.

EXAMPLES:

sage: pari(2).agm(2)                  
2.00000000000000
sage: pari(0).agm(1)
0
sage: pari(1).agm(2)
1.45679103104691
sage: C.<i> = ComplexField()
sage: pari(1+i).agm(-3)
-0.964731722290876 + 1.15700282952632*I

algdep(n, bit=0)¶

EXAMPLES:

sage: n = pari.set_real_precision (200)
sage: w1 = pari('z1=2-sqrt(26); (z1+I)/(z1-I)')
sage: f = w1.algdep(12); f
545*x^11 - 297*x^10 - 281*x^9 + 48*x^8 - 168*x^7 + 690*x^6 - 168*x^5 + 48*x^4 - 281*x^3 - 297*x^2 + 545*x
sage: f(w1)
7.75513996 E-200 + 5.70672991 E-200*I     # 32-bit
3.780069700150794274 E-209 - 9.362977321012524836 E-211*I   # 64-bit
sage: f.factor()
[x, 1; x + 1, 2; x^2 + 1, 1; x^2 + x + 1, 1; 545*x^4 - 1932*x^3 + 2790*x^2 - 1932*x + 545, 1]
sage: pari.set_real_precision(n)
200

arg(x, precision=0)¶

arg(x): argument of x,such that $-\pi < \arg(x) \leq \pi$ .

If $x$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If $x$ is inexact (e.g. real), its own precision is used in the computation, and the parameter precision is ignored.

EXAMPLES:

sage: C.<i> = ComplexField()
sage: pari(2+i).arg()               
0.463647609000806

asin(x, precision=0)¶

The principal branch of $\sin^{-1}(x)$ , so that $\RR e(\mathrm{asin}(x))$ belongs to $[-\pi/2,\pi/2]$ . If $x$ is real and $|x| > 1$ then $\mathrm{asin}(x)$ is complex.

If $x$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If $x$ is inexact (e.g. real), its own precision is used in the computation, and the parameter precision is ignored.

EXAMPLES:

sage: pari(pari(0.5).sin()).asin()
0.500000000000000
sage: pari(2).asin()
1.57079632679490 + 1.31695789692482*I

asinh(x, precision=0)¶

The principal branch of $\sinh^{-1}(x)$ , so that $\Im(\mathrm{asinh}(x))$ belongs to $[-\pi/2,\pi/2]$ .

If $x$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If $x$ is inexact (e.g. real), its own precision is used in the computation, and the parameter precision is ignored.

EXAMPLES:

sage: pari(2).asinh()
1.44363547517881
sage: C.<i> = ComplexField()
sage: pari(2+i).asinh()
1.52857091948100 + 0.427078586392476*I

atan(x, precision=0)¶

The principal branch of $\tan^{-1}(x)$ , so that $\RR e(\mathrm{atan}(x))$ belongs to $]-\pi/2, \pi/2[$ .

If $x$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If $x$ is inexact (e.g. real), its own precision is used in the computation, and the parameter precision is ignored.

EXAMPLES:

sage: pari(1).atan()
0.785398163397448
sage: C.<i> = ComplexField()
sage: pari(1.5+i).atan()
1.10714871779409 + 0.255412811882995*I

atanh(x, precision=0)¶

The principal branch of $\tanh^{-1}(x)$ , so that $\Im(\mathrm{atanh}(x))$ belongs to $]-\pi/2,\pi/2]$ . If $x$ is real and $|x| > 1$ then $\mathrm{atanh}(x)$ is complex.

If $x$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If $x$ is inexact (e.g. real), its own precision is used in the computation, and the parameter precision is ignored.

EXAMPLES:

sage: pari(0).atanh()
0.E-19
sage: pari(2).atanh()
0.549306144334055 + 1.57079632679490*I

bernfrac(x)¶

The Bernoulli number $B_x$ , where $B_0 = 1$ , $B_1 = -1/2$ , $B_2 = 1/6,\ldots,$ expressed as a rational number. The argument $x$ should be of type integer.

EXAMPLES:

sage: pari(18).bernfrac()
43867/798
sage: [pari(n).bernfrac() for n in range(10)]
[1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0]

bernreal(x)¶

The Bernoulli number $B_x$ , as for the function bernfrac, but $B_x$ is returned as a real number (with the current precision).

EXAMPLES:

sage: pari(18).bernreal()
54.9711779448622

bernvec(x)¶

Creates a vector containing, as rational numbers, the Bernoulli numbers $B_0, B_2,\ldots, B_{2x}$ . This routine is obsolete. Use bernfrac instead each time you need a Bernoulli number in exact form.

Note: this routine is implemented using repeated independent calls to bernfrac, which is faster than the standard recursion in exact arithmetic.

EXAMPLES:

sage: pari(8).bernvec()
[1, 1/6, -1/30, 1/42, -1/30, 5/66, -691/2730, 7/6, -3617/510]
sage: [pari(2*n).bernfrac() for n in range(9)]
[1, 1/6, -1/30, 1/42, -1/30, 5/66, -691/2730, 7/6, -3617/510]

besselh1(nu, x, precision=0)¶

The $H^1$ -Bessel function of index $\nu$ and argument $x$ .

If $nu$ or $x$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If the arguments are inexact (e.g. real), the smallest of their precisions is used in the computation, and the parameter precision is ignored.

EXAMPLES:

sage: pari(2).besselh1(3)
0.486091260585891 - 0.160400393484924*I

besselh2(nu, x, precision=0)¶

The $H^2$ -Bessel function of index $\nu$ and argument $x$ .

If $nu$ or $x$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If the arguments are inexact (e.g. real), the smallest of their precisions is used in the computation, and the parameter precision is ignored.

EXAMPLES:

sage: pari(2).besselh2(3)
0.486091260585891 + 0.160400393484924*I

besseli(nu, x, precision=0)¶

Bessel I function (Bessel function of the second kind), with index $\nu$ and argument $x$ . If $x$ converts to a power series, the initial factor $(x/2)^{\nu}/\Gamma(\nu+1)$ is omitted (since it cannot be represented in PARI when $\nu$ is not integral).

If $nu$ or $x$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If the arguments are inexact (e.g. real), the smallest of their precisions is used in the computation, and the parameter precision is ignored.

EXAMPLES:

sage: pari(2).besseli(3)
2.24521244092995
sage: C.<i> = ComplexField()
sage: pari(2).besseli(3+i)
1.12539407613913 + 2.08313822670661*I

besselj(nu, x, precision=0)¶

Bessel J function (Bessel function of the first kind), with index $\nu$ and argument $x$ . If $x$ converts to a power series, the initial factor $(x/2)^{\nu}/\Gamma(\nu+1)$ is omitted (since it cannot be represented in PARI when $\nu$ is not integral).

If $nu$ or $x$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If the arguments are inexact (e.g. real), the smallest of their precisions is used in the computation, and the parameter precision is ignored.

EXAMPLES:

sage: pari(2).besselj(3)
0.486091260585891

besseljh(nu, x, precision=0)¶

J-Bessel function of half integral index (Spherical Bessel function of the first kind). More precisely, besseljh(n,x) computes $J_{n+1/2}(x)$ where n must an integer, and x is any complex value. In the current implementation (PARI, version 2.2.11), this function is not very accurate when $x$ is small.

If $nu$ or $x$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If the arguments are inexact (e.g. real), the smallest of their precisions is used in the computation, and the parameter precision is ignored.

EXAMPLES:

sage: pari(2).besseljh(3)
0.4127100324          # 32-bit
0.412710032209716     # 64-bit

besselk(nu, x, flag=0, precision=0)¶

nu.besselk(x, flag=0): K-Bessel function (modified Bessel function of the second kind) of index nu, which can be complex, and argument x.

If $nu$ or $x$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If the arguments are inexact (e.g. real), the smallest of their precisions is used in the computation, and the parameter precision is ignored.

INPUT:

nu - a complex number
x - real number (positive or negative)
flag - default: 0 or 1: use hyperu (hyperu is much slower for small x, and doesn’t work for negative x).

EXAMPLES:

sage: C.<i> = ComplexField()
sage: pari(2+i).besselk(3)
0.0455907718407551 + 0.0289192946582081*I

sage: pari(2+i).besselk(-3)
-4.34870874986752 - 5.38744882697109*I

sage: pari(2+i).besselk(300, flag=1)
3.74224603319728 E-132 + 2.49071062641525 E-134*I

besseln(nu, x, precision=0)¶

nu.besseln(x): Bessel N function (Spherical Bessel function of the second kind) of index nu and argument x.

If $nu$ or $x$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If the arguments are inexact (e.g. real), the smallest of their precisions is used in the computation, and the parameter precision is ignored.

EXAMPLES:

sage: C.<i> = ComplexField()
sage: pari(2+i).besseln(3)
-0.280775566958244 - 0.486708533223726*I

bezout(x, y)¶

binary(x)¶

binary(x): gives the vector formed by the binary digits of abs(x), where x is of type t_INT.

INPUT:

x - gen of type t_INT

OUTPUT:

gen - of type t_VEC

EXAMPLES:

sage: pari(0).binary()
[0]
sage: pari(-5).binary()
[1, 0, 1]
sage: pari(5).binary()
[1, 0, 1]
sage: pari(2005).binary()
[1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1]

sage: pari('"2"').binary()
...
TypeError: x (=2) must be of type t_INT, but is of type t_STR.

binomial(x, k)¶

binomial(x, k): return the binomial coefficient “x choose k”.

INPUT:

x - any PARI object (gen)
k - integer

EXAMPLES:

sage: pari(6).binomial(2)
15
sage: pari('x+1').binomial(3)
1/6*x^3 - 1/6*x
sage: pari('2+x+O(x^2)').binomial(3)
1/3*x + O(x^2)

bitand(x, y)¶

bitand(x,y): Bitwise and of two integers x and y. Negative numbers behave as if modulo some large power of 2.

INPUT:

x - gen (of type t_INT)
y - coercible to gen (of type t_INT)

OUTPUT:

gen - of type type t_INT

EXAMPLES:

sage: pari(8).bitand(4)
0
sage: pari(8).bitand(8)
8
sage: pari(6).binary()
[1, 1, 0]
sage: pari(7).binary()
[1, 1, 1]
sage: pari(6).bitand(7)
6
sage: pari(19).bitand(-1)
19
sage: pari(-1).bitand(-1)
-1

bitneg(x, n=-1)¶

bitneg(x,n=-1): Bitwise negation of the integer x truncated to n bits. n=-1 (the default) represents an infinite sequence of the bit 1. Negative numbers behave as if modulo some large power of 2.

With n=-1, this function returns -n-1. With n = 0, it returns a number a such that $a\cong -n-1 \pmod{2^n}$ .

INPUT:

x - gen (t_INT)
n - long, default = -1

OUTPUT:

gen - t_INT

EXAMPLES:

sage: pari(10).bitneg()
-11
sage: pari(1).bitneg()
-2
sage: pari(-2).bitneg()
1
sage: pari(-1).bitneg()
0
sage: pari(569).bitneg()
-570
sage: pari(569).bitneg(10)
454
sage: 454 % 2^10
454
sage: -570 % 2^10
454

bitnegimply(x, y)¶

bitnegimply(x,y): Bitwise negated imply of two integers x and y, in other words, x BITAND BITNEG(y). Negative numbers behave as if modulo big power of 2.

INPUT:

x - gen (of type t_INT)
y - coercible to gen (of type t_INT)

OUTPUT:

gen - of type type t_INT

EXAMPLES:

sage: pari(14).bitnegimply(0)    
14
sage: pari(8).bitnegimply(8)
0
sage: pari(8+4).bitnegimply(8)
4

bitor(x, y)¶

bitor(x,y): Bitwise or of two integers x and y. Negative numbers behave as if modulo big power of 2.

INPUT:

x - gen (of type t_INT)
y - coercible to gen (of type t_INT)

OUTPUT:

gen - of type type t_INT

EXAMPLES:

sage: pari(14).bitor(0)
14
sage: pari(8).bitor(4)
12
sage: pari(12).bitor(1)
13
sage: pari(13).bitor(1)
13

bittest(x, n)¶

bittest(x, long n): Returns bit number n (coefficient of $2^n$ in binary) of the integer x. Negative numbers behave as if modulo a big power of 2.

INPUT:

x - gen (pari integer)

OUTPUT:

bool - a Python bool

EXAMPLES:

sage: x = pari(6)
sage: x.bittest(0)
False
sage: x.bittest(1)
True
sage: x.bittest(2)
True
sage: x.bittest(3)
False
sage: pari(-3).bittest(0)
True
sage: pari(-3).bittest(1)
False
sage: [pari(-3).bittest(n) for n in range(10)]
[True, False, True, True, True, True, True, True, True, True]

bitxor(x, y)¶

bitxor(x,y): Bitwise exclusive or of two integers x and y. Negative numbers behave as if modulo big power of 2.

INPUT:

x - gen (of type t_INT)
y - coercible to gen (of type t_INT)

OUTPUT:

gen - of type type t_INT

EXAMPLES:

sage: pari(6).bitxor(4)
2
sage: pari(0).bitxor(4)
4
sage: pari(6).bitxor(0)
6

bnfcertify()¶: bnf being as output by bnfinit, checks whether the result is correct, i.e. whether the calculation of the contents of self are correct without assuming the Generalized Riemann Hypothesis. If it is correct, the answer is 1. If not, the program may output some error message, but more probably will loop indefinitely. In no occasion can the program give a wrong answer (barring bugs of course): if the program answers 1, the answer is certified.

Warning

By default, most of the bnf routines depend on the correctness of a heuristic assumption which is stronger than GRH. In order to obtain a provably-correct result you must specify $c=c_2=12$ for the technical optional parameters to the function. There are known counterexamples for smaller $c$ (which is the default).

bnfinit(flag=0, tech=None)¶

bnfisintnorm(x)¶

bnfisprincipal(x, flag=1)¶

bnfisunit(x)¶

bnfnarrow()¶

bnfunit()¶

ceil(x)¶

For real x: return the smallest integer = x. For rational functions: the quotient of numerator by denominator. For lists: apply componentwise.

INPUT:

x - gen

OUTPUT:

gen - depends on type of x

EXAMPLES:

sage: pari(1.4).ceil()
2
sage: pari(-1.4).ceil()
-1
sage: pari(3/4).ceil()
1
sage: pari(x).ceil()
x
sage: pari((x^2+x+1)/x).ceil()
x + 1

This may be unexpected: but it is correct, treating the argument as a rational function in RR(x).

sage: pari(x^2+5*x+2.5).ceil()
x^2 + 5*x + 2.50000000000000

centerlift(x, v=-1)¶

centerlift(x,v): Centered lift of x. This function returns exactly the same thing as lift, except if x is an integer mod.

