This is probably obvious but why is type='gap' not standard for Galois groups?
On Sunday, October 15, 2017 at 12:25:11 PM UTC-4, John Cremona wrote: > > Extracting information about a Galois group is more painful than it > should be. After > > sage: K.<z> = CyclotomicField(5) > sage: G = K.galois_group(type='pari') > sage: G > Galois group PARI group [4, -1, 1, "C(4) = 4"] of degree 4 of the > Cyclotomic Field of order 5 and degree 4 > > we have > > sage: type(G) > <class 'sage.rings.number_field.galois_group.GaloisGroup_v1'> > > (other types are returned if other options for the galois_group() > method are chosen). There is not a lot you can do with this G except > get its order (G.order()) without going deeper: > > sage: GG=G.group() > sage: type(GG) > <class 'sage.groups.pari_group.PariGroup_with_category'> > sage: GG > PARI group [4, -1, 1, "C(4) = 4"] of degree 4 > > This type has "forgotten" that it is a Galois group but has many more > methods; sadly most not implemented. At least one might want to > extract the 4 elements of the underlying list which are the order (4) > which in this example happens to also be the degree (4), meaning that > GG is a subgroup of S_4 (degree=4) of order 4. The second entry -1 is > the sign (-1 means odd, i.e. not a subgroup of A_4), the third is the > "T-number" which identifies this group in some classification of > transitive groups. > > As far as I know the only way to get the sign and T-number is to > retrieve the underlying PARI list via GG.__pari__() (which until > recently was GG._pari_() with single underscores). I would like to > implement > > GG.sign() # returns GG.__pari__()[1] > GG.t_number() # returns GG.__pari__()[2] > > and perhaps more. I have been looking in the PARI/gp documentation on > Galois groups and what it says about this 4-tuple is > > "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 A_d, 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. > > k is more arbitrary and the choice made up to version 2.2.3 of PARI is > rather unfortunate: for d > 7, k is the numbering of the group among > all transitive subgroups of S_d, as given in "The transitive groups of > degree up to eleven", G. Butler and J. McKay, Communications in > Algebra, vol. 11, 1983, pp. 863--911 (group k is denoted T_k there). > And for d ≤ 7, it was ad hoc, so as to ensure that a given triple > would denote a unique group. Specifically, for polynomials of degree d > ≤ 7, the groups are coded as follows, using standard notations (etc)" > > Despite the ad hoc nature of this parameter k I still think we should > allow users to get at it more easily. > > John > -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at https://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
