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\begin{verbatim}
Magma V2.23-1     Wed Jun  8 2022 17:37:46 on Tool     [Seed = 3861225522]

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Type ? for help.  Type <Ctrl>-D to quit.

Loading file "D4.txt"

//////////////////////////////
//                          //
//   Dolgachev type-4 dP1   //
//                          //
//////////////////////////////

This program analyzes the Dolgachev type-4 del Pezzo surface of degree 1 over Q,
defined by
    f := -x0^6 - x0*x1^5 - x2^3 + x3^2

Loading "D4_ECs_Fq.txt"

/////////////////////////////////////////////
//                                         //
//   Intersection matrix for (C1,...,C9)   //
//                                         //
/////////////////////////////////////////////

[-1  0  0  0  0  0  0  0  0]
[ 0 -1  0  0  0  0  0  0  0]
[ 0  0 -1  0  0  0  0  0  0]
[ 0  0  0 -1  0  0  0  0  0]
[ 0  0  0  0 -1  0  0  0  0]
[ 0  0  0  0  0 -1  0  0  0]
[ 0  0  0  0  0  0 -1  0  1]
[ 0  0  0  0  0  0  0 -1  1]
[ 0  0  0  0  0  0  1  1 -1]

Loading "D4_GalGenMats.txt"

///////////////////////////////////////
//                                   //
//   Analysis : K|K0 and Gal(K|K0)   //
//                                   //
///////////////////////////////////////

Base field K0 : Rational Field

Splitting field K obtained iteratively from K0 using the following polynomials:
p_{0} = t^2 - 5
p_{1} = t^2 + 1/5*(-174*theta0 - 390)
p_{2} = t^2 + -3*theta1 + 9*theta0 + 18
p_{3} = t^3 + -3*theta1 + 9*theta0 + 19
p_{4} = t^2 + 1/2*(39*theta0 - 87)*theta1 - 9*theta0 + 18
p_{5} = t^5 + (1/36*(-23*theta0 + 44)*theta1 + 1/60*(49*theta0 + 45))*theta2

For each i,
(*) theta_{i} is a root of p_{i}.
(*) sigma_{i} is a matrix representation of the Galois element sending theta_{i}
    to another root of p_{i} (with consequences for p_{j}, and hence the image
    of theta_{j}, for j > i).

The subgroup G of Gal(K|K0) is generated by the following 6 elements :

sigma_{0} =
[-2 -2 -1 -1 -1 -1 -2 -1 -1]
[-1 -1 -1  0  0 -1 -1  0 -1]
[-2 -2 -2 -1 -2 -2 -2 -1 -2]
[-1 -2 -1  0 -1 -1 -1 -1 -1]
[-2 -1 -1  0 -1 -1 -1 -1 -1]
[-2 -2 -2 -1 -1 -1 -1 -1 -2]
[ 3  3  3  1  2  2  3  2  2]
[ 3  3  2  1  2  2  2  1  3]
[ 5  5  4  2  3  4  4  3  4]

sigma_{1} =
[-1  0  0  0  0 -1  0  0 -1]
[-2 -1 -1 -1 -2 -2 -1 -1 -2]
[-2 -1 -1 -2 -1 -2 -1 -1 -2]
[-2 -1  0 -1 -1 -1 -1 -1 -1]
[-2  0 -1 -1 -1 -1 -1 -1 -1]
[-1  0  0 -1 -1 -1  0 -1 -1]
[ 3  1  1  2  2  2  1  1  3]
[ 3  1  1  2  2  3  2  2  2]
[ 5  2  2  3  3  4  2  3  4]

sigma_{2} =
[-2 -1 -2 -2 -1 -1 -1 -1 -2]
[-1 -1 -2 -1 -1 -1  0 -1 -2]
[-2 -2 -2 -2 -2 -1 -1 -2 -2]
[-2 -1 -2 -1 -2 -1 -1 -1 -2]
[-1 -1 -2 -2 -2 -1 -1 -1 -2]
[-1 -1 -1 -1 -1  0  0  0 -2]
[ 3  3  4  3  3  2  2  2  4]
[ 3  2  3  3  3  2  1  2  4]
[ 4  3  5  4  4  2  2  3  5]