INPUT:

x - gen
v - var (default: x)

OUTPUT: gen

EXAMPLES:

sage: x = pari(-2).Mod(5)
sage: x.centerlift()
-2
sage: x.lift()
3
sage: f = pari('x-1').Mod('x^2 + 1')
sage: f.centerlift()
x - 1
sage: f.lift()
x - 1
sage: f = pari('x-y').Mod('x^2+1')
sage: f
Mod(x - y, x^2 + 1)
sage: f.centerlift('x')
x - y
sage: f.centerlift('y')
Mod(x - y, x^2 + 1)

changevar(x, y)¶

changevar(gen x, y): change variables of x according to the vector y.

Warning

This doesn’t seem to work right at all in Sage (!). Use with caution. STRANGE

INPUT:

x - gen
y - gen (or coercible to gen)

OUTPUT: gen

EXAMPLES:

sage: pari('x^3+1').changevar(pari(['y']))
y^3 + 1

charpoly(var=-1, flag=0)¶: charpoly(A,v=x,flag=0): det(v*Id-A) = characteristic polynomial of A using the comatrix. flag is optional and may be set to 1 (use Lagrange interpolation) or 2 (use Hessenberg form), 0 being the default.

chinese(y)¶

component(x, n)¶

component(x, long n): Return n’th component of the internal representation of x. This function is 1-based instead of 0-based.

Note

For vectors or matrices, it is simpler to use x[n-1]. For list objects such as is output by nfinit, it is easier to use member functions.

INPUT:

x - gen
n - C long (coercible to)

OUTPUT: gen

EXAMPLES:

sage: pari([0,1,2,3,4]).component(1)
0
sage: pari([0,1,2,3,4]).component(2)
1
sage: pari([0,1,2,3,4]).component(4)
3
sage: pari('x^3 + 2').component(1)
2
sage: pari('x^3 + 2').component(2)
0
sage: pari('x^3 + 2').component(4)
1

sage: pari('x').component(0)
...
PariError:  (8)

concat(y)¶

conj(x)¶

conj(x): Return the algebraic conjugate of x.

INPUT:

x - gen

OUTPUT: gen

EXAMPLES:

sage: pari('x+1').conj()
x + 1
sage: pari('x+I').conj()
x - I
sage: pari('1/(2*x+3*I)').conj()
1/(2*x - 3*I)
sage: pari([1,2,'2-I','Mod(x,x^2+1)', 'Mod(x,x^2-2)']).conj()
[1, 2, 2 + I, Mod(-x, x^2 + 1), Mod(-x, x^2 - 2)]
sage: pari('Mod(x,x^2-2)').conj()
Mod(-x, x^2 - 2)
sage: pari('Mod(x,x^3-3)').conj()
...
PariError: incorrect type (20)

conjvec(x)¶

conjvec(x): Returns the vector of all conjugates of the algebraic number x. An algebraic number is a polynomial over Q modulo an irreducible polynomial.

INPUT:

x - gen

OUTPUT: gen

EXAMPLES:

sage: pari('Mod(1+x,x^2-2)').conjvec()
[-0.414213562373095, 2.41421356237310]~
sage: pari('Mod(x,x^3-3)').conjvec()
[1.44224957030741, -0.721124785153704 + 1.24902476648341*I, -0.721124785153704 - 1.24902476648341*I]~

contfrac(x, b=0, lmax=0)¶: contfrac(x,b,lmax): continued fraction expansion of x (x rational, real or rational function). b and lmax are both optional, where b is the vector of numerators of the continued fraction, and lmax is a bound for the number of terms in the continued fraction expansion.

contfracpnqn(x, b=0, lmax=0)¶: contfracpnqn(x): [p_n,p_n-1; q_n,q_n-1] corresponding to the continued fraction x.

cos(x, precision=0)¶

The cosine function.

If $x$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If $x$ is inexact (e.g. real), its own precision is used in the computation, and the parameter precision is ignored.

EXAMPLES:

sage: pari(1.5).cos()
0.0707372016677029
sage: C.<i> = ComplexField()
sage: pari(1+i).cos()
0.833730025131149 - 0.988897705762865*I
sage: pari('x+O(x^8)').cos()
1 - 1/2*x^2 + 1/24*x^4 - 1/720*x^6 + 1/40320*x^8 + O(x^9)

cosh(x, precision=0)¶

The hyperbolic cosine function.

If $x$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If $x$ is inexact (e.g. real), its own precision is used in the computation, and the parameter precision is ignored.

EXAMPLES:

sage: pari(1.5).cosh()
2.35240961524325
sage: C.<i> = ComplexField()
sage: pari(1+i).cosh()
0.833730025131149 + 0.988897705762865*I
sage: pari('x+O(x^8)').cosh()
1 + 1/2*x^2 + 1/24*x^4 + 1/720*x^6 + O(x^8)

cotan(x, precision=0)¶

The cotangent of x.

If $x$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If $x$ is inexact (e.g. real), its own precision is used in the computation, and the parameter precision is ignored.

EXAMPLES:

sage: pari(5).cotan()
-0.295812915532746

Computing the cotangent of $\pi$ doesn’t raise an error, but instead just returns a very large (positive or negative) number.

sage: x = RR(pi)
sage: pari(x).cotan()         # random
-8.17674825 E15

denominator(x)¶

denominator(x): Return the denominator of x. When x is a vector, this is the least common multiple of the denominators of the components of x.

what about poly? INPUT:

x - gen

OUTPUT: gen

EXAMPLES:

sage: pari('5/9').denominator()
9
sage: pari('(x+1)/(x-2)').denominator()
x - 2
sage: pari('2/3 + 5/8*x + 7/3*x^2 + 1/5*y').denominator()
1
sage: pari('2/3*x').denominator()
1
sage: pari('[2/3, 5/8, 7/3, 1/5]').denominator()
120

deriv(v=-1)¶

dilog(x, precision=0)¶

The principal branch of the dilogarithm of $x$ , i.e. the analytic continuation of the power series $\log_2(x) = \sum_{n>=1} x^n/n^2$ .

If $x$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If $x$ is inexact (e.g. real), its own precision is used in the computation, and the parameter precision is ignored.

EXAMPLES:

sage: pari(1).dilog()
1.64493406684823
sage: C.<i> = ComplexField()
sage: pari(1+i).dilog()
0.616850275068085 + 1.46036211675312*I

dirzetak(n)¶

disc()¶

e.disc(): return the discriminant of the elliptic curve e.

EXAMPLES:

sage: e = pari([0, -1, 1, -10, -20]).ellinit()
sage: e.disc()
-161051
sage: _.factor()
[-1, 1; 11, 5]

divrem(x, y, var=-1)¶: divrem(x, y, v): Euclidean division of x by y giving as a 2-dimensional column vector the quotient and the remainder, with respect to v (to main variable if v is omitted).

eint1(x, n=0, precision=0)¶

x.eint1(n): exponential integral E1(x):

$\int_{x}^{\infty} \frac{e^{-t}}{t} dt$

If n is present, output the vector [eint1(x), eint1(2*x), ..., eint1(n*x)]. This is faster than repeatedly calling eint1(i*x).

If $x$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If $x$ is inexact (e.g. real), its own precision is used in the computation, and the parameter precision is ignored.

REFERENCE:

See page 262, Prop 5.6.12, of Cohen’s book “A Course in Computational Algebraic Number Theory”.

EXAMPLES:

elementval(x, p)¶

elladd(z0, z1)¶

e.elladd(z0, z1): return the sum of the points z0 and z1 on this elliptic curve.

INPUT:

e - elliptic curve E
z0 - point on E
z1 - point on E

OUTPUT: point on E

EXAMPLES: First we create an elliptic curve:

sage: e = pari([0, 1, 1, -2, 0]).ellinit()
sage: str(e)[:65]   # first part of output
'[0, 1, 1, -2, 0, 4, -4, 1, -3, 112, -856, 389, 1404928/389, [0.90'

Next we add two points on the elliptic curve. Notice that the Python lists are automatically converted to PARI objects so you don’t have to do that explicitly in your code.

sage: e.elladd([1,0], [-1,1])
[-3/4, -15/8]

ellak(n)¶

e.ellak(n): Returns the coefficient $a_n$ of the $L$ -function of the elliptic curve e, i.e. the $n$ -th Fourier coefficient of the weight 2 newform associated to e (according to Shimura-Taniyama).

The curve $e$ must be a medium or long vector of the type given by ellinit. For this function to work for every n and not just those prime to the conductor, e must be a minimal Weierstrass equation. If this is not the case, use the function ellminimalmodel first before using ellak (or you will get INCORRECT RESULTS!)

INPUT:

e - a PARI elliptic curve.
n - integer.

EXAMPLES:

sage: e = pari([0, -1, 1, -10, -20]).ellinit()
sage: e.ellak(6)
2
sage: e.ellak(2005)        
2
sage: e.ellak(-1)
0
sage: e.ellak(0)
0

ellan(n, python_ints=False)¶

Return the first $n$ Fourier coefficients of the modular form attached to this elliptic curve. See ellak for more details.

INPUT:

n - a long integer
python_ints - bool (default is False); if True, return a list of Python ints instead of a PARI gen wrapper.

EXAMPLES:

sage: e = pari([0, -1, 1, -10, -20]).ellinit()
sage: e.ellan(3)
[1, -2, -1]
sage: e.ellan(20)
[1, -2, -1, 2, 1, 2, -2, 0, -2, -2, 1, -2, 4, 4, -1, -4, -2, 4, 0, 2]
sage: e.ellan(-1)
[]
sage: v = e.ellan(10, python_ints=True); v
[1, -2, -1, 2, 1, 2, -2, 0, -2, -2]
sage: type(v)
<type 'list'>
sage: type(v[0])
<type 'int'>

ellap(p)¶

e.ellap(p): Returns the prime-indexed coefficient $a_p$ of the $L$ -function of the elliptic curve $e$ , i.e. the $p$ -th Fourier coefficient of the newform attached to e.

The computation uses the baby-step giant-step method and a trick due to Mestre, and requires $O(p^{1/4})$ time and $O(p^{1/4})$ storage.

If p is not prime, this function will return an incorrect answer.

The curve e must be a medium or long vector of the type given by ellinit. For this function to work for every n and not just those prime to the conductor, e must be a minimal Weierstrass equation. If this is not the case, use the function ellminimalmodel first before using ellap (or you will get INCORRECT RESULTS!)

INPUT:

e - a PARI elliptic curve.
p - prime integer

EXAMPLES:

sage: e = pari([0, -1, 1, -10, -20]).ellinit()
sage: e.ellap(2)
-2
sage: e.ellap(2003)
4
sage: e.ellak(-1)
0

ellaplist(n, python_ints=False)¶

e.ellaplist(n): Returns a PARI list of all the prime-indexed coefficients $a_p$ (up to n) of the $L$ -function of the elliptic curve $e$ , i.e. the Fourier coefficients of the newform attached to $e$ .

INPUT:

n - a long integer
python_ints - bool (default is False); if True, return a list of Python ints instead of a PARI gen wrapper.

The curve e must be a medium or long vector of the type given by ellinit. For this function to work for every n and not just those prime to the conductor, e must be a minimal Weierstrass equation. If this is not the case, use the function ellminimalmodel first before using ellaplist (or you will get INCORRECT RESULTS!)

INPUT:

e - a PARI elliptic curve.
n - an integer

EXAMPLES:

sage: e = pari([0, -1, 1, -10, -20]).ellinit()
sage: v = e.ellaplist(10); v
[-2, -1, 1, -2]
sage: type(v)
<type 'sage.libs.pari.gen.gen'>
sage: v.type()
't_VEC'
sage: e.ellan(10)
[1, -2, -1, 2, 1, 2, -2, 0, -2, -2]
sage: v = e.ellaplist(10, python_ints=True); v
[-2, -1, 1, -2]
sage: type(v)
<type 'list'>
sage: type(v[0])
<type 'int'>

ellbil(z0, z1)¶

e.ellbil(z0, z1): return the value of the canonical bilinear form on z0 and z1.

INPUT:

e - elliptic curve (assumed integral given by a minimal model, as returned by ellminimalmodel)
z0, z1 - rational points on e

EXAMPLES:

sage: e = pari([0,1,1,-2,0]).ellinit().ellminimalmodel()[0]
sage: e.ellbil([1, 0], [-1, 1])
0.418188984498861

ellchangecurve(ch)¶

e.ellchangecurve(ch): return the new model (equation) for the elliptic curve e given by the change of coordinates ch.

The change of coordinates is specified by a vector ch=[u,r,s,t]; if $x'$ and $y'$ are the new coordinates, then $x = u^2 x' + r$ and $y = u^3 y' + su^2 x' + t$ .

INPUT:

e - elliptic curve
ch - change of coordinates vector with 4 entries

EXAMPLES:

sage: e = pari([1,2,3,4,5]).ellinit()
sage: e.ellglobalred()
[10351, [1, -1, 0, -1], 1]
sage: f = e.ellchangecurve([1,-1,0,-1])
sage: f[:5]
[1, -1, 0, 4, 3]

ellchangepoint(y)¶

self.ellchangepoint(y): change data on point or vector of points self on an elliptic curve according to y=[u,r,s,t]

EXAMPLES:

sage: e = pari([0,1,1,-2,0]).ellinit()
sage: x = pari([1,0])
sage: e.ellisoncurve([1,4])
False
sage: e.ellisoncurve(x)
True
sage: f = e.ellchangecurve([1,2,3,-1])
sage: f[:5]   # show only first five entries
[6, -2, -1, 17, 8]
sage: x.ellchangepoint([1,2,3,-1])
[-1, 4]
sage: f.ellisoncurve([-1,4])
True

elleisnum(k, flag=0)¶

om.elleisnum(k, flag=0): om=[om1,om2] being a 2-component vector giving a basis of a lattice L and k an even positive integer, computes the numerical value of the Eisenstein series of weight k. When flag is non-zero and k=4 or 6, this gives g2 or g3 with the correct normalization.

INPUT:

om - gen, 2-component vector giving a basis of a lattice L
k - int (even positive)
flag - int (default 0)

OUTPUT:

gen - numerical value of E_k

EXAMPLES:

sage: e = pari([0,1,1,-2,0]).ellinit()
sage: om = e.omega()
sage: om
[2.49021256085506, 1.97173770155165*I]
sage: om.elleisnum(2)
-5.28864933965426
sage: om.elleisnum(4)
112.000000000000
sage: om.elleisnum(100)
2.15314248576078 E50

elleta()¶

e.elleta(): return the vector [eta1,eta2] of quasi-periods associated with the period lattice e.omega() of the elliptic curve e.

