Wrong data in Number field 8.0.3036027392.1

[kwa...@chosun.ac.kr]

Dear the editor of LMFDB

My name is Kwang-Seob Kim, working at Chosun University.

I found some wrong data in the following page.

https://www.lmfdb.org/NumberField/8.0.3036027392.1

Here, we can find that a sibling field of degree 16 is defined by the polynomial

x^16-2*x^15+17*x^14-30*x^13+154*x^12-276*x^11+881*x^10-1668*x^9+3748*x^8-6968*x^7+11608*x^6-19360*x^5+27752*x^4-39296*x^3+42576*x^2-33408*x+17312.

But we can check that the Galois group of the splitting field of the above sibling field is isomorphic to PGL_2(F_7) x C_2.

The correct polynomial is

x^16 + 2*x^15 - 2891*x^14 - 5056*x^13 + 3659476*x^12 + 5489380*x^11 - 2649070364*x^10 - 3318090524*x^9 + 1199467722791*x^8 + 1205917321886*x^7 - 347859732638477*x^6 - 263505489039148*x^5 + 63101818112963557*x^4 + 32051392053372766*x^3 - 6546161605185997903*x^2 - 1673942566359575964*x + 297344248441326445466

Sincerely,

Kwang-Seob Kim.

[John Jones]

Hi,

According to pari/gp, these two degree 16 polynomials define the same field and according to magma, they have the same Galois group.  How did you determine the Galois group of the one listed in the lmfdb?

John Jones

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[Edgar Costa]

Perhaps there is a typo on your side regarding the computation of the Galois group:

> _<x> := PolynomialRing(Rationals());
> f := x^16-2*x^15+17*x^14-30*x^13+154*x^12-276*x^11+881*x^10-1668*x^9+3748*x^8-6968*x^7+11608*x^6-19360*x^5+27752*x^4-39296*x^3+42576*x^2-33408*x+17312;
> Gf, _, dataf := GaloisGroup(f);
> GaloisProof(f, dataf);
true
> TransitiveGroupIdentification(Gf);
713 16
> g := x^16 + 2*x^15 - 2891*x^14 - 5056*x^13 + 3659476*x^12 + 5489380*x^11 - 2649070364*x^10 - 3318090524*x^9 + 1199467722791*x^8 + 1205917321886*x^7 - 347859732638477*x^6 - 263505489039148*x^5 + 631\
01818112963557*x^4 + 32051392053372766*x^3 - 6546161605185997903*x^2 - 1673942566359575964*x + 297344248441326445466;
> Gg, _, datag := GaloisGroup(g);
> GaloisProof(g, datag);
true
> TransitiveGroupIdentification(Gg);
713 16
> gabs := Polredabs(g); // this calls polredabs in gp
> gabs eq f;
true