sigma_{3} =
[-2 -1 -2 -2 -1 -1 -1 -1 -2]
[-2 -2 -2 -2 -1 -2 -2 -1 -2]
[-1 -1 -2 -2 -1 -1 -2 -1 -1]
[-1 -1 -1 -2 -1 -1 -1  0 -2]
[-2 -2 -2 -2 -2 -1 -2 -1 -2]
[-2 -1 -2 -2 -2 -2 -2 -1 -2]
[ 3  3  4  4  3  3  3  2  4]
[ 3  2  3  3  2  2  3  1  3]
[ 5  4  5  6  4  4  5  3  5]

sigma_{4} =
[ 0 -1  0  0 -1 -1 -1 -1  0]
[-1 -1  0 -1 -1 -1 -2 -1  0]
[ 0  0  0  0  0 -1 -1  0  0]
[ 0 -1  0 -1  0 -1 -1 -1  0]
[-1 -1  0  0  0 -1 -1 -1  0]
[-1 -1 -1 -1 -1 -2 -2 -2  0]
[ 1  1  0  1  1  2  2  2  0]
[ 1  2  1  1  1  2  3  2  0]
[ 2  3  1  2  2  4  4  3  1]

sigma_{5} =
[-1 -1 -1 -2 -1 -1 -2 -2  0]
[ 0 -1 -1 -1 -1 -1 -2 -1  0]
[-1 -1 -1 -2 -1 -2 -2 -1 -1]
[ 0  0  0 -1 -1 -1 -1 -1  0]
[-1 -1 -2 -2 -2 -2 -2 -2 -1]
[-1 -1 -1 -1 -1 -2 -2 -2  0]
[ 1  2  2  3  2  3  3  3  1]
[ 2  2  2  3  3  3  4  3  1]
[ 2  2  3  4  3  4  5  4  1]

[K : K0] = 240
#(G)     = 240
[K : K0] = #(G) ? true
Thus G = Gal(K|K0).


//////////////////////////////////////////
//                                      //
//   Summary : Subgroups of Gal(K|K0)   //
//                                      //
//////////////////////////////////////////

The following analysis examines one representative from each conjugacy class of 
subgroup in Gal(K|K0), except for the conjugacy class of the trivial subgroup.

Orders of subgroups :
{* 2^^2, 3, 4^^3, 5, 6^^2, 8^^3, 10^^2, 12^^3, 15, 16, 20^^3, 24^^3, 30^^2, 
40^^3, 48, 60^^3, 80, 120^^3, 240 *}

First cohomology groups :
{*
    []^^27,
    [ 2 ],
    [ 2, 2 ]^^3,
    [ 2, 2, 2, 2 ]^^3,
    [ 2, 2, 2, 2, 2, 2, 2, 2 ],
    [ 3, 3 ],
    [ 3, 3, 3, 3 ],
    [ 5 ],
    [ 5, 5 ]
*}

///////////////////////////////////////////////////////
//                                                   //
//   Analysis of particular subgroups of Gal(K|K0)   //
//                                                   //
///////////////////////////////////////////////////////

Subgroup H generated by sigma_{0},sigma_{1},sigma_{2},sigma_{3},sigma_{4},sigma_{5} :
    #Gal(K|FixedField(H)) = 240
    H^1(H,Pic X_{K}) moduli : []

Subgroup H generated by sigma_{1},sigma_{2},sigma_{3},sigma_{4},sigma_{5} :
    #Gal(K|FixedField(H)) = 120
    H^1(H,Pic X_{K}) moduli : []

Subgroup H generated by sigma_{2},sigma_{3},sigma_{4},sigma_{5} :
    #Gal(K|FixedField(H)) = 60
    H^1(H,Pic X_{K}) moduli : []

Subgroup H generated by sigma_{2},sigma_{3},sigma_{5} :
    #Gal(K|FixedField(H)) = 30
    H^1(H,Pic X_{K}) moduli : []

Subgroup H generated by sigma_{2},sigma_{4},sigma_{5} :
    #Gal(K|FixedField(H)) = 20
    H^1(H,Pic X_{K}) moduli : []

Subgroup H generated by sigma_{2},sigma_{5} :
    #Gal(K|FixedField(H)) = 10
    H^1(H,Pic X_{K}) moduli : [ 5 ]

Subgroup H generated by sigma_{5} :
    #Gal(K|FixedField(H)) = 5
    H^1(H,Pic X_{K}) moduli : [ 5, 5 ]


//////////////////////
//                  //
//   Error alerts   //
//                  //
//////////////////////

No errors were flagged during this run.

Total time: 6.690 seconds, Total memory usage: 64.12MB
\end{verbatim}
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