EXAMPLES:

sage: e = pari([0,0,0,-82,0]).ellinit()
sage: e.elleta()
[3.60546360143265, 10.8163908042980*I]

ellglobalred()¶

e.ellglobalred(): return information related to the global minimal model of the elliptic curve e.

INPUT:

e - elliptic curve (returned by ellinit)

OUTPUT:

gen - the (arithmetic) conductor of e
gen - a vector giving the coordinate change over Q from e to its minimal integral model (see also ellminimalmodel)
gen - the product of the local Tamagawa numbers of e

EXAMPLES:

sage: e = pari([0, 5, 2, -1, 1]).ellinit()
sage: e.ellglobalred()
[20144, [1, -2, 0, -1], 1]
sage: e = pari(EllipticCurve('17a').a_invariants()).ellinit()
sage: e.ellglobalred()
[17, [1, 0, 0, 0], 4]

ellheight(a, flag=2, precision=0)¶

e.ellheight(a, flag=2): return the global Neron-Tate height of the point a on the elliptic curve e.

INPUT:

e - elliptic curve over $\QQ$ , assumed to be in a standard minimal integral model (as given by ellminimalmodel)
a - rational point on e
flag (optional) - specifies which algorithm to be used for computing the archimedean local height:
- 0 - uses sigma- and theta-functions and a trick
  
  due to J. Silverman
- 1 - uses Tate’s $4^n$ algorithm
- 2 - uses Mestre’s AGM algorithm (this is the default, being faster than the other two)
precision (optional) - the precision of the result, in bits.

Note that in order to achieve the desired precision, the elliptic curve must have been created using ellinit with the desired precision.

EXAMPLES:

sage: e = pari([0,1,1,-2,0]).ellinit().ellminimalmodel()[0]
sage: e.ellheight([1,0])
0.476711659343740
sage: e.ellheight([1,0], flag=0)
0.476711659343740
sage: e.ellheight([1,0], flag=1)
0.476711659343740

ellheightmatrix(x)¶

e.ellheightmatrix(x): return the height matrix for the vector x of points on the elliptic curve e.

In other words, it returns the Gram matrix of x with respect to the height bilinear form on e (see ellbil).

INPUT:

e - elliptic curve over $\QQ$ , assumed to be in a standard minimal integral model (as given by ellminimalmodel)
x - vector of rational points on e

EXAMPLES:

sage: e = pari([0,1,1,-2,0]).ellinit().ellminimalmodel()[0]
sage: e.ellheightmatrix([[1,0], [-1,1]])
[0.476711659343740, 0.418188984498861; 0.418188984498861, 0.686667083305587]

ellinit(flag=0, precision=0)¶

Return the Pari elliptic curve object with Weierstrass coefficients given by self, a list with 5 elements.

INPUT:

self - a list of 5 coefficients
flag (optional, default: 0) - if 0, ask for a Pari ell structure with 19 components; if 1, ask for a Pari sell structure with only the first 13 components
precision (optional, default: 0) - the real precision to be used in the computation of the components of the Pari (s)ell structure; if 0, use the default 53 bits.

Note

the parameter precision in ellinit() controls not only the real precision of the resulting (s)ell structure, but also the precision of most subsequent computations with this elliptic curve. You should therefore set it from the start to the value you require.

OUTPUT:

gen - either a Pari ell structure with 19 components (if flag=0), or a Pari sell structure with 13 components (if flag=1)

EXAMPLES: An elliptic curve with integer coefficients:

sage: e = pari([0,1,0,1,0]).ellinit(); e
[0, 1, 0, 1, 0, 4, 2, 0, -1, -32, 224, -48, 2048/3, [0.E-28, -0.500000000000000 - 0.866025403784439*I, -0.500000000000000 + 0.866025403784439*I]~, 3.37150070962519, 1.68575035481260 + 2.15651564749964*I, -0.687257278928726 + 7.57138254 E-30*I, -0.343628639464363 - 1.37139930484298*I, 7.27069403586288] # 32-bit
[0, 1, 0, 1, 0, 4, 2, 0, -1, -32, 224, -48, 2048/3, [0.E-38, -0.500000000000000 - 0.866025403784439*I, -0.500000000000000 + 0.866025403784439*I]~, 3.37150070962519, 1.68575035481260 + 2.15651564749964*I, -0.687257278928726 + 1.76284987179941 E-39*I, -0.343628639464363 - 1.37139930484298*I, 7.27069403586288] # 64-bit

Its inexact components have the default precision of 53 bits:

sage: RR(e[14])
3.37150070962519

We can compute this to higher precision:

sage: R = RealField(150)
sage: e = pari([0,1,0,1,0]).ellinit(precision=150)
sage: R(e[14])
3.3715007096251920857424073155981539790016018

Using flag=1 returns a short elliptic curve Pari object:

sage: pari([0,1,0,1,0]).ellinit(flag=1)
[0, 1, 0, 1, 0, 4, 2, 0, -1, -32, 224, -48, 2048/3]

The coefficients can be any ring elements that convert to Pari:

sage: pari([0,1/2,0,-3/4,0]).ellinit(flag=1)
[0, 1/2, 0, -3/4, 0, 2, -3/2, 0, -9/16, 40, -116, 117/4, 256000/117]
sage: pari([0,0.5,0,-0.75,0]).ellinit(flag=1)
[0, 0.500000000000000, 0, -0.750000000000000, 0, 2.00000000000000, -1.50000000000000, 0, -0.562500000000000, 40.0000000000000, -116.000000000000, 29.2500000000000, 2188.03418803419]
sage: pari([0,I,0,1,0]).ellinit(flag=1)
[0, I, 0, 1, 0, 4*I, 2, 0, -1, -64, 352*I, -80, 16384/5]
sage: pari([0,x,0,2*x,1]).ellinit(flag=1)
[0, x, 0, 2*x, 1, 4*x, 4*x, 4, -4*x^2 + 4*x, 16*x^2 - 96*x, -64*x^3 + 576*x^2 - 864, 64*x^4 - 576*x^3 + 576*x^2 - 432, (256*x^6 - 4608*x^5 + 27648*x^4 - 55296*x^3)/(4*x^4 - 36*x^3 + 36*x^2 - 27)]

ellisoncurve(x)¶

e.ellisoncurve(x): return True if the point x is on the elliptic curve e, False otherwise.

If the point or the curve have inexact coefficients, an attempt is made to take this into account.

EXAMPLES:

sage: e = pari([0,1,1,-2,0]).ellinit()
sage: e.ellisoncurve([1,0])
True
sage: e.ellisoncurve([1,1])
False
sage: e.ellisoncurve([1,0.00000000000000001])
False
sage: e.ellisoncurve([1,0.000000000000000001])
True
sage: e.ellisoncurve([0])
True

ellj()¶

elllocalred(p)¶

e.elllocalred(p): computes the data of local reduction at the prime p on the elliptic curve e

For more details on local reduction and Kodaira types, see IV.8 and IV.9 in J. Silverman’s book “Advanced topics in the arithmetic of elliptic curves”.

INPUT:

e - elliptic curve with coefficients in $\ZZ$
p - prime number

OUTPUT:

gen - the exponent of p in the arithmetic conductor of e
gen - the Kodaira type of e at p, encoded as an integer:
1 - type $I_0$ : good reduction, nonsingular curve of genus 1
2 - type $II$ : rational curve with a cusp
3 - type $III$ : two nonsingular rational curves intersecting tangentially at one point
4 - type $IV$ : three nonsingular rational curves intersecting at one point
5 - type $I_1$ : rational curve with a node
6 or larger - think of it as $4+v$ , then it is type $I_v$ : $v$ nonsingular rational curves arranged as a $v$ -gon
-1 - type $I_0^*$ : nonsingular rational curve of multiplicity two with four nonsingular rational curves of multiplicity one attached
-2 - type $II^*$ : nine nonsingular rational curves in a special configuration
-3 - type $III^*$ : eight nonsingular rational curves in a special configuration
-4 - type $IV^*$ : seven nonsingular rational curves in a special configuration
-5 or smaller - think of it as $-4-v$ , then it is type $I_v^*$ : chain of $v+1$ nonsingular rational curves of multiplicity two, with two nonsingular rational curves of multiplicity one attached at either end
gen - a vector with 4 components, giving the coordinate changes done during the local reduction; if the first component is 1, then the equation for e was already minimal at p
gen - the local Tamagawa number $c_p$

EXAMPLES:

Type $I_0$ :

sage: e = pari([0,0,0,0,1]).ellinit()
sage: e.elllocalred(7)
[0, 1, [1, 0, 0, 0], 1]

Type $II$ :

sage: e = pari(EllipticCurve('27a3').a_invariants()).ellinit()
sage: e.elllocalred(3)
[3, 2, [1, -1, 0, 1], 1]

Type $III$ :

sage: e = pari(EllipticCurve('24a4').a_invariants()).ellinit()
sage: e.elllocalred(2)
[3, 3, [1, 1, 0, 1], 2]

Type $IV$ :

sage: e = pari(EllipticCurve('20a2').a_invariants()).ellinit()
sage: e.elllocalred(2)
[2, 4, [1, 1, 0, 1], 3]

Type $I_1$ :

sage: e = pari(EllipticCurve('11a2').a_invariants()).ellinit()
sage: e.elllocalred(11)
[1, 5, [1, 0, 0, 0], 1]

Type $I_2$ :

sage: e = pari(EllipticCurve('14a4').a_invariants()).ellinit()
sage: e.elllocalred(2)
[1, 6, [1, 0, 0, 0], 2]

Type $I_6$ :

sage: e = pari(EllipticCurve('14a1').a_invariants()).ellinit()
sage: e.elllocalred(2)
[1, 10, [1, 0, 0, 0], 2]

Type $I_0^*$ :

sage: e = pari(EllipticCurve('32a3').a_invariants()).ellinit()
sage: e.elllocalred(2)
[5, -1, [1, 1, 1, 0], 1]

Type $II^*$ :

sage: e = pari(EllipticCurve('24a5').a_invariants()).ellinit()
sage: e.elllocalred(2)
[3, -2, [1, 2, 1, 4], 1]

Type $III^*$ :

sage: e = pari(EllipticCurve('24a2').a_invariants()).ellinit()
sage: e.elllocalred(2)
[3, -3, [1, 2, 1, 4], 2]

Type $IV^*$ :

sage: e = pari(EllipticCurve('20a1').a_invariants()).ellinit()
sage: e.elllocalred(2)
[2, -4, [1, 0, 1, 2], 3]

Type $I_1^*$ :

sage: e = pari(EllipticCurve('24a1').a_invariants()).ellinit()
sage: e.elllocalred(2)
[3, -5, [1, 0, 1, 2], 4]

Type $I_6^*$ :

sage: e = pari(EllipticCurve('90c2').a_invariants()).ellinit()
sage: e.elllocalred(3)
[2, -10, [1, 96, 1, 316], 4]

elllseries(s, A=1)¶

e.elllseries(s, A=1): return the value of the $L$ -series of the elliptic curve e at the complex number s.

This uses an $O(N^{1/2})$ algorithm in the conductor N of e, so it is impractical for large conductors (say greater than $10^{12}$ ).

INPUT:

e - elliptic curve defined over $\QQ$
s - complex number
A (optional) - cutoff point for the integral, which must be chosen close to 1 for best speed.

EXAMPLES:

sage: e = pari([0,1,1,-2,0]).ellinit()
sage: e.elllseries(2.1)
0.402838047956645
sage: e.elllseries(1)   # random, close to 0
1.822829333527862 E-19
sage: e.elllseries(-2)
0

The following example differs for the last digit on 32 vs. 64 bit systems

sage: e.elllseries(2.1, A=1.1)
0.402838047956645

ellminimalmodel()¶

ellminimalmodel(e): return the standard minimal integral model of the rational elliptic curve e and the corresponding change of variables. INPUT:

e - gen (that defines an elliptic curve)

OUTPUT:

gen - minimal model
gen - change of coordinates

EXAMPLES:

sage: e = pari([1,2,3,4,5]).ellinit()
sage: F, ch = e.ellminimalmodel()
sage: F[:5]
[1, -1, 0, 4, 3]
sage: ch
[1, -1, 0, -1]
sage: e.ellchangecurve(ch)[:5]
[1, -1, 0, 4, 3]

ellorder(x)¶

e.ellorder(x): return the order of the point x on the elliptic curve e (return 0 if x is not a torsion point)

INPUT:

e - elliptic curve defined over $\QQ$
x - point on e

EXAMPLES:

sage: e = pari(EllipticCurve('65a1').a_invariants()).ellinit()

A point of order two:

sage: e.ellorder([0,0])
2

And a point of infinite order:

sage: e.ellorder([1,0])
0

ellordinate(x)¶

e.ellordinate(x): return the $y$ -coordinates of the points on the elliptic curve e having x as $x$ -coordinate.

INPUT:

e - elliptic curve
x - x-coordinate (can be a complex or p-adic number, or a more complicated object like a power series)

EXAMPLES:

sage: e = pari([0,1,1,-2,0]).ellinit()
sage: e.ellordinate(0)
[0, -1]
sage: C.<i> = ComplexField()
sage: e.ellordinate(i)
[0.582203589721741 - 1.38606082464177*I, -1.58220358972174 + 1.38606082464177*I]
sage: e.ellordinate(1+3*5^1+O(5^3))
[4*5 + 5^2 + O(5^3), 4 + 3*5^2 + O(5^3)]
sage: e.ellordinate('z+2*z^2+O(z^4)')
[-2*z - 7*z^2 - 23*z^3 + O(z^4), -1 + 2*z + 7*z^2 + 23*z^3 + O(z^4)]

ellpointtoz(P, precision=0)¶

e.ellpointtoz(P): return the complex number (in the fundamental parallelogram) corresponding to the point P on the elliptic curve e, under the complex uniformization of e given by the Weierstrass p-function.

The complex number z returned by this function lies in the parallelogram formed by the real and complex periods of e, as given by e.omega().

EXAMPLES:

sage: e = pari([0,0,0,1,0]).ellinit()
sage: e.ellpointtoz([0,0])
1.85407467730137

The point at infinity is sent to the complex number 0:

sage: e.ellpointtoz([0])
0

ellpow(z, n)¶

e.ellpow(z, n): return n times the point z on the elliptic curve e.

