# Interface to the GP calculator of PARI/GP#

Type `gp.[tab]` for a list of all the functions available from your Gp install. Type `gp.[tab]?` for Gp’s help about a given function. Type `gp(...)` to create a new Gp object, and `gp.eval(...)` to evaluate a string using Gp (and get the result back as a string).

EXAMPLES: We illustrate objects that wrap GP objects (gp is the PARI interpreter):

```sage: M = gp('[1,2;3,4]')
sage: M
[1, 2; 3, 4]
sage: M * M
[7, 10; 15, 22]
sage: M + M
[2, 4; 6, 8]
sage: M.matdet()
-2
```
```>>> from sage.all import *
>>> M = gp('[1,2;3,4]')
>>> M
[1, 2; 3, 4]
>>> M * M
[7, 10; 15, 22]
>>> M + M
[2, 4; 6, 8]
>>> M.matdet()
-2
```
```sage: E = gp.ellinit([1,2,3,4,5])
sage: E.ellglobalred()
[10351, [1, -1, 0, -1], 1, [11, 1; 941, 1], [[1, 5, 0, 1], [1, 5, 0, 1]]]
sage: E.ellan(20)
[1, 1, 0, -1, -3, 0, -1, -3, -3, -3, -1, 0, 1, -1, 0, -1, 5, -3, 4, 3]
```
```>>> from sage.all import *
>>> E = gp.ellinit([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)])
>>> E.ellglobalred()
[10351, [1, -1, 0, -1], 1, [11, 1; 941, 1], [[1, 5, 0, 1], [1, 5, 0, 1]]]
>>> E.ellan(Integer(20))
[1, 1, 0, -1, -3, 0, -1, -3, -3, -3, -1, 0, 1, -1, 0, -1, 5, -3, 4, 3]
```
```sage: primitive_root(7)
3
sage: x = gp("znlog( Mod(2,7), Mod(3,7))")
sage: 3^x % 7
2
```
```>>> from sage.all import *
>>> primitive_root(Integer(7))
3
>>> x = gp("znlog( Mod(2,7), Mod(3,7))")
>>> Integer(3)**x % Integer(7)
2
```
```sage: print(gp("taylor(sin(x),x)"))
x - 1/6*x^3 + 1/120*x^5 - 1/5040*x^7 + 1/362880*x^9 - 1/39916800*x^11 + 1/6227020800*x^13 - 1/1307674368000*x^15 + O(x^16)
```
```>>> from sage.all import *
>>> print(gp("taylor(sin(x),x)"))
x - 1/6*x^3 + 1/120*x^5 - 1/5040*x^7 + 1/362880*x^9 - 1/39916800*x^11 + 1/6227020800*x^13 - 1/1307674368000*x^15 + O(x^16)
```

GP has a powerful very efficient algorithm for numerical computation of integrals.

```sage: gp("a = intnum(x=0,6,sin(x))")
0.03982971334963397945434770208               # 32-bit
0.039829713349633979454347702077075594548     # 64-bit
sage: gp("a")
0.03982971334963397945434770208               # 32-bit
0.039829713349633979454347702077075594548     # 64-bit
sage: gp.kill("a")
sage: gp("a")
a
```
```>>> from sage.all import *
>>> gp("a = intnum(x=0,6,sin(x))")
0.03982971334963397945434770208               # 32-bit
0.039829713349633979454347702077075594548     # 64-bit
>>> gp("a")
0.03982971334963397945434770208               # 32-bit
0.039829713349633979454347702077075594548     # 64-bit
>>> gp.kill("a")
>>> gp("a")
a
```

Note that gp ASCII plots do work in Sage, as follows:

```sage: print(gp.eval("plot(x=0,6,sin(x))"))

0.9988963 |''''''''''''_x...x_''''''''''''''''''''''''''''''''''''''''''|
|          x"        "x                                        |
|        _"            "_                                      |
|       x                x                                     |
|      "                  "                                    |
|     "                    "                                   |
|   _"                      "_                                 |
|  _                          _                                |
| _                            _                               |
|_                              _                              |
_                                                              |
`````````````````````````````````"``````````````````````````````
|                                 "                            |
|                                  "                           |
|                                   "                          "
|                                    "_                      _"|
|                                      _                    _  |
|                                       _                  _   |
|                                        x                x    |
|                                         "_            _"     |
|                                           x_        _x       |
-0.998955 |............................................."x____x".........|
0                                                              6
```
```>>> from sage.all import *
>>> print(gp.eval("plot(x=0,6,sin(x))"))
<BLANKLINE>
0.9988963 |''''''''''''_x...x_''''''''''''''''''''''''''''''''''''''''''|
|          x"        "x                                        |
|        _"            "_                                      |
|       x                x                                     |
|      "                  "                                    |
|     "                    "                                   |
|   _"                      "_                                 |
|  _                          _                                |
| _                            _                               |
|_                              _                              |
_                                                              |
`````````````````````````````````"``````````````````````````````
|                                 "                            |
|                                  "                           |
|                                   "                          "
|                                    "_                      _"|
|                                      _                    _  |
|                                       _                  _   |
|                                        x                x    |
|                                         "_            _"     |
|                                           x_        _x       |
-0.998955 |............................................."x____x".........|
0                                                              6
```

