# PARI Group Labelling

I have a Galois group that I'm tying to determine:

f = x^5 - 792*x^4 + 71280*x^3 + 39517632*x^2 - 7519640832*x + 314605513728
d = 3
A = -9504*d^2
B = 365904*d^3
K.<a> = NumberField(f)
R.<y> = K[]
g = y^2 - a^3 - A*a - B
L.<z> = NumberField(y^2 - a^3-A*a-B)
G = L.galois_group()


I have a couple of issues: When I type G[1], it says

GaloisGroup_v1 is not subscriptable


The second is the output for G says

[10,-1,1,'C(10)=5[x]2']


I know this is the PARI label and the last entry is the GAP 4 label. But for the life of me I cannot find anywhere where it explicitly tells me what C(10)=5[x]2 means.

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The names (for degrees up to 15) are from the paper:

Basically they describe the particular permutation action, not just the abstract isomorphism type. The group in the question is abstractly a C(10) (cyclic), but the [x] (brackets indicate permutational actions) indicates that it is the product action of C(5) with C(2) on 10 points (not the intransitive action on 7 points).

[Related notes.]

The Sage documentation page:

points to the PARI documentation page:

Note that it is after a previous similar question here:

There is an open ticket to make it easier to extract information about a PARI group:

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