INPUT:

e - elliptic curve
z - point on e
n - integer, or a complex quadratic integer of complex multiplication for e (CM case is currently broken in pari)

EXAMPLES: We consider a CM curve:

sage: e = pari([0,0,0,1,0]).ellinit()

Multiplication by two:

sage: e.ellpow([0,0], 2)
[0]

Complex multiplication (this is broken at the moment):

sage: e.ellpow([0,0], I+1) # optional

ellrootno(p=1)¶

e.ellrootno(p): return the (local or global) root number of the $L$ -series of the elliptic curve e

If p is a prime number, the local root number at p is returned. If p is 1, the global root number is returned. Note that the global root number is the sign of the functional equation of the $L$ -series, and therefore conjecturally equal to the parity of the rank of e.

INPUT:

e - elliptic curve over $\QQ$
p (default = 1) - 1 or a prime number

OUTPUT: 1 or -1

EXAMPLES: Here is a curve of rank 3:

sage: e = pari([0,0,0,-82,0]).ellinit()
sage: e.ellrootno()
-1
sage: e.ellrootno(2)
1
sage: e.ellrootno(1009)
1

ellsigma(z, flag=0)¶

e.ellsigma(z, flag=0): return the value at the complex point z of the Weierstrass $\sigma$ function associated to the elliptic curve e.

EXAMPLES:

sage: e = pari([0,0,0,1,0]).ellinit()
sage: C.<i> = ComplexField()
sage: e.ellsigma(2+i)
1.43490215804166 + 1.80307856719256*I

ellsub(z0, z1)¶

e.ellsub(z0, z1): return z0-z1 on this elliptic curve.

INPUT:

e - elliptic curve E
z0 - point on E
z1 - point on E

OUTPUT: point on E

EXAMPLES:

sage: e = pari([0, 1, 1, -2, 0]).ellinit()
sage: e.ellsub([1,0], [-1,1])
[0, 0]

elltaniyama()¶

elltors(flag=0)¶

e.elltors(flag = 0): return information about the torsion subgroup of the elliptic curve e

INPUT:

e - elliptic curve over $\QQ$
flag (optional) - specify which algorithm to use:
0 (default) - use Doud’s algorithm: bound torsion by computing the cardinality of e(GF(p)) for small primes of good reduction, then look for torsion points using Weierstrass parametrization and Mazur’s classification
1 - use algorithm given by the Nagell-Lutz theorem (this is much slower)

OUTPUT:

gen - the order of the torsion subgroup, a.k.a. the number of points of finite order
gen - vector giving the structure of the torsion subgroup as a product of cyclic groups, sorted in non-increasing order
gen - vector giving points on e generating these cyclic groups

EXAMPLES:

sage: e = pari([1,0,1,-19,26]).ellinit()
sage: e.elltors()
[12, [6, 2], [[-2, 8], [3, -2]]]

ellwp(z='z', n=20, flag=0)¶

ellwp(E, z,flag=0): Return the complex value of the Weierstrass P-function at z on the lattice defined by e.

INPUT:

E - list OR elliptic curve
list - [om1, om2], which are Z-generators for a lattice
elliptic curve - created using ellinit
z - (optional) complex number OR string (default = “z”)
complex number - any number in the complex plane
string (or PARI variable) - name of a variable.
n - int (optional: default 20) if z is a variable, compute up to at least $o(z^n)$ .
flag - int: 0 (default): compute only P(z) 1 compute [P(z),P’(z)] 2 consider om or E as an elliptic curve and use P-function to compute the point on E (with the Weierstrass equation for E) P(z) for that curve (identical to ellztopoint in this case).

OUTPUT:

gen - complex number or list of two complex numbers

EXAMPLES:

We first define the elliptic curve X_0(11):

sage: E = pari([0,-1,1,-10,-20]).ellinit()

Compute P(1).

sage: E.ellwp(1)
13.9658695257485 + 0.E-18*I

Compute P(1+i), where i = sqrt(-1).

sage: C.<i> = ComplexField()
sage: E.ellwp(pari(1+i))
-1.11510682565555 + 2.33419052307470*I
sage: E.ellwp(1+i)
-1.11510682565555 + 2.33419052307470*I

The series expansion, to the default 20 precision:

sage: E.ellwp()
z^-2 + 31/15*z^2 + 2501/756*z^4 + 961/675*z^6 + 77531/41580*z^8 + 1202285717/928746000*z^10 + 2403461/2806650*z^12 + 30211462703/43418875500*z^14 + 3539374016033/7723451736000*z^16 + 413306031683977/1289540602350000*z^18 + O(z^20)

Compute the series for wp to lower precision:

sage: E.ellwp(n=4)
z^-2 + 31/15*z^2 + O(z^4)

Next we use the version where the input is generators for a lattice:

sage: pari([1.2692, 0.63 + 1.45*i]).ellwp(1)
13.9656146936689 + 0.000644829272810...*I

With flag 1 compute the pair P(z) and P’(z):

sage: E.ellwp(1, flag=1)
[13.9658695257485 + 0.E-18*I, 50.5619300880073 ... E-18*I]

With flag=2, the computed pair is (x,y) on the curve instead of [P(z),P’(z)]:

sage: E.ellwp(1, flag=2)
[14.2992028590818 + 0.E-18*I, 50.0619300880073 ... E-18*I] 

ellzeta(z)¶

e.ellzeta(z): return the value at the complex point z of the Weierstrass $\zeta$ function associated with the elliptic curve e.

Note

This function has infinitely many poles (one of which is at z=0); attempting to evaluate it too close to one of the poles will result in a PariError.

INPUT:

e - elliptic curve
z - complex number

EXAMPLES:

sage: e = pari([0,0,0,1,0]).ellinit()
sage: e.ellzeta(1)
1.06479841295883 + 0.E-19*I                # 32-bit
1.06479841295883 - 5.42101086242752 E-20*I # 64-bit
sage: C.<i> = ComplexField()
sage: e.ellzeta(i-1)
-0.350122658523049 - 0.350122658523049*I

ellztopoint(z)¶

e.ellztopoint(z): return the point on the elliptic curve e corresponding to the complex number z, under the usual complex uniformization of e by the Weierstrass p-function.

INPUT:

e - elliptic curve
z - complex number

OUTPUT point on e

EXAMPLES:

sage: e = pari([0,0,0,1,0]).ellinit()
sage: C.<i> = ComplexField()
sage: e.ellztopoint(1+i)
[0.E-19 - 1.02152286795670*I, -0.149072813701096 - 0.149072813701096*I] # 32-bit
[8.67655312026478 E-20 - 1.02152286795670*I, -0.149072813701096 - 0.149072813701096*I] # 64-bit

Complex numbers belonging to the period lattice of e are of course sent to the point at infinity on e:

sage: e.ellztopoint(0)
[0]

erfc(x, precision=0)¶

Return the complementary error function:

$(2/\sqrt{\pi}) \int_{x}^{\infty} e^{-t^2} dt.$

If $x$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If $x$ is inexact (e.g. real), its own precision is used in the computation, and the parameter precision is ignored.

EXAMPLES:

sage: pari(1).erfc()
0.157299207050285

eta(x, flag=0, precision=0)¶

x.eta(flag=0): if flag=0, $\eta$ function without the $q^{1/24}$ ; otherwise $\eta$ of the complex number $x$ in the upper half plane intelligently computed using $\mathrm{SL}(2,\ZZ)$ transformations.

DETAILS: This functions computes the following. If the input $x$ is a complex number with positive imaginary part, the result is $\prod_{n=1}^{\infty} (q-1^n)$ , where $q=e^{2 i \pi x}$ . If $x$ is a power series (or can be converted to a power series) with positive valuation, the result is $\prod_{n=1}^{\infty} (1-x^n)$ .

If $x$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If $x$ is inexact (e.g. real), its own precision is used in the computation, and the parameter precision is ignored.

EXAMPLES:

sage: C.<i> = ComplexField()
sage: pari(i).eta()
0.998129069925959 + 0.E-21*I

eval(x)¶

exp(precision=0)¶

x.exp(): exponential of x.

If $x$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If $x$ is inexact (e.g. real), its own precision is used in the computation, and the parameter precision is ignored.

EXAMPLES:

sage: pari(0).exp()
1.00000000000000
sage: pari(1).exp()
2.71828182845905
sage: pari('x+O(x^8)').exp()
1 + x + 1/2*x^2 + 1/6*x^3 + 1/24*x^4 + 1/120*x^5 + 1/720*x^6 + 1/5040*x^7 + O(x^8)

factor(limit=-1, proof=1)¶

Return the factorization of x.

INPUT:

limit - (default: -1) is optional and can be set whenever x is of (possibly recursive) rational type. If limit is set return partial factorization, using primes up to limit (up to primelimit if limit=0).

proof - (default: True) optional. If False (not the default), returned factors $<10^{15}$ may only be pseudoprimes.

Note

In the standard PARI/GP interpreter and C-library the factor command always has proof=False, so beware!

EXAMPLES:

sage: pari('x^10-1').factor()
[x - 1, 1; x + 1, 1; x^4 - x^3 + x^2 - x + 1, 1; x^4 + x^3 + x^2 + x + 1, 1]
sage: pari(2^100-1).factor()
[3, 1; 5, 3; 11, 1; 31, 1; 41, 1; 101, 1; 251, 1; 601, 1; 1801, 1; 4051, 1; 8101, 1; 268501, 1]
sage: pari(2^100-1).factor(proof=False)
[3, 1; 5, 3; 11, 1; 31, 1; 41, 1; 101, 1; 251, 1; 601, 1; 1801, 1; 4051, 1; 8101, 1; 268501, 1]

We illustrate setting a limit:

sage: pari(next_prime(10^50)*next_prime(10^60)*next_prime(10^4)).factor(10^5)
[10007, 1; 100000000000000000000000000000000000000000000000151000000000700000000000000000000000000000000000000000000001057, 1]

PARI doesn’t have an algorithm for factoring multivariate polynomials:

sage: pari('x^3 - y^3').factor()
...
PariError: sorry, (15)

factormod(p, flag=0)¶: x.factormod(p,flag=0): factorization mod p of the polynomial x using Berlekamp. flag is optional, and can be 0: default or 1: simple factormod, same except that only the degrees of the irreducible factors are given.

factorpadic(p, r=20, flag=0)¶: self.factorpadic(p,r=20,flag=0): p-adic factorization of the polynomial x to precision r. flag is optional and may be set to 0 (use round 4) or 1 (use Buchmann-Lenstra)

fibonacci(x)¶

Return the Fibonacci number of index x.

EXAMPLES:

sage: pari(18).fibonacci()
2584
sage: [pari(n).fibonacci() for n in range(10)]
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34]

floor(x)¶

For real x: return the largest integer = x. For rational functions: the quotient of numerator by denominator. For lists: apply componentwise.

INPUT:

x - gen

OUTPUT: gen

EXAMPLES:

sage: pari(5/9).floor()
0
sage: pari(11/9).floor()
1
sage: pari(1.17).floor()
1
sage: pari([1.5,2.3,4.99]).floor()
[1, 2, 4]
sage: pari([[1.1,2.2],[3.3,4.4]]).floor()
[[1, 2], [3, 4]]
sage: pari(x).floor()
x
sage: pari((x^2+x+1)/x).floor()
x + 1
sage: pari(x^2+5*x+2.5).floor()
x^2 + 5*x + 2.50000000000000

sage: pari('"hello world"').floor()
...
PariError: incorrect type (20)

frac(x)¶

frac(x): Return the fractional part of x, which is x - floor(x).

INPUT:

x - gen

OUTPUT: gen

EXAMPLES:

sage: pari(1.75).frac()
0.750000000000000
sage: pari(sqrt(2)).frac()
0.414213562373095
sage: pari('sqrt(-2)').frac()
...
PariError: incorrect type (20)

galoisapply(aut, x)¶

galoisconj()¶

galoisfixedfield(v, flag, y)¶

galoisinit(den=None)¶: galoisinit(K{,den}): calculate Galois group of number field K; see PARI manual for meaning of den

galoispermtopol(perm)¶

gamma(s, precision=0)¶

s.gamma(precision): Gamma function at s.

If $s$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If $s$ is inexact (e.g. real), its own precision is used in the computation, and the parameter precision is ignored.

EXAMPLES:

sage: pari(2).gamma()
1.00000000000000
sage: pari(5).gamma()
24.0000000000000
sage: C.<i> = ComplexField()
sage: pari(1+i).gamma()
0.498015668118356 - 0.154949828301811*I

TESTS:

sage: pari(-1).gamma()
...
PariError:  (8)

gammah(s, precision=0)¶

s.gammah(): Gamma function evaluated at the argument x+1/2.

If $s$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If $s$ is inexact (e.g. real), its own precision is used in the computation, and the parameter precision is ignored.

EXAMPLES:

sage: pari(2).gammah()
1.32934038817914
sage: pari(5).gammah()
52.3427777845535
sage: C.<i> = ComplexField()
sage: pari(1+i).gammah()
0.575315188063452 + 0.0882106775440939*I

gcd(x, y, flag=0)¶: gcd(x,y,flag=0): greatest common divisor of x and y. flag is optional, and can be 0: default, 1: use the modular gcd algorithm (x and y must be polynomials), 2 use the subresultant algorithm (x and y must be polynomials)

getattr(attr)¶

hilbert(x, y, p)¶

hyperu(a, b, x, precision=0)¶

a.hyperu(b,x): U-confluent hypergeometric function.

If $a$ , $b$ , or $x$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If the arguments are inexact (e.g. real), the smallest of their precisions is used in the computation, and the parameter precision is ignored.

EXAMPLES:

sage: pari(1).hyperu(2,3)
0.333333333333333

idealadd(x, y)¶

idealaddtoone(x, y)¶

idealappr(x, flag=0)¶

idealcoprime(x, y)¶

Given two integral ideals x and y of a pari number field self, return an element a of the field (expressed in the integral basis of self) such that a*x is an integral ideal coprime to y.

EXAMPLES:

sage: F = NumberField(x^3-2, 'alpha')
sage: nf = F._pari_()
sage: x = pari('[1, -1, 2]~')
sage: y = pari('[1, -1, 3]~')
sage: nf.idealcoprime(x, y)
[1, 0, 0]~

sage: y = pari('[2, -2, 4]~')
sage: nf.idealcoprime(x, y)
[5/43, 9/43, -1/43]~

idealdiv(x, y, flag=0)¶

idealfactor(x)¶

idealhnf(a, b=None)¶

idealintersection(x, y)¶

ideallog(x, bid)¶

Return the discrete logarithm of the unit x in (ring of integers)/bid.