The GP interface reads in even very long input (using files) in a robust manner, as long as you are creating a new object.

```sage: t = '"%s"'%10^10000   # ten thousand character string.
sage: a = gp.eval(t)
sage: a = gp(t)
```
```>>> from sage.all import *
>>> t = '"%s"'%Integer(10)**Integer(10000)   # ten thousand character string.
>>> a = gp.eval(t)
>>> a = gp(t)
```

In Sage, the PARI large Galois groups datafiles should be installed by default:

```sage: f = gp('x^9 - x - 2')
sage: f.polgalois()
[362880, -1, 34, "S9"]
```
```>>> from sage.all import *
>>> f = gp('x^9 - x - 2')
>>> f.polgalois()
[362880, -1, 34, "S9"]
```

AUTHORS:

• William Stein

• David Joyner: some examples

• William Stein (2006-03-01): added tab completion for methods: gp.[tab] and x = gp(blah); x.[tab]

• William Stein (2006-03-01): updated to work with PARI 2.2.12-beta

• William Stein (2006-05-17): updated to work with PARI 2.2.13-beta

class sage.interfaces.gp.Gp(stacksize=10000000, maxread=None, script_subdirectory=None, logfile=None, server=None, server_tmpdir=None, init_list_length=1024, seed=None)[source]#

Bases: `ExtraTabCompletion`, `Expect`

Interface to the PARI gp interpreter.

Type `gp.[tab]` for a list of all the functions available from your Gp install. Type `gp.[tab]?` for Gp’s help about a given function. Type `gp(...)` to create a new Gp object, and `gp.eval(...)` to evaluate a string using Gp (and get the result back as a string).

INPUT:

• `stacksize` (int, default 10000000) – the initial PARI stacksize in bytes (default 10MB)

• `script_subdirectory` (string, default None) – name of the subdirectory of SAGE_EXTCODE/pari from which to read scripts

• `logfile` (string, default None) – log file for the pexpect interface

• `server` – name of remote server

• `server_tmpdir` – name of temporary directory on remote server

• `init_list_length` (int, default 1024) – length of initial list of local variables.

• `seed` (int, default random) – random number generator seed for pari

EXAMPLES:

```sage: Gp()
PARI/GP interpreter
```
```>>> from sage.all import *
>>> Gp()
PARI/GP interpreter
```
console()[source]#

Spawn a new GP command-line session.

EXAMPLES:

```sage: gp.console()  # not tested
GP/PARI CALCULATOR Version 2.4.3 (development svn-12577)
amd64 running linux (x86-64/GMP-4.2.1 kernel) 64-bit version
compiled: Jul 21 2010, gcc-4.6.0 20100705 (experimental) (GCC)
(readline v6.0 enabled, extended help enabled)
```
```>>> from sage.all import *
>>> gp.console()  # not tested
GP/PARI CALCULATOR Version 2.4.3 (development svn-12577)
amd64 running linux (x86-64/GMP-4.2.1 kernel) 64-bit version
compiled: Jul 21 2010, gcc-4.6.0 20100705 (experimental) (GCC)
(readline v6.0 enabled, extended help enabled)
```
cputime(t=None)[source]#

cputime for pari - cputime since the pari process was started.

INPUT:

• `t` – (default: None); if not None, then returns time since t

Warning

If you call gettime explicitly, e.g., gp.eval(‘gettime’), you will throw off this clock.

