# What is a PARI group?

Take for instance the example:

G = NumberField(x^5 + 15*x + 12, 't').galois_group(type="pari"); G


This gives an output:

Galois group PARI group [20, -1, 3, "F(5) = 5:4"] of degree 5 of the Number Field in t with defining polynomial x^5 + 15*x + 12


So I wonder what is the meaning of [20, -1, 3, "F(5) = 5:4"]. Is it some kind of permutation?

The relevant page of the documentation doesn't clarify it.

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@Jacques Saliba -- Would you suggest adding a link from "the relevant page of the documentation" (which page do you have in mind?) to the PARI/GP documentation page pointed to by @vdelecroix?

( 2017-12-30 19:49:52 +0200 )edit

@slelievre I didn't have the privilege to post a link that's why. i believe the page given in the answer has more information in it:

The output is a 4-component vector [n,s,k,name] with the following meaning: n is the cardinality of the group, s is its signature (s = 1 if the group is a subgroup of the alternating group Ad, s = -1 otherwise) and name is a character string containing name of the transitive group according to the GAP 4 transitive groups library by Alexander Hulpke.

That's the relevant part i think

( 2017-12-30 19:52:04 +0200 )edit

I opened SageMath trac ticket #24452 for that.

( 2017-12-31 00:36:52 +0200 )edit

See also a previous request for making it easier to extract information about a Galois group, posted by John Cremona on sage-devel and other lists on 2017-10-15.

( 2018-01-03 06:19:48 +0200 )edit

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This is described in PARI/GP documentation at http://pari.math.u-bordeaux.fr/dochtm...

And to have a more direct hand on the underlying PARI/GP object you can do

sage: P = pari(G.group())
sage: P
[20, -1, 3, "F(5) = 5:4"]

more