INPUT:

self - a pari number field
bid - a big ideal structure (corresponding to an ideal I of self) output by idealstar
x - an element of self with valuation zero at all primes dividing I

OUTPUT:

the discrete logarithm of x on the generators given in bid[2]

EXAMPLE:

sage: F = NumberField(x^3-2, 'alpha')
sage: nf = F._pari_()
sage: I = pari('[1, -1, 2]~')
sage: bid = nf.idealstar(I)
sage: x = pari('5')
sage: nf.ideallog(x, bid)
[25]~

idealmul(x, y, flag=0)¶

idealnorm(x)¶

idealred(I, vdir=0)¶

idealstar(I, flag=1)¶

Return the big ideal (bid) structure of modulus I.

INPUT:

self - a pari number field
I – an ideal of self, or a row vector whose first component is an ideal and whose second component is a row vector of r_1 0 or 1.
flag - determines the amount of computation and the shape of the output:
- 1 (default): return a bid structure without generators
- 2: return a bid structure with generators (slower)
- 0 (deprecated): only outputs units of (ring of integers/I) as an abelian group, i.e as a 3-component vector [h,d,g]: h is the order, d is the vector of SNF cyclic components and g the corresponding generators. This flag is deprecated: it is in fact slightly faster to compute a true bid structure, which contains much more information.

EXAMPLE:

sage: F = NumberField(x^3-2, 'alpha')
sage: nf = F._pari_()
sage: I = pari('[1, -1, 2]~')
sage: nf.idealstar(I)
[[[43, 9, 5; 0, 1, 0; 0, 0, 1], [0]], [42, [42]], Mat([[43, [9, 1, 0]~, 1, 1, [-5, -9, 1]~], 1]), [[[[42], [[3, 0, 0]~], [[3, 0, 0]~], [[]~], 1]], [[], [], [;]]], Mat(1)]

idealtwoelt(x, a=None)¶

idealval(x, p)¶

imag(x)¶

imag(x): Return the imaginary part of x. This function also works component-wise.

INPUT:

x - gen

OUTPUT: gen

EXAMPLES:

sage: pari('1+2*I').imag()
2
sage: pari(sqrt(-2)).imag()
1.41421356237310
sage: pari('x+I').imag()
1
sage: pari('x+2*I').imag()
2
sage: pari('(1+I)*x^2+2*I').imag()
x^2 + 2
sage: pari('[1,2,3] + [4*I,5,6]').imag()
[4, 0, 0]

incgam(s, x, y=None, precision=0)¶

s.incgam(x, y, precision): incomplete gamma function. y is optional and is the precomputed value of gamma(s).

If $s$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If $s$ is inexact (e.g. real), its own precision is used in the computation, and the parameter precision is ignored.

EXAMPLES:

sage: C.<i> = ComplexField()
sage: pari(1+i).incgam(3-i)
-0.0458297859919946 + 0.0433696818726677*I

incgamc(s, x, precision=0)¶

s.incgamc(x): complementary incomplete gamma function.

The arguments $x$ and $s$ are complex numbers such that $s$ is not a pole of $\Gamma$ and $|x|/(|s|+1)$ is not much larger than $1$ (otherwise, the convergence is very slow). The function returns the value of the integral $\int_{0}^{x} e^{-t} t^{s-1} dt.$

If $s$ or $x$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If the arguments are inexact (e.g. real), the smallest of their precisions is used in the computation, and the parameter precision is ignored.

EXAMPLES:

sage: pari(1).incgamc(2)
0.864664716763387

int_unsafe()¶

Returns int form of self, but raises an exception if int does not fit into a long integer.

This is about 5 times faster than the usual int conversion.

intformal(y=-1)¶: x.intformal(y): formal integration of x with respect to the main variable of y, or to the main variable of x if y is omitted

intvec_unsafe()¶

Returns Python int list form of entries of self, but raises an exception if int does not fit into a long integer. Here self must be a vector.

EXAMPLES:

sage: pari('[3,4,5]').type()
't_VEC'
sage: pari('[3,4,5]').intvec_unsafe()
[3, 4, 5]
sage: type(pari('[3,4,5]').intvec_unsafe()[0])
<type 'int'>

TESTS:

sage: pari(3).intvec_unsafe()
...
TypeError: gen must be of PARI type t_VEC
sage: pari('[2^150,1]').intvec_unsafe()
...
PariError: impossible assignment I-->S (23)

ispower(k=None)¶

Determine whether or not self is a perfect k-th power. If k is not specified, find the largest k so that self is a k-th power.

INPUT:

k - int (optional)

OUTPUT:

power - int, what power it is
g - what it is a power of

EXAMPLES:

sage: pari(9).ispower()
(2, 3)
sage: pari(17).ispower()
(1, 17)
sage: pari(17).ispower(2)
(False, None)
sage: pari(17).ispower(1)
(1, 17)
sage: pari(2).ispower()
(1, 2)

isprime(flag=0)¶

isprime(x, flag=0): Returns True if x is a PROVEN prime number, and False otherwise.

INPUT:

flag - int 0 (default): use a combination of algorithms. 1: certify primality using the Pocklington-Lehmer Test. 2: certify primality using the APRCL test.

OUTPUT:

bool - True or False

EXAMPLES:

sage: pari(9).isprime()
False
sage: pari(17).isprime()
True
sage: n = pari(561)    # smallest Carmichael number
sage: n.isprime()      # not just a pseudo-primality test!
False
sage: n.isprime(1)
False
sage: n.isprime(2)
False

ispseudoprime(flag=0)¶

ispseudoprime(x, flag=0): Returns True if x is a pseudo-prime number, and False otherwise.

INPUT:

flag - int 0 (default): checks whether x is a Baillie-Pomerance-Selfridge-Wagstaff pseudo prime (strong Rabin-Miller pseudo prime for base 2, followed by strong Lucas test for the sequence (P,-1), P smallest positive integer such that $P^2 - 4$ is not a square mod x). 0: checks whether x is a strong Miller-Rabin pseudo prime for flag randomly chosen bases (with end-matching to catch square roots of -1).

OUTPUT:

bool - True or False

EXAMPLES:

sage: pari(9).ispseudoprime()
False
sage: pari(17).ispseudoprime()
True
sage: n = pari(561)     # smallest Carmichael number
sage: n.ispseudoprime() # not just any old pseudo-primality test!
False
sage: n.ispseudoprime(2)
False

issquare(x, find_root=False)¶: issquare(x,n): true(1) if x is a square, false(0) if not. If find_root is given, also returns the exact square root if it was computed.

issquarefree()¶

EXAMPLES:

sage: pari(10).issquarefree()
True
sage: pari(20).issquarefree()
False

j()¶

e.j(): return the j-invariant of the elliptic curve e.

EXAMPLES:

sage: e = pari([0, -1, 1, -10, -20]).ellinit()
sage: e.j()
-122023936/161051
sage: _.factor()
[-1, 1; 2, 12; 11, -5; 31, 3]

kronecker(y)¶

lcm(x, y)¶

Return the least common multiple of x and y. EXAMPLES:

sage: pari(10).lcm(15)
30

length()¶

lex(x, y)¶: lex(x,y): Compare x and y lexicographically (1 if xy, 0 if x==y, -1 if xy)

lift(x, v=-1)¶

lift(x,v): Returns the lift of an element of Z/nZ to Z or R[x]/(P) to R[x] for a type R if v is omitted. If v is given, lift only polymods with main variable v. If v does not occur in x, lift only intmods.

INPUT:

x - gen
v - (optional) variable

OUTPUT: gen

EXAMPLES:

sage: x = pari("x")
sage: a = x.Mod('x^3 + 17*x + 3')
sage: a
Mod(x, x^3 + 17*x + 3)
sage: b = a^4; b
Mod(-17*x^2 - 3*x, x^3 + 17*x + 3)
sage: b.lift()
-17*x^2 - 3*x

??? more examples

lindep(flag=0)¶

list()¶

list_str()¶: Return str that might correctly evaluate to a Python-list.

listinsert(obj, n)¶

listput(obj, n)¶

lllgram()¶

lllgramint()¶

lngamma(x, precision=0)¶

This method is deprecated, please use log_gamma() instead.

See the log_gamma() method for documentation and examples.

EXAMPLES:

sage: pari(100).lngamma()
doctest:...: DeprecationWarning: The method lngamma() is deprecated. Use log_gamma() instead.
359.134205369575

log(x, precision=0)¶

x.log(): natural logarithm of x.

This function returns the principal branch of the natural logarithm of $x$ , i.e., the branch such that $\Im(\log(x)) \in ]-\pi, \pi].$ The result is complex (with imaginary part equal to $\pi$ ) if $x\in \RR$ and $x<0$ . In general, the algorithm uses the formula

$\log(x) \simeq \frac{\pi}{2{\rm agm}(1,4/s)} - m\log(2),$

if $s=x 2^m$ is large enough. (The result is exact to $B$ bits provided that $s>2^{B/2}$ .) At low accuracies, this function computes $\log$ using the series expansion near $1$ .

If $x$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If $x$ is inexact (e.g. real), its own precision is used in the computation, and the parameter precision is ignored.

Note that $p$ -adic arguments can also be given as input, with the convention that $\log(p)=0$ . Hence, in particular, $\exp(\log(x))/x$ is not in general equal to $1$ but instead to a $(p-1)$ -st root of unity (or $\pm 1$ if $p=2$ ) times a power of $p$ .

EXAMPLES:

sage: pari(5).log()
1.60943791243410
sage: C.<i> = ComplexField()
sage: pari(i).log()
0.E-19 + 1.57079632679490*I

log_gamma(x, precision=0)¶

Logarithm of the gamma function of x.

This function returns the principal branch of the logarithm of the gamma function of $x$ . The function $\log(\Gamma(x))$ is analytic on the complex plane with non-positive integers removed. This function can have much larger inputs than $\Gamma$ itself.

The $p$ -adic analogue of this function is unfortunately not implemented.

If $x$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If $x$ is inexact (e.g. real), its own precision is used in the computation, and the parameter precision is ignored.

EXAMPLES:

sage: pari(100).log_gamma()
359.134205369575

matadjoint()¶

matadjoint(x): adjoint matrix of x.

EXAMPLES:

sage: pari('[1,2,3; 4,5,6;  7,8,9]').matadjoint()
[-3, 6, -3; 6, -12, 6; -3, 6, -3]
sage: pari('[a,b,c; d,e,f; g,h,i]').matadjoint()
[(i*e - h*f), (-i*b + h*c), (f*b - e*c); (-i*d + g*f), i*a - g*c, -f*a + d*c; (h*d - g*e), -h*a + g*b, e*a - d*b]

matdet(flag=0)¶

Return the determinant of this matrix.

INPUT:

flag - (optional) flag 0: using Gauss-Bareiss. 1: use classical Gaussian elimination (slightly better for integer entries)

EXAMPLES:

sage: pari('[1,2; 3,4]').matdet(0)
-2
sage: pari('[1,2; 3,4]').matdet(1)
-2

matfrobenius(flag=0)¶

M.matfrobenius(flag=0): Return the Frobenius form of the square matrix M. If flag is 1, return only the elementary divisors (a list of polynomials). If flag is 2, return a two-components vector [F,B] where F is the Frobenius form and B is the basis change so that $M=B^{-1} F B$ .

EXAMPLES:

sage: a = pari('[1,2;3,4]')
sage: a.matfrobenius()
[0, 2; 1, 5]
sage: a.matfrobenius(flag=1)
[x^2 - 5*x - 2]
sage: a.matfrobenius(2)
[[0, 2; 1, 5], [1, -1/3; 0, 1/3]]
sage: v = a.matfrobenius(2)
sage: v[0]
[0, 2; 1, 5]
sage: v[1]^(-1)*v[0]*v[1]
[1, 2; 3, 4]

We let t be the matrix of $T_2$ acting on modular symbols of level 43, which was computed using ModularSymbols(43,sign=1).T(2).matrix():

sage: t = pari('[3, -2, 0, 0; 0, -2, 0, 1; 0, -1, -2, 2; 0, -2, 0, 2]')
sage: t.matfrobenius()
[0, 0, 0, -12; 1, 0, 0, -2; 0, 1, 0, 8; 0, 0, 1, 1]
sage: t.charpoly('x')
x^4 - x^3 - 8*x^2 + 2*x + 12
sage: t.matfrobenius(1)
[x^4 - x^3 - 8*x^2 + 2*x + 12]

AUTHORS:

Martin Albrect (2006-04-02)

mathnf(flag=0)¶

A.mathnf(flag=0): (upper triangular) Hermite normal form of A, basis for the lattice formed by the columns of A.

INPUT:

flag - optional, value range from 0 to 4 (0 if omitted), meaning : 0: naive algorithm
1: Use Batut's algorithm - output 2-component vector [H,U] such that H is the HNF of A, and U is a unimodular matrix such that xU=H. 3: Use Batut’s algorithm. Output [H,U,P] where P is a permutation matrix such that P A U = H. 4: As 1, using a heuristic variant of LLL reduction along the way.

EXAMPLES:

sage: pari('[1,2,3; 4,5,6;  7,8,9]').mathnf()
[6, 1; 3, 1; 0, 1]

mathnfmod(d)¶

Returns the Hermite normal form if d is a multiple of the determinant

Beware that PARI’s concept of a Hermite normal form is an upper triangular matrix with the same column space as the input matrix.

INPUT:

d - multiple of the determinant of self

EXAMPLES:

       sage: M=matrix([[1,2,3],[4,5,6],[7,8,11]])
sage: d=M.det()
sage: pari(M).mathnfmod(d)
       [6, 4, 3; 0, 1, 0; 0, 0, 1]

Note that d really needs to be a multiple of the discriminant, not just of the exponent of the cokernel:

       sage: M=matrix([[1,0,0],[0,2,0],[0,0,6]])
sage: pari(M).mathnfmod(6)
[1, 0, 0; 0, 1, 0; 0, 0, 6]
sage: pari(M).mathnfmod(12)
[1, 0, 0; 0, 2, 0; 0, 0, 6]

mathnfmodid(d)¶

Returns the Hermite Normal Form of M concatenated with d*Identity

Beware that PARI’s concept of a Hermite normal form is a maximal rank upper triangular matrix with the same column space as the input matrix.