EXAMPLES:

```sage: gp.cputime()          # random output
0.0080000000000000002
sage: gp.factor('2^157-1')
[852133201, 1; 60726444167, 1; 1654058017289, 1; 2134387368610417, 1]
sage: gp.cputime()          # random output
0.26900000000000002
```
```>>> from sage.all import *
>>> gp.cputime()          # random output
0.0080000000000000002
>>> gp.factor('2^157-1')
[852133201, 1; 60726444167, 1; 1654058017289, 1; 2134387368610417, 1]
>>> gp.cputime()          # random output
0.26900000000000002
```
get(var)[source]#

Get the value of the GP variable var.

INPUT:

• `var` (string) – a valid GP variable identifier

EXAMPLES:

```sage: gp.set('x', '2')
sage: gp.get('x')
'2'
```
```>>> from sage.all import *
>>> gp.set('x', '2')
>>> gp.get('x')
'2'
```
get_default(var)[source]#

Return the current value of a PARI gp configuration variable.

INPUT:

• `var` (string) – the name of a PARI gp configuration variable. (See `gp.default()` for a list.)

OUTPUT:

(string) the value of the variable.

EXAMPLES:

```sage: gp.get_default('log')
0
'.../share/pari'
sage: gp.get_default('seriesprecision')
16
sage: gp.get_default('realprecision')
28              # 32-bit
38              # 64-bit
```
```>>> from sage.all import *
>>> gp.get_default('log')
0
'.../share/pari'
>>> gp.get_default('seriesprecision')
16
>>> gp.get_default('realprecision')
28              # 32-bit
38              # 64-bit
```
get_precision()[source]#

Return the current PARI precision for real number computations.

EXAMPLES:

```sage: gp.get_precision()
28              # 32-bit
38              # 64-bit
```
```>>> from sage.all import *
>>> gp.get_precision()
28              # 32-bit
38              # 64-bit
```
get_real_precision()[source]#

Return the current PARI precision for real number computations.

EXAMPLES:

```sage: gp.get_precision()
28              # 32-bit
38              # 64-bit
```
```>>> from sage.all import *
>>> gp.get_precision()
28              # 32-bit
38              # 64-bit
```
get_series_precision()[source]#

Return the current PARI power series precision.

EXAMPLES:

```sage: gp.get_series_precision()
16
```
```>>> from sage.all import *
>>> gp.get_series_precision()
16
```
help(command)[source]#

Returns GP’s help for `command`.

EXAMPLES:

```sage: gp.help('gcd')
'gcd(x,{y}): greatest common divisor of x and y.'
```
```>>> from sage.all import *
>>> gp.help('gcd')
'gcd(x,{y}): greatest common divisor of x and y.'
```
kill(var)[source]#

Kill the value of the GP variable var.

INPUT:

• `var` (string) – a valid GP variable identifier

EXAMPLES:

```sage: gp.set('xx', '22')
sage: gp.get('xx')
'22'
sage: gp.kill('xx')
sage: gp.get('xx')
'xx'
```
```>>> from sage.all import *
>>> gp.set('xx', '22')
>>> gp.get('xx')
'22'
>>> gp.kill('xx')
>>> gp.get('xx')
'xx'
```
new_with_bits_prec(s, precision=0)[source]#

Creates a GP object from s with `precision` bits of precision. GP actually automatically increases this precision to the nearest word (i.e. the next multiple of 32 on a 32-bit machine, or the next multiple of 64 on a 64-bit machine).

EXAMPLES:

```sage: # needs sage.symbolic
sage: pi_def = gp(pi); pi_def
3.141592653589793238462643383                  # 32-bit
3.1415926535897932384626433832795028842        # 64-bit
sage: pi_def.precision()
28                                             # 32-bit
38                                             # 64-bit
sage: pi_150 = gp.new_with_bits_prec(pi, 150)
sage: new_prec = pi_150.precision(); new_prec
48                                             # 32-bit
57                                             # 64-bit
sage: old_prec = gp.set_precision(new_prec); old_prec
28                                             # 32-bit
38                                             # 64-bit
sage: pi_150
3.14159265358979323846264338327950288419716939938  # 32-bit
3.14159265358979323846264338327950288419716939937510582098  # 64-bit
sage: gp.set_precision(old_prec)
48                                             # 32-bit
57                                             # 64-bit
sage: gp.get_precision()
28                                             # 32-bit
38                                             # 64-bit
```
```>>> from sage.all import *
>>> # needs sage.symbolic
>>> pi_def = gp(pi); pi_def
3.141592653589793238462643383                  # 32-bit
3.1415926535897932384626433832795028842        # 64-bit
>>> pi_def.precision()
28                                             # 32-bit
38                                             # 64-bit
>>> pi_150 = gp.new_with_bits_prec(pi, Integer(150))
>>> new_prec = pi_150.precision(); new_prec
48                                             # 32-bit
57                                             # 64-bit
>>> old_prec = gp.set_precision(new_prec); old_prec
28                                             # 32-bit
38                                             # 64-bit
>>> pi_150
3.14159265358979323846264338327950288419716939938  # 32-bit
3.14159265358979323846264338327950288419716939937510582098  # 64-bit
>>> gp.set_precision(old_prec)
48                                             # 32-bit
57                                             # 64-bit
>>> gp.get_precision()
28                                             # 32-bit
38                                             # 64-bit
```
set(var, value)[source]#

