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.
@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?
@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
I opened SageMath trac ticket #24452 for that.
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.