INPUT:

d - Determines

EXAMPLES:

       sage: M=matrix([[1,0,0],[0,2,0],[0,0,6]])
sage: pari(M).mathnfmodid(6)
       [1, 0, 0; 0, 2, 0; 0, 0, 6]

This routine is not completely equivalent to mathnfmod:

sage: pari(M).mathnfmod(6)
[1, 0, 0; 0, 1, 0; 0, 0, 6]

matker(flag=0)¶

Return a basis of the kernel of this matrix.

INPUT:

flag - optional; may be set to 0: default; non-zero: x is known to have integral entries.

EXAMPLES:

sage: pari('[1,2,3;4,5,6;7,8,9]').matker()
[1; -2; 1]

With algorithm 1, even if the matrix has integer entries the kernel need not be saturated (which is weird):

sage: pari('[1,2,3;4,5,6;7,8,9]').matker(1)
[3; -6; 3]
sage: pari('matrix(3,3,i,j,i)').matker()
[-1, -1; 1, 0; 0, 1]            
sage: pari('[1,2,3;4,5,6;7,8,9]*Mod(1,2)').matker()
[Mod(1, 2); Mod(0, 2); 1]

matkerint(flag=0)¶

Return the integer kernel of a matrix.

This is the LLL-reduced Z-basis of the kernel of the matrix x with integral entries.

INPUT:

flag - optional, and may be set to 0: default, uses a modified LLL, 1: uses matrixqz.

EXAMPLES:

sage: pari('[2,1;2,1]').matker()
[-1/2; 1]
sage: pari('[2,1;2,1]').matkerint()
[-1; 2]

This is worrisome (so be careful!):

sage: pari('[2,1;2,1]').matkerint(1)
Mat(1)

matsnf(flag=0)¶

x.matsnf(flag=0): Smith normal form (i.e. elementary divisors) of the matrix x, expressed as a vector d. Binary digits of flag mean 1: returns [u,v,d] where d=u*x*v, otherwise only the diagonal d is returned, 2: allow polynomial entries, otherwise assume x is integral, 4: removes all information corresponding to entries equal to 1 in d.

EXAMPLES:

sage: pari('[1,2,3; 4,5,6;  7,8,9]').matsnf()
[0, 3, 1]

matsolve(B)¶

matsolve(B): Solve the linear system Mx=B for an invertible matrix M

matsolve(B) uses Gaussian elimination to solve Mx=B, where M is invertible and B is a column vector.

The corresponding pari library routine is gauss. The gp-interface name matsolve has been given preference here.

INPUT:

B - a column vector of the same dimension as the square matrix self

EXAMPLES:

sage: pari('[1,1;1,-1]').matsolve(pari('[1;0]'))
[1/2; 1/2]

mattranspose()¶

Transpose of the matrix self.

EXAMPLES:

sage: pari('[1,2,3; 4,5,6;  7,8,9]').mattranspose()
[1, 4, 7; 2, 5, 8; 3, 6, 9]

max(x, y)¶: max(x,y): Return the maximum of x and y.

min(x, y)¶: min(x,y): Return the minimum of x and y.

modreverse()¶

modreverse(x): reverse polymod of the polymod x, if it exists.

EXAMPLES:

moebius(x)¶: moebius(x): Moebius function of x.

ncols()¶

Return the number of columns of self.

EXAMPLES:

sage: pari('matrix(19,8)').ncols()
8

newtonpoly(p)¶

x.newtonpoly(p): Newton polygon of polynomial x with respect to the prime p.

EXAMPLES:

sage: x = pari('y^8+6*y^6-27*y^5+1/9*y^2-y+1')
sage: x.newtonpoly(3)
[1, 1, -1/3, -1/3, -1/3, -1/3, -1/3, -1/3]

nextprime(add_one=0)¶

nextprime(x): smallest pseudoprime = x

EXAMPLES:

sage: pari(1).nextprime()
2
sage: pari(2^100).nextprime()
1267650600228229401496703205653

nfbasis(flag=0, p=0)¶

nfbasis_d(flag=0, p=0)¶

nfbasis_d(x): Return a basis of the number field defined over QQ by x and its discriminant.

EXAMPLES:

sage: F = NumberField(x^3-2,'alpha')
sage: F._pari_()[0].nfbasis_d()
([1, x, x^2], -108)

sage: G = NumberField(x^5-11,'beta')
sage: G._pari_()[0].nfbasis_d()
([1, x, x^2, x^3, x^4], 45753125)

sage: pari([-2,0,0,1]).Polrev().nfbasis_d()
([1, x, x^2], -108)

nfdisc(flag=0, p=0)¶

nfdisc(x): Return the discriminant of the number field defined over QQ by x.

EXAMPLES:

sage: F = NumberField(x^3-2,'alpha')
sage: F._pari_()[0].nfdisc()
-108

sage: G = NumberField(x^5-11,'beta')
sage: G._pari_()[0].nfdisc()
45753125

sage: f = x^3-2
sage: f._pari_()
x^3 - 2
sage: f._pari_().nfdisc()
-108

nfeltreduce(x, I)¶

Given an ideal I in Hermite normal form and an element x of the pari number field self, finds an element r in self such that x-r belongs to the ideal and r is small.

EXAMPLES:

sage: k.<a> = NumberField(x^2 + 5)
sage: I = k.ideal(a)
sage: kp = pari(k)
sage: kp.nfeltreduce(12, I.pari_hnf())
[2, 0]~
sage: 12 - k(kp.nfeltreduce(12, I.pari_hnf())) in I
True

nffactor(x)¶

nfgaloisconj()¶

Edited from the pari documentation:

nfgaloisconj(nf): list of conjugates of a root of the polynomial x=nf.pol in the same number field.

Uses a combination of Allombert’s algorithm and nfroots.

EXAMPLES:

sage: x = QQ['x'].0; nf = pari(x^2 + 2).nfinit()
sage: nf.nfgaloisconj()
[-x, x]~
sage: nf = pari(x^3 + 2).nfinit()
sage: nf.nfgaloisconj()
[x]~
sage: nf = pari(x^4 + 2).nfinit()
sage: nf.nfgaloisconj()
[-x, x]~

nfgenerator()¶

nfinit(flag=0)¶

nfisisom(other)¶

nfisisom(x, y): Determine if the number fields defined by x and y are isomorphic. According to the PARI documentation, this is much faster if at least one of x or y is a number field. If they are isomorphic, it returns an embedding for the generators. If not, returns 0.

EXAMPLES:

sage: F = NumberField(x^3-2,'alpha')
sage: G = NumberField(x^3-2,'beta')
sage: F._pari_().nfisisom(G._pari_())
[x]

sage: GG = NumberField(x^3-4,'gamma')
sage: F._pari_().nfisisom(GG._pari_())
[1/2*x^2]

sage: F._pari_().nfisisom(GG.pari_nf())
[1/2*x^2]

sage: F.pari_nf().nfisisom(GG._pari_()[0])
[x^2]

sage: H = NumberField(x^2-2,'alpha')
sage: F._pari_().nfisisom(H._pari_())
0

nfroots(poly)¶

Return the roots of $poly$ in the number field self without multiplicity.

EXAMPLES:

sage: y = QQ['yy'].0; _ = pari(y) # pari has variable ordering rules
sage: x = QQ['zz'].0; nf = pari(x^2 + 2).nfinit()
sage: nf.nfroots(y^2 + 2)
[Mod(-zz, zz^2 + 2), Mod(zz, zz^2 + 2)]
sage: nf = pari(x^3 + 2).nfinit()
sage: nf.nfroots(y^3 + 2)
[Mod(zz, zz^3 + 2)]
sage: nf = pari(x^4 + 2).nfinit()
sage: nf.nfroots(y^4 + 2)
[Mod(-zz, zz^4 + 2), Mod(zz, zz^4 + 2)]

nfrootsof1()¶

nf.nfrootsof1()

number of roots of unity and primitive root of unity in the number field nf.

EXAMPLES:

sage: nf = pari('x^2 + 1').nfinit()
sage: nf.nfrootsof1()
[4, [0, 1]~]

nfsubfields(d=0)¶

Find all subfields of degree d of number field nf (all subfields if d is null or omitted). Result is a vector of subfields, each being given by [g,h], where g is an absolute equation and h expresses one of the roots of g in terms of the root x of the polynomial defining nf.

INPUT:

self - nf number field
d - integer

norm()¶

nrows()¶

Return the number of rows of self.

EXAMPLES:

sage: pari('matrix(19,8)').nrows()
19

numbpart(x)¶

numbpart(x): returns the number of partitions of x.

EXAMPLES:

sage: pari(20).numbpart()
627
sage: pari(100).numbpart()
190569292

numdiv(n)¶

Return the number of divisors of the integer n.

EXAMPLES:

sage: pari(10).numdiv()
4

numerator(x)¶

numerator(x): Returns the numerator of x.

INPUT:

x - gen

OUTPUT: gen

EXAMPLES:

numtoperm(k, n)¶

numtoperm(k, n): Return the permutation number k (mod n!) of n letters, where n is an integer.

INPUT:

k - gen, integer
n - int

OUTPUT:

gen - vector (permutation of 1,...,n)

EXAMPLES:

omega()¶

e.omega(): return basis for the period lattice of the elliptic curve e.

EXAMPLES:

sage: e = pari([0, -1, 1, -10, -20]).ellinit()
sage: e.omega()
[1.26920930427955, 0.634604652139777 + 1.45881661693850*I]

order()¶

padicappr(a)¶: x.padicappr(a): p-adic roots of the polynomial x congruent to a mod p

padicprec(x, p)¶

padicprec(x,p): Return the absolute p-adic precision of the object x.

INPUT:

x - gen

OUTPUT: int

EXAMPLES:

parent()¶

permtonum(x)¶

permtonum(x): Return the ordinal (between 1 and n!) of permutation vector x. ??? Huh ??? say more. what is a perm vector. 0 to n-1 or 1-n.

INPUT:

x - gen (vector of integers)

OUTPUT:

gen - integer

EXAMPLES:

phi(n)¶

Return the Euler phi function of n. EXAMPLES:

sage: pari(10).phi()
4

polcoeff(n, var=-1)¶

EXAMPLES:

sage: f = pari("x^2 + y^3 + x*y")
sage: f
x^2 + y*x + y^3
sage: f.polcoeff(1)
y
sage: f.polcoeff(3)
0
sage: f.polcoeff(3, "y")
1
sage: f.polcoeff(1, "y")
x

polcompositum(pol2, flag=0)¶

poldegree(var=-1)¶: f.poldegree(var=x): Return the degree of this polynomial.

poldisc(var=-1)¶: f.poldist(var=x): Return the discriminant of this polynomial.

poldiscreduced()¶

polgalois()¶: f.polgalois(): Galois group of the polynomial f

polhensellift(y, p, e)¶: self.polhensellift(y, p, e): lift the factorization y of self modulo p to a factorization modulo $p^e$ using Hensel lift. The factors in y must be pairwise relatively prime modulo p.

polinterpolate(ya, x)¶: self.polinterpolate(ya,x,e): polynomial interpolation at x according to data vectors self, ya (i.e. return P such that P(self[i]) = ya[i] for all i). Also return an error estimate on the returned value.

polisirreducible()¶: f.polisirreducible(): Returns True if f is an irreducible non-constant polynomial, or False if f is reducible or constant.

pollead(v=-1)¶: self.pollead(v): leading coefficient of polynomial or series self, or self itself if self is a scalar. Error otherwise. With respect to the main variable of self if v is omitted, with respect to the variable v otherwise

polrecip()¶

polred(flag=0, fa=None)¶

polredabs(flag=0)¶

polresultant(y, var=-1, flag=0)¶

polroots(flag=0, precision=0)¶: polroots(x,flag=0): complex roots of the polynomial x. flag is optional, and can be 0: default, uses Schonhage’s method modified by Gourdon, or 1: uses a modified Newton method.

polrootsmod(p, flag=0)¶

polrootspadic(p, r=20)¶

polrootspadicfast(p, r=20)¶

polsturm(a, b)¶

polsturm_full()¶

polsylvestermatrix(g)¶

polsym(n)¶

polylog(x, m, flag=0, precision=0)¶

x.polylog(m,flag=0): m-th polylogarithm of x. flag is optional, and can be 0: default, 1: D_m -modified m-th polylog of x, 2: D_m-modified m-th polylog of x, 3: P_m-modified m-th polylog of x.

If $x$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If $x$ is inexact (e.g. real), its own precision is used in the computation, and the parameter precision is ignored.

TODO: Add more explanation, copied from the PARI manual.

EXAMPLES:

sage: pari(10).polylog(3)
5.64181141475134 - 8.32820207698027*I
sage: pari(10).polylog(3,0)
5.64181141475134 - 8.32820207698027*I
sage: pari(10).polylog(3,1)
0.523778453502411
sage: pari(10).polylog(3,2)
-0.400459056163451

precision(x, n=-1)¶

precision(x,n): Change the precision of x to be n, where n is a C-integer). If n is omitted, output the real precision of x.

INPUT:

x - gen
n - (optional) int

OUTPUT: nothing or gen if n is omitted

EXAMPLES:

primepi()¶

Return the number of primes less than or equal to self.

EXAMPLES:

sage: pari(7).primepi()
4
sage: pari(100).primepi()
25
sage: pari(1000).primepi()
168
sage: pari(100000).primepi()
9592
sage: pari(0).primepi()
0
sage: pari(-15).primepi()
0
sage: pari(500509).primepi()
41581

printtex(x)¶

psi(x, precision=0)¶

x.psi(): psi-function at x.

Return the $\psi$ -function of $x$ , i.e., the logarithmic derivative $\Gamma'(x)/\Gamma(x)$ .

If $x$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If $x$ is inexact (e.g. real), its own precision is used in the computation, and the parameter precision is ignored.

EXAMPLES:

sage: pari(1).psi()
-0.577215664901533

python(locals=None)¶

Return Python eval of self.

Note: is self is a real (type t_REAL) the result will be a RealField element of the equivalent precision; if self is a complex (type t_COMPLEX) the result will be a ComplexField element of precision the minimum precision of the real and imaginary parts.

EXAMPLES:

sage: pari('389/17').python()
389/17
sage: f = pari('(2/3)*x^3 + x - 5/7 + y'); f
2/3*x^3 + x + (y - 5/7)
sage: var('x,y')
(x, y)
sage: f.python({'x':x, 'y':y})
2/3*x^3 + x + y - 5/7

You can also use .sage, which is a psynonym:

sage: f.sage({'x':x, 'y':y})
2/3*x^3 + x + y - 5/7

python_list()¶

Return a Python list of the PARI gens. This object must be of type t_VEC

INPUT: NoneOUTPUT:

list - Python list whose elements are the elements of the input gen.