Set the GP variable var to the given value.

INPUT:

• `var` (string) – a valid GP variable identifier

• `value` – a value for the variable

EXAMPLES:

```sage: gp.set('x', '2')
sage: gp.get('x')
'2'
```
```>>> from sage.all import *
>>> gp.set('x', '2')
>>> gp.get('x')
'2'
```
set_default(var, value)[source]#

Set a PARI gp configuration variable, and return the old value.

INPUT:

• `var` (string) – the name of a PARI gp configuration variable. (See `gp.default()` for a list.)

• `value` – the value to set the variable to.

EXAMPLES:

```sage: old_prec = gp.set_default('realprecision', 110)
sage: gp.get_default('realprecision')
115
sage: gp.set_default('realprecision', old_prec)
115
sage: gp.get_default('realprecision')
28              # 32-bit
38              # 64-bit
```
```>>> from sage.all import *
>>> old_prec = gp.set_default('realprecision', Integer(110))
>>> gp.get_default('realprecision')
115
>>> gp.set_default('realprecision', old_prec)
115
>>> gp.get_default('realprecision')
28              # 32-bit
38              # 64-bit
```
set_precision(prec)[source]#

Sets the PARI precision (in decimal digits) for real computations, and returns the old value.

Note

PARI/GP rounds up precisions to the nearest machine word, so the result of `get_precision()` is not always the same as the last value inputted to `set_precision()`.

EXAMPLES:

```sage: old_prec = gp.set_precision(53); old_prec
28              # 32-bit
38              # 64-bit
sage: gp.get_precision()
57
sage: gp.set_precision(old_prec)
57
sage: gp.get_precision()
28              # 32-bit
38              # 64-bit
```
```>>> from sage.all import *
>>> old_prec = gp.set_precision(Integer(53)); old_prec
28              # 32-bit
38              # 64-bit
>>> gp.get_precision()
57
>>> gp.set_precision(old_prec)
57
>>> gp.get_precision()
28              # 32-bit
38              # 64-bit
```
set_real_precision(prec)[source]#

Sets the PARI precision (in decimal digits) for real computations, and returns the old value.

Note

PARI/GP rounds up precisions to the nearest machine word, so the result of `get_precision()` is not always the same as the last value inputted to `set_precision()`.

EXAMPLES:

```sage: old_prec = gp.set_precision(53); old_prec
28              # 32-bit
38              # 64-bit
sage: gp.get_precision()
57
sage: gp.set_precision(old_prec)
57
sage: gp.get_precision()
28              # 32-bit
38              # 64-bit
```
```>>> from sage.all import *
>>> old_prec = gp.set_precision(Integer(53)); old_prec
28              # 32-bit
38              # 64-bit
>>> gp.get_precision()
57
>>> gp.set_precision(old_prec)
57
>>> gp.get_precision()
28              # 32-bit
38              # 64-bit
```
set_seed(seed=None)[source]#

Set the seed for gp interpreter.

The seed should be an integer.

EXAMPLES:

```sage: g = Gp()
sage: g.set_seed(1)
1
sage: [g.random() for i in range(5)]
[1546275796, 879788114, 1745191708, 771966234, 1247963869]
```
```>>> from sage.all import *
>>> g = Gp()
>>> g.set_seed(Integer(1))
1
>>> [g.random() for i in range(Integer(5))]
[1546275796, 879788114, 1745191708, 771966234, 1247963869]
```
set_series_precision(prec=None)[source]#

Sets the PARI power series precision, and returns the old precision.

EXAMPLES:

```sage: old_prec = gp.set_series_precision(50); old_prec
16
sage: gp.get_series_precision()
50
sage: gp.set_series_precision(old_prec)
50
sage: gp.get_series_precision()
16
```
```>>> from sage.all import *
>>> old_prec = gp.set_series_precision(Integer(50)); old_prec
16
>>> gp.get_series_precision()
50
>>> gp.set_series_precision(old_prec)
50
>>> gp.get_series_precision()
16
```
version()[source]#

Returns the version of GP being used.