EXAMPLES:

sage: v=pari([1,2,3,10,102,10])
sage: w = v.python_list()
sage: w
[1, 2, 3, 10, 102, 10]
sage: type(w[0])
<type 'sage.libs.pari.gen.gen'>
sage: pari("[1,2,3]").python_list()
[1, 2, 3]

python_list_small()¶

Return a Python list of the PARI gens. This object must be of type t_VECSMALL, and the resulting list contains python ‘int’s

EXAMPLES:

sage: v=pari([1,2,3,10,102,10]).Vecsmall()
sage: w = v.python_list_small()
sage: w
[1, 2, 3, 10, 102, 10]
sage: type(w[0])
<type 'int'>

qfbhclassno(n)¶

Computes the Hurwitz-Kronecker class number of n.

If n is large (more than $5*10^5$ ), the result is conditional upon GRH.

EXAMPLES:

sage: pari(-10007).qfbhclassno()
77
sage: pari(-3).qfbhclassno()
1/3

qflll(flag=0, precision=0)¶: qflll(x,flag=0): LLL reduction of the vectors forming the matrix x (gives the unimodular transformation matrix). The columns of x must be linearly independent, unless specified otherwise below. flag is optional, and can be 0: default, 1: assumes x is integral, columns may be dependent, 2: assumes x is integral, returns a partially reduced basis, 4: assumes x is integral, returns [K,I] where K is the integer kernel of x and I the LLL reduced image, 5: same as 4 but x may have polynomial coefficients, 8: same as 0 but x may have polynomial coefficients.

qflllgram(flag=0, precision=0)¶: qflllgram(x,flag=0): LLL reduction of the lattice whose gram matrix is x (gives the unimodular transformation matrix). flag is optional and can be 0: default,1: lllgramint algorithm for integer matrices, 4: lllgramkerim giving the kernel and the LLL reduced image, 5: lllgramkerimgen same when the matrix has polynomial coefficients, 8: lllgramgen, same as qflllgram when the coefficients are polynomials.

qfminim(B, max, flag=0)¶: qfminim(x,bound,maxnum,flag=0): number of vectors of square norm = bound, maximum norm and list of vectors for the integral and definite quadratic form x; minimal non-zero vectors if bound=0. flag is optional, and can be 0: default; 1: returns the first minimal vector found (ignore maxnum); 2: as 0 but uses a more robust, slower implementation, valid for non integral quadratic forms.

qfrep(B, flag=0)¶: qfrep(x,B,flag=0): vector of (half) the number of vectors of norms from 1 to B for the integral and definite quadratic form x. Binary digits of flag mean 1: count vectors of even norm from 1 to 2B, 2: return a t_VECSMALL instead of a t_VEC.

random(N)¶

random(N=2^31): Return a pseudo-random integer between 0 and $N-1$ .

INPUT:

-N - gen, integer

OUTPUT:

gen - integer

EXAMPLES:

real(x)¶

real(x): Return the real part of x.

INPUT:

x - gen

OUTPUT: gen

EXAMPLES:

reverse()¶: Return the polynomial obtained by reversing the coefficients of this polynomial.

rnfcharpoly(T, a, v='x')¶

rnfdisc(x)¶

rnfeltabstorel(x)¶

rnfeltreltoabs(x)¶

rnfequation(poly, flag=0)¶

rnfidealabstorel(x)¶

rnfidealdown(x)¶

rnfidealdown(rnf,x): finds the intersection of the ideal x with the base field.

EXAMPLES:: sage: x = ZZ[‘xx1’].0; pari(x) xx1 sage: y = ZZ[‘yy1’].0; pari(y) yy1 sage: nf = pari(y^2 - 6*y + 24).nfinit() sage: rnf = nf.rnfinit(x^2 - pari(y))

This is the relative HNF of the inert ideal (2) in rnf:

sage: P = pari(‘[[[1, 0]~, [0, 0]~; [0, 0]~, [1, 0]~], [[2, 0; 0, 2], [2, 0; 0, 1/2]]]’)

And this is the HNF of the inert ideal (2) in nf:

sage: rnf.rnfidealdown(P) [2, 0; 0, 2]

rnfidealhnf(x)¶

rnfidealnormrel(x)¶

rnfidealreltoabs(x)¶

rnfidealtwoelt(x)¶

rnfinit(poly)¶

EXAMPLES: We construct a relative number field.

sage: f = pari('y^3+y+1')
sage: K = f.nfinit()
sage: x = pari('x'); y = pari('y')
sage: g = x^5 - x^2 + y
sage: L = K.rnfinit(g)

rnfisfree(poly)¶

round(x, estimate=False)¶

round(x,estimate=False): If x is a real number, returns x rounded to the nearest integer (rounding up). If the optional argument estimate is True, also returns the binary exponent e of the difference between the original and the rounded value (the “fractional part”) (this is the integer ceiling of log_2(error)).

When x is a general PARI object, this function returns the result of rounding every coefficient at every level of PARI object. Note that this is different than what the truncate function does (see the example below).

One use of round is to get exact results after a long approximate computation, when theory tells you that the coefficients must be integers.

INPUT:

x - gen
estimate - (optional) bool, False by default

OUTPUT:

if estimate is False, return a single gen.
if estimate is True, return rounded version of x and error estimate in bits, both as gens.

EXAMPLES:

sage: pari('1.5').round()
2
sage: pari('1.5').round(True)
(2, -1)
sage: pari('1.5 + 2.1*I').round()
2 + 2*I
sage: pari('1.0001').round(True)
(1, -14)
sage: pari('(2.4*x^2 - 1.7)/x').round()
(2*x^2 - 2)/x
sage: pari('(2.4*x^2 - 1.7)/x').truncate()
2.40000000000000*x

sage(locals=None)¶

Return Python eval of self.

Note: is self is a real (type t_REAL) the result will be a RealField element of the equivalent precision; if self is a complex (type t_COMPLEX) the result will be a ComplexField element of precision the minimum precision of the real and imaginary parts.

EXAMPLES:

sage: pari('389/17').python()
389/17
sage: f = pari('(2/3)*x^3 + x - 5/7 + y'); f
2/3*x^3 + x + (y - 5/7)
sage: var('x,y')
(x, y)
sage: f.python({'x':x, 'y':y})
2/3*x^3 + x + y - 5/7

You can also use .sage, which is a psynonym:

sage: f.sage({'x':x, 'y':y})
2/3*x^3 + x + y - 5/7

serconvol(g)¶

serlaplace()¶

serreverse()¶

serreverse(f): reversion of the power series f.

If f(t) is a series in t with valuation 1, find the series g(t) such that g(f(t)) = t.

EXAMPLES:

sage: f = pari('x+x^2+x^3+O(x^4)'); f
x + x^2 + x^3 + O(x^4)
sage: g = f.serreverse(); g
x - x^2 + x^3 + O(x^4)
sage: f.subst('x',g)
x + O(x^4)
sage: g.subst('x',f)
x + O(x^4)

shift(x, n)¶: shift(x,n): shift x left n bits if n=0, right -n bits if n0.

shiftmul(x, n)¶: shiftmul(x,n): Return the product of x by $2^n$ .

sign(x)¶: sign(x): Return the sign of x, where x is of type integer, real or fraction.

simplify(x)¶

simplify(x): Simplify the object x as much as possible, and return the result.

A complex or quadratic number whose imaginary part is an exact 0 (i.e., not an approximate one such as O(3) or 0.E-28) is converted to its real part, and a a polynomial of degree 0 is converted to its constant term. Simplification occurs recursively.

This function is useful before using arithmetic functions, which expect integer arguments:

EXAMPLES:

sage: y = pari('y')
sage: x = pari('9') + y - y
sage: x
9
sage: x.type()
't_POL'
sage: x.factor()
matrix(0,2)
sage: pari('9').factor()
Mat([3, 2])
sage: x.simplify()
9
sage: x.simplify().factor()
Mat([3, 2])
sage: x = pari('1.5 + 0*I')
sage: x.type()
't_COMPLEX'
sage: x.simplify()
1.50000000000000
sage: y = x.simplify()
sage: y.type()
't_REAL'

sin(x, precision=0)¶

x.sin(): The sine of x.

If $x$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If $x$ is inexact (e.g. real), its own precision is used in the computation, and the parameter precision is ignored.

EXAMPLES:

sage: pari(1).sin() 
0.841470984807897
sage: C.<i> = ComplexField()
sage: pari(1+i).sin()
1.29845758141598 + 0.634963914784736*I

sinh(x, precision=0)¶

The hyperbolic sine function.

If $x$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If $x$ is inexact (e.g. real), its own precision is used in the computation, and the parameter precision is ignored.

EXAMPLES:

sage: pari(0).sinh()
0.E-19
sage: C.<i> = ComplexField()
sage: pari(1+i).sinh()
0.634963914784736 + 1.29845758141598*I

sizebyte(x)¶

sizebyte(x): Return the total number of bytes occupied by the complete tree of the object x. Note that this number depends on whether the computer is 32-bit or 64-bit (see examples).

INPUT:

x - gen

OUTPUT: int (a Python int)

EXAMPLES:

sage: pari('1').sizebyte()
12           # 32-bit
24           # 64-bit
sage: pari('10').sizebyte()
12           # 32-bit
24           # 64-bit
sage: pari('10000000000000').sizebyte()
16           # 32-bit
24           # 64-bit
sage: pari('10^100').sizebyte()
52           # 32-bit
64           # 64-bit
sage: pari('x').sizebyte()
36           # 32-bit
72           # 64-bit
sage: pari('x^20').sizebyte()
264          # 32-bit
528          # 64-bit
sage: pari('[x, 10^100]').sizebyte()
100          # 32-bit
160          # 64-bit

sizedigit(x)¶

sizedigit(x): Return a quick estimate for the maximal number of decimal digits before the decimal point of any component of x.

INPUT:

x - gen

OUTPUT:

int - Python integer

EXAMPLES:

sage: x = pari('10^100')
sage: x.Str().length()
101
sage: x.sizedigit()
101

Note that digits after the decimal point are ignored.

sage: x = pari('1.234')
sage: x
1.23400000000000
sage: x.sizedigit()
1

The estimate can be one too big:

sage: pari('7234.1').sizedigit()
4
sage: pari('9234.1').sizedigit()
5

sqr(x)¶

x.sqr(): square of x. Faster than, and most of the time (but not always - see the examples) identical to x*x.

EXAMPLES:

sage: pari(2).sqr()
4

For $2$ -adic numbers, x.sqr() may not be identical to x*x (squaring a $2$ -adic number increases its precision):

sage: pari("1+O(2^5)").sqr()
1 + O(2^6)
sage: pari("1+O(2^5)")*pari("1+O(2^5)")
1 + O(2^5)

However:

sage: x = pari("1+O(2^5)"); x*x
1 + O(2^6)

sqrt(x, precision=0)¶

x.sqrt(precision): The square root of x.

If $x$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If $x$ is inexact (e.g. real), its own precision is used in the computation, and the parameter precision is ignored.

EXAMPLES:

sage: pari(2).sqrt()
1.41421356237310

sqrtn(x, n, precision=0)¶

x.sqrtn(n): return the principal branch of the n-th root of x, i.e., the one such that $\arg(\sqrt(x)) \in ]-\pi/n, \pi/n]$ . Also returns a second argument which is a suitable root of unity allowing one to recover all the other roots. If it was not possible to find such a number, then this second return value is 0. If the argument is present and no square root exists, return 0 instead of raising an error.

If $x$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If $x$ is inexact (e.g. real), its own precision is used in the computation, and the parameter precision is ignored.

Note

intmods (modulo a prime) and $p$ -adic numbers are allowed as arguments.

INPUT:

x - gen
n - integer

OUTPUT:

gen - principal n-th root of x
gen - root of unity z that gives the other roots

EXAMPLES:

sage: s, z = pari(2).sqrtn(5)
sage: z
0.309016994374947 + 0.951056516295154*I
sage: s
1.14869835499704
sage: s^5
2.00000000000000
sage: z^5
1.00000000000000 + 5.42101086 E-19*I        # 32-bit
1.00000000000000 + 4.87890977618477 E-19*I  # 64-bit
sage: (s*z)^5
2.00000000000000 + 1.409462824 E-18*I       # 32-bit
2.00000000000000 + 7.58941520739853 E-19*I  # 64-bit

subst(var, y)¶

EXAMPLES:

sage: x = pari("x"); y = pari("y")
sage: f = pari('x^3 + 17*x + 3')
sage: f.subst(x, y)
y^3 + 17*y + 3
sage: f.subst(x, "z")
z^3 + 17*z + 3
sage: f.subst(x, "z")^2
z^6 + 34*z^4 + 6*z^3 + 289*z^2 + 102*z + 9
sage: f.subst(x, "x+1")
x^3 + 3*x^2 + 20*x + 21
sage: f.subst(x, "xyz")
xyz^3 + 17*xyz + 3
sage: f.subst(x, "xyz")^2
xyz^6 + 34*xyz^4 + 6*xyz^3 + 289*xyz^2 + 102*xyz + 9

substpol(y, z)¶

sumdiv(n)¶

Return the sum of the divisors of $n$ .

EXAMPLES:

sage: pari(10).sumdiv()
18

sumdivk(n, k)¶

Return the sum of the k-th powers of the divisors of n.

EXAMPLES:

sage: pari(10).sumdivk(2)
130

tan(x, precision=0)¶

x.tan() - tangent of x

If $x$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If $x$ is inexact (e.g. real), its own precision is used in the computation, and the parameter precision is ignored.

EXAMPLES:

sage: pari(2).tan()
-2.18503986326152
sage: C.<i> = ComplexField()
sage: pari(i).tan()
0.E-19 + 0.761594155955765*I

tanh(x, precision=0)¶

x.tanh() - hyperbolic tangent of x

If $x$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If $x$ is inexact (e.g. real), its own precision is used in the computation, and the parameter precision is ignored.

EXAMPLES:

sage: pari(1).tanh()
0.761594155955765
sage: C.<i> = ComplexField()
sage: pari(i).tanh()
0.E-19 + 1.55740772465490*I

taylor(v=-1)¶

teichmuller(x)¶

teichmuller(x): teichmuller character of p-adic number x.

This is the unique $(p-1)$ -st root of unity congruent to $x/p^{v_p(x)}$ modulo $p$ .

EXAMPLES:

sage: pari('2+O(7^5)').teichmuller()
2 + 4*7 + 6*7^2 + 3*7^3 + O(7^5)

theta(q, z, precision=0)¶

q.theta(z): Jacobi sine theta-function.