EXAMPLES:

```sage: gp.version()  # not tested
((2, 4, 3), 'GP/PARI CALCULATOR Version 2.4.3 (development svn-12577)')
```
```>>> from sage.all import *
>>> gp.version()  # not tested
((2, 4, 3), 'GP/PARI CALCULATOR Version 2.4.3 (development svn-12577)')
```
class sage.interfaces.gp.GpElement(parent, value, is_name=False, name=None)[source]#

EXAMPLES: This example illustrates dumping and loading GP elements to compressed strings.

```sage: a = gp(39393)
True
```
```>>> from sage.all import *
>>> a = gp(Integer(39393))
True
```

Since dumping and loading uses the string representation of the object, it need not result in an identical object from the point of view of PARI:

```sage: E = gp('ellinit([1,2,3,4,5])')
True
sage: x = gp.Pi()/3
False
sage: x
1.047197551196597746154214461            # 32-bit
1.0471975511965977461542144610931676281  # 64-bit
1.047197551196597746154214461            # 32-bit
1.0471975511965977461542144610931676281  # 64-bit
```
```>>> from sage.all import *
>>> E = gp('ellinit([1,2,3,4,5])')
True
>>> x = gp.Pi()/Integer(3)
False
>>> x
1.047197551196597746154214461            # 32-bit
1.0471975511965977461542144610931676281  # 64-bit
1.047197551196597746154214461            # 32-bit
1.0471975511965977461542144610931676281  # 64-bit
```

The two elliptic curves look the same, but internally the floating point numbers are slightly different.

is_string()[source]#

Tell whether this element is a string.

EXAMPLES:

```sage: gp('"abc"').is_string()
True
sage: gp('[1,2,3]').is_string()
False
```
```>>> from sage.all import *
>>> gp('"abc"').is_string()
True
>>> gp('[1,2,3]').is_string()
False
```
sage.interfaces.gp.gp_console()[source]#

Spawn a new GP command-line session.

EXAMPLES:

```sage: gp.console()  # not tested
GP/PARI CALCULATOR Version 2.4.3 (development svn-12577)
amd64 running linux (x86-64/GMP-4.2.1 kernel) 64-bit version
compiled: Jul 21 2010, gcc-4.6.0 20100705 (experimental) (GCC)
(readline v6.0 enabled, extended help enabled)
```
```>>> from sage.all import *
>>> gp.console()  # not tested
GP/PARI CALCULATOR Version 2.4.3 (development svn-12577)
amd64 running linux (x86-64/GMP-4.2.1 kernel) 64-bit version
compiled: Jul 21 2010, gcc-4.6.0 20100705 (experimental) (GCC)
(readline v6.0 enabled, extended help enabled)
```
sage.interfaces.gp.gp_version()[source]#

EXAMPLES:

```sage: gp.version()  # not tested
((2, 4, 3), 'GP/PARI CALCULATOR Version 2.4.3 (development svn-12577)')
```
```>>> from sage.all import *
>>> gp.version()  # not tested
((2, 4, 3), 'GP/PARI CALCULATOR Version 2.4.3 (development svn-12577)')
```
sage.interfaces.gp.is_GpElement(x)[source]#

Return True if `x` is of type `GpElement`

This function is deprecated; use `isinstance()` (of `sage.interfaces.abc.GpElement`) instead.

EXAMPLES:

```sage: from sage.interfaces.gp import is_GpElement
sage: is_GpElement(gp(2))
doctest:...: DeprecationWarning: the function is_GpElement is deprecated; use isinstance(x, sage.interfaces.abc.GpElement) instead
See https://github.com/sagemath/sage/issues/34804 for details.
True
sage: is_GpElement(2)
False
```
```>>> from sage.all import *
>>> from sage.interfaces.gp import is_GpElement
>>> is_GpElement(gp(Integer(2)))
doctest:...: DeprecationWarning: the function is_GpElement is deprecated; use isinstance(x, sage.interfaces.abc.GpElement) instead
See https://github.com/sagemath/sage/issues/34804 for details.
True
>>> is_GpElement(Integer(2))
False
```

Returns the GP interface object defined in sage.interfaces.gp.

EXAMPLES:

```sage: from sage.interfaces.gp import reduce_load_GP
```>>> from sage.all import *