If $q$ or $z$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If the arguments are inexact (e.g. real), the smallest of their precisions is used in the computation, and the parameter precision is ignored.

EXAMPLES:

sage: pari(0.5).theta(2)
1.63202590295260

thetanullk(q, k, precision=0)¶

q.thetanullk(k): return the k-th derivative at z=0 of theta(q,z).

If $q$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If $q$ is inexact (e.g. real), its own precision is used in the computation, and the parameter precision is ignored.

EXAMPLES:

sage: pari(0.5).thetanullk(1)
0.548978532560341

thue(rhs, ne)¶

thueinit(flag=0)¶

trace()¶

Return the trace of this PARI object.

EXAMPLES:

sage: pari('[1,2; 3,4]').trace()  
5

truncate(x, estimate=False)¶

truncate(x,estimate=False): Return the truncation of x. If estimate is True, also return the number of error bits.

When x is in the real numbers, this means that the part after the decimal point is chopped away, e is the binary exponent of the difference between the original and truncated value (the “fractional part”). If x is a rational function, the result is the integer part (Euclidean quotient of numerator by denominator) and if requested the error estimate is 0.

When truncate is applied to a power series (in X), it transforms it into a polynomial or a rational function with denominator a power of X, by chopping away the $O(X^k)$ . Similarly, when applied to a p-adic number, it transforms it into an integer or a rational number by chopping away the $O(p^k)$ .

INPUT:

x - gen
estimate - (optional) bool, which is False by default

OUTPUT:

if estimate is False, return a single gen.
if estimate is True, return rounded version of x and error estimate in bits, both as gens.

EXAMPLES:

sage: pari('(x^2+1)/x').round()
(x^2 + 1)/x
sage: pari('(x^2+1)/x').truncate()
x
sage: pari('1.043').truncate()
1
sage: pari('1.043').truncate(True)
(1, -5)
sage: pari('1.6').truncate()
1
sage: pari('1.6').round()
2
sage: pari('1/3 + 2 + 3^2 + O(3^3)').truncate()
34/3
sage: pari('sin(x+O(x^10))').truncate()
1/362880*x^9 - 1/5040*x^7 + 1/120*x^5 - 1/6*x^3 + x
sage: pari('sin(x+O(x^10))').round()   # each coefficient has abs < 1
x + O(x^10)

type()¶

Return the Pari type of self as a string.

Note

In Cython, it is much faster to simply use typ(self.g) for checking Pari types.

EXAMPLES:

sage: pari(7).type()
't_INT'
sage: pari('x').type()
't_POL'

valuation(x, p)¶

valuation(x,p): Return the valuation of x with respect to p.

The valuation is the highest exponent of p dividing x.

If p is an integer, x must be an integer, an intmod whose modulus is divisible by p, a rational number, a p-adic number, or a polynomial or power series in which case the valuation is the minimum of the valuations of the coefficients.
If p is a polynomial, x must be a polynomial or a rational function. If p is a monomial then x may also be a power series.
If x is a vector, complex or quadratic number, then the valuation is the minimum of the component valuations.
If x = 0, the result is $2^31-1$ on 32-bit machines or $2^63-1$ on 64-bit machines if x is an exact object. If x is a p-adic number or power series, the result is the exponent of the zero.

INPUT:

x - gen
p - coercible to gen

OUTPUT:

gen - integer

EXAMPLES:

sage: pari(9).valuation(3)
2
sage: pari(9).valuation(9)
1
sage: x = pari(9).Mod(27); x.valuation(3)
2
sage: pari('5/3').valuation(3)
-1
sage: pari('9 + 3*x + 15*x^2').valuation(3)
1
sage: pari([9,3,15]).valuation(3)
1
sage: pari('9 + 3*x + 15*x^2 + O(x^5)').valuation(3)
1

sage: pari('x^2*(x+1)^3').valuation(pari('x+1'))
3
sage: pari('x + O(x^5)').valuation('x')
1
sage: pari('2*x^2 + O(x^5)').valuation('x')
2

sage: pari(0).valuation(3)   
2147483647            # 32-bit
9223372036854775807   # 64-bit

variable(x)¶

variable(x): Return the main variable of the object x, or p if x is a p-adic number.

This function raises a TypeError exception on scalars, i.e., on objects with no variable associated to them.

INPUT:

x - gen

OUTPUT: gen

EXAMPLES:

sage: pari('x^2 + x -2').variable()
x
sage: pari('1+2^3 + O(2^5)').variable()
2
sage: pari('x+y0').variable()
x
sage: pari('y0+z0').variable()
y0

vecextract(y, z=None)¶: self.vecextract(y,z): extraction of the components of the matrix or vector x according to y and z. If z is omitted, y designates columns, otherwise y corresponds to rows and z to columns. y and z can be vectors (of indices), strings (indicating ranges as in”1..10”) or masks (integers whose binary representation indicates the indices to extract, from left to right 1, 2, 4, 8, etc.)

Note

This function uses the PARI row and column indexing, so the first row or column is indexed by 1 instead of 0.

vecmax(x)¶: vecmax(x): Return the maximum of the elements of the vector/matrix x.

vecmin(x)¶: vecmin(x): Return the maximum of the elements of the vector/matrix x.

weber(x, flag=0, precision=0)¶

x.weber(flag=0): One of Weber’s f functions of x. flag is optional, and can be 0: default, function f(x)=exp(-i*Pi/24)*eta((x+1)/2)/eta(x) such that $j=(f^{24}-16)^3/f^{24}$ , 1: function f1(x)=eta(x/2)/eta(x) such that $j=(f1^24+16)^3/f2^{24}$ , 2: function f2(x)=sqrt(2)*eta(2*x)/eta(x) such that $j=(f2^{24}+16)^3/f2^{24}$ .

If $x$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If $x$ is inexact (e.g. real), its own precision is used in the computation, and the parameter precision is ignored.

TODO: Add further explanation from PARI manual.

EXAMPLES:

sage: C.<i> = ComplexField()
sage: pari(i).weber()
1.18920711500272 + 0.E-19*I
sage: pari(i).weber(1)    
1.09050773266526 + 0.E-19*I
sage: pari(i).weber(2)
1.09050773266526 + 0.E-19*I

xgcd(x, y)¶

Returns u,v,d such that d=gcd(x,y) and u*x+v*y=d.

EXAMPLES:

sage: pari(10).xgcd(15)
(5, -1, 1)

zeta(s, precision=0)¶

zeta(s): zeta function at s with s a complex or a p-adic number.

If $s$ is a complex number, this is the Riemann zeta function $\zeta(s)=\sum_{n\geq 1} n^{-s}$ , computed either using the Euler-Maclaurin summation formula (if $s$ is not an integer), or using Bernoulli numbers (if $s$ is a negative integer or an even nonnegative integer), or using modular forms (if $s$ is an odd nonnegative integer).

If $s$ is a $p$ -adic number, this is the Kubota-Leopoldt zeta function, i.e. the unique continuous $p$ -adic function on the $p$ -adic integers that interpolates the values of $(1-p^{-k})\zeta(k)$ at negative integers $k$ such that $k\equiv 1\pmod{p-1}$ if $p$ is odd, and at odd $k$ if $p=2$ .

If $x$ is an exact argument, it is first converted to a real or complex number using the optional parameter precision (in bits). If $x$ is inexact (e.g. real), its own precision is used in the computation, and the parameter precision is ignored.

INPUT:

s - gen (real, complex, or p-adic number)

OUTPUT:

gen - value of zeta at s.

EXAMPLES:

sage: pari(2).zeta()
1.64493406684823
sage: x = RR(pi)^2/6
sage: pari(x)
1.64493406684823
sage: pari(3).zeta()
1.20205690315959
sage: pari('1+5*7+2*7^2+O(7^3)').zeta()
4*7^-2 + 5*7^-1 + O(7^0)

znprimroot()¶

Return a primitive root modulo self, whenever it exists.

This is a generator of the group $(\ZZ/n\ZZ)^*$ , whenever this group is cyclic, i.e. if $n=4$ or $n=p^k$ or $n=2p^k$ , where $p$ is an odd prime and $k$ is a natural number.

INPUT:

self - positive integer equal to 4, or a power of an odd prime, or twice a power of an odd prime

OUTPUT: gen

EXAMPLES:

sage: pari(4).znprimroot()
Mod(3, 4)
sage: pari(10007^3).znprimroot()
Mod(5, 1002101470343)
sage: pari(2*109^10).znprimroot()
Mod(236736367459211723407, 473472734918423446802)

sage.libs.pari.gen.init_pari_stack(size=8000000)¶

Change the PARI scratch stack space to the given size.

The main application of this command is that you’ve done some individual PARI computation that used a lot of stack space. As a result the PARI stack may have doubled several times and is now quite large. That object you computed is copied off to the heap, but the PARI stack is never automatically shrunk back down. If you call this function you can shrink it back down.

If you set this too small then it will automatically be increased if it is exceeded, which could make some calculations initially slower (since they have to be redone until the stack is big enough).

INPUT:

size - an integer (default: 8000000)

EXAMPLES:

sage: get_memory_usage()                       # random output
'122M+'
sage: a = pari('2^100000000')
sage: get_memory_usage()                       # random output
'157M+'
sage: del a
sage: get_memory_usage()                       # random output
'145M+'

Hey, I want my memory back!

sage: sage.libs.pari.gen.init_pari_stack()
sage: get_memory_usage()                       # random output
'114M+'

Ahh, that’s better.

sage.libs.pari.gen.max(x, y)¶: max(x,y): Return the maximum of x and y.

sage.libs.pari.gen.min(x, y)¶: min(x,y): Return the minimum of x and y.

sage.libs.pari.gen.pbw(prec_in_bits=0)¶

Convert from precision expressed in bits to pari real precision expressed in words. Note: this rounds up to the nearest word, adjusts for the two codewords of a pari real, and is architecture-dependent.

EXAMPLES:

sage: import sage.libs.pari.gen as gen
sage: gen.prec_bits_to_words(70)
5   # 32-bit
4   # 64-bit

sage: [(32*n,gen.prec_bits_to_words(32*n)) for n in range(1,9)]
[(32, 3), (64, 4), (96, 5), (128, 6), (160, 7), (192, 8), (224, 9), (256, 10)] # 32-bit
[(32, 3), (64, 3), (96, 4), (128, 4), (160, 5), (192, 5), (224, 6), (256, 6)] # 64-bit

sage.libs.pari.gen.prec_bits_to_dec(prec_in_bits)¶

Convert from precision expressed in bits to precision expressed in decimal.

EXAMPLES:

sage: import sage.libs.pari.gen as gen
sage: gen.prec_bits_to_dec(53)
15
sage: [(32*n,gen.prec_bits_to_dec(32*n)) for n in range(1,9)]
[(32, 9),
(64, 19),
(96, 28),
(128, 38),
(160, 48),
(192, 57),
(224, 67),
(256, 77)]

sage.libs.pari.gen.prec_bits_to_words(prec_in_bits=0)¶

Convert from precision expressed in bits to pari real precision expressed in words. Note: this rounds up to the nearest word, adjusts for the two codewords of a pari real, and is architecture-dependent.

EXAMPLES:

sage: import sage.libs.pari.gen as gen
sage: gen.prec_bits_to_words(70)
5   # 32-bit
4   # 64-bit

sage: [(32*n,gen.prec_bits_to_words(32*n)) for n in range(1,9)]
[(32, 3), (64, 4), (96, 5), (128, 6), (160, 7), (192, 8), (224, 9), (256, 10)] # 32-bit
[(32, 3), (64, 3), (96, 4), (128, 4), (160, 5), (192, 5), (224, 6), (256, 6)] # 64-bit

sage.libs.pari.gen.prec_dec_to_bits(prec_in_dec)¶

Convert from precision expressed in decimal to precision expressed in bits.

EXAMPLES:

sage: import sage.libs.pari.gen as gen
sage: gen.prec_dec_to_bits(15)
49
sage: [(n,gen.prec_dec_to_bits(n)) for n in range(10,100,10)]
[(10, 33),
(20, 66),
(30, 99),
(40, 132),
(50, 166),
(60, 199),
(70, 232),
(80, 265),
(90, 298)]

sage.libs.pari.gen.prec_dec_to_words(prec_in_dec)¶

Convert from precision expressed in decimal to precision expressed in words. Note: this rounds up to the nearest word, adjusts for the two codewords of a pari real, and is architecture-dependent.

EXAMPLES:

sage: import sage.libs.pari.gen as gen
sage: gen.prec_dec_to_words(38)
6   # 32-bit
4   # 64-bit
sage: [(n,gen.prec_dec_to_words(n)) for n in range(10,90,10)]
[(10, 4), (20, 5), (30, 6), (40, 7), (50, 8), (60, 9), (70, 10), (80, 11)] # 32-bit
[(10, 3), (20, 4), (30, 4), (40, 5), (50, 5), (60, 6), (70, 6), (80, 7)] # 64-bit

sage.libs.pari.gen.prec_words_to_bits(prec_in_words)¶

Convert from pari real precision expressed in words to precision expressed in bits. Note: this adjusts for the two codewords of a pari real, and is architecture-dependent.

EXAMPLES:

sage: import sage.libs.pari.gen as gen
sage: gen.prec_words_to_bits(10)
256   # 32-bit
512   # 64-bit
sage: [(n,gen.prec_words_to_bits(n)) for n in range(3,10)]
[(3, 32), (4, 64), (5, 96), (6, 128), (7, 160), (8, 192), (9, 224)]  # 32-bit
[(3, 64), (4, 128), (5, 192), (6, 256), (7, 320), (8, 384), (9, 448)] # 64-bit

sage.libs.pari.gen.prec_words_to_dec(prec_in_words)¶

Convert from precision expressed in words to precision expressed in decimal. Note: this adjusts for the two codewords of a pari real, and is architecture-dependent.

EXAMPLES:

sage: import sage.libs.pari.gen as gen
sage: gen.prec_words_to_dec(5)
28   # 32-bit
57   # 64-bit
sage: [(n,gen.prec_words_to_dec(n)) for n in range(3,10)]
[(3, 9), (4, 19), (5, 28), (6, 38), (7, 48), (8, 57), (9, 67)] # 32-bit
[(3, 19), (4, 38), (5, 57), (6, 77), (7, 96), (8, 115), (9, 134)] # 64-bit

sage.libs.pari.gen.vecsmall_to_intlist(v)¶

INPUT:

v - a gen of type Vecsmall

OUTPUT: a Python list of Python ints

PARI C-library interface¶

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