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\begin{verbatim}
Magma V2.23-1     Fri Jun 24 2022 16:32:30 on Tool     [Seed = 3795234277]

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

Loading file "D13.txt"

///////////////////////////////
//                           //
//   Dolgachev type-13 dP1   //
//                           //
///////////////////////////////

This program analyzes the Dolgachev type-13 del Pezzo surface of degree 1 over 
Q, defined by
    f := -x0^6 - x0*x1^5 - x0^4*x2 - x2^3 + x3^2

///////////////////////////////////////
//                                   //
//   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^12 + 62/5*t^10 + 76*t^9 - 21*t^8 + 48*t^7 - 108*t^6 - 696/5*t^5 - 
409*t^4 - 336*t^3 - 58*t^2 - 148*t - 287/5

p_{1} = t^10 + 1/3844*(-75*theta0^11 + 45*theta0^10 - 935*theta0^9 - 
5112*theta0^8 + 4893*theta0^7 - 4578*theta0^6 + 12159*theta0^5 + 1425*theta0^4 +
25470*theta0^3 + 4347*theta0^2 - 140*theta0 + 5553)*t^9 + 
1/76880*(1350*theta0^11 - 600*theta0^10 + 18990*theta0^9 + 92085*theta0^8 - 
47700*theta0^7 + 210320*theta0^6 - 468960*theta0^5 - 72950*theta0^4 - 
742010*theta0^3 + 384080*theta0^2 - 256070*theta0 + 929417)*t^8 + 
1/19220*(-6000*theta0^11 + 5490*theta0^10 - 77835*theta0^9 - 387414*theta0^8 + 
497816*theta0^7 - 652776*theta0^6 + 990418*theta0^5 - 87260*theta0^4 + 
2594640*theta0^3 + 135214*theta0^2 + 165825*theta0 + 2014906)*t^7 + 
1/9610*(-7470*theta0^11 + 2070*theta0^10 - 92028*theta0^9 - 542387*theta0^8 + 
323610*theta0^7 - 360694*theta0^6 + 892052*theta0^5 + 1004150*theta0^4 + 
2628752*theta0^3 + 1713284*theta0^2 - 525596*theta0 + 713669)*t^6 + 
1/19220*(-26940*theta0^11 + 7980*theta0^10 - 343881*theta0^9 - 1941153*theta0^8 
+ 1051150*theta0^7 - 2114576*theta0^6 + 4070384*theta0^5 + 2342596*theta0^4 + 
11519270*theta0^3 + 6667440*theta0^2 + 1945337*theta0 + 4976385)*t^5 + 
1/38440*(-62250*theta0^11 + 26820*theta0^10 - 767400*theta0^9 - 4398187*theta0^8
+ 3406940*theta0^7 - 3205192*theta0^6 + 7996028*theta0^5 + 6442786*theta0^4 + 
21129798*theta0^3 + 9302748*theta0^2 - 5246540*theta0 + 7589409)*t^4 + 
1/19220*(-61740*theta0^11 + 16950*theta0^10 - 771441*theta0^9 - 4472300*theta0^8
+ 2519064*theta0^7 - 3650096*theta0^6 + 8433950*theta0^5 + 6760824*theta0^4 + 
23403324*theta0^3 + 14274370*theta0^2 - 156853*theta0 + 5343820)*t^3 + 
1/9610*(-16800*theta0^11 + 1320*theta0^10 - 209355*theta0^9 - 1257662*theta0^8 +
444806*theta0^7 - 882802*theta0^6 + 2166678*theta0^5 + 2384772*theta0^4 + 
6506462*theta0^3 + 5401278*theta0^2 + 335113*theta0 + 1131888)*t^2 + 
1/19220*(-17745*theta0^11 + 9555*theta0^10 - 213388*theta0^9 - 1232333*theta0^8 
+ 1184145*theta0^7 - 575278*theta0^6 + 2094749*theta0^5 + 2022011*theta0^4 + 
4849976*theta0^3 + 963665*theta0^2 - 3560013*theta0 - 290884)*t + 
1/76880*(-57090*theta0^11 + 30320*theta0^10 - 703206*theta0^9 - 3963847*theta0^8
+ 3563164*theta0^7 - 3025760*theta0^6 + 7457880*theta0^5 + 5128002*theta0^4 + 
18858622*theta0^3 + 5744376*theta0^2 - 6005386*theta0 + 5182285)

p_{2} = t^2 + -theta0^3 - theta0 - 1

p_{3} = t^2 + (1/1891371008*(-408240*theta0^11 + 242445*theta0^10 - 
5081616*theta0^9 - 27905277*theta0^8 + 26695224*theta0^7 - 24828302*theta0^6 + 
64112088*theta0^5 + 21493190*theta0^4 + 136001432*theta0^3 + 30437089*theta0^2 -
65040232*theta0 + 18457239)*theta1^9 + 1/1891371008*(-211155*theta0^11 - 
187920*theta0^10 - 2436497*theta0^9 - 18502038*theta0^8 - 7558110*theta0^7 + 
5973600*theta0^6 + 1040190*theta0^5 + 62127972*theta0^4 + 78514065*theta0^3 + 
143682144*theta0^2 + 5053043*theta0 + 71323554)*theta1^8 + 
1/2364213760*(-8287920*theta0^11 + 5528885*theta0^10 - 104089128*theta0^9 - 
559062261*theta0^8 + 575846232*theta0^7 - 586846886*theta0^6 + 
1335465864*theta0^5 + 283313870*theta0^4 + 2864079896*theta0^3 + 
455138417*theta0^2 - 904989856*theta0 + 494600967)*theta1^7 + 
1/2364213760*(-38369815*theta0^11 + 17104440*theta0^10 - 472083181*theta0^9 - 
2699422734*theta0^8 + 2148635690*theta0^7 - 1872008880*theta0^6 + 
5208738790*theta0^5 + 3226324156*theta0^4 + 12592518125*theta0^3 + 
5614967032*theta0^2 - 6689875641*theta0 + 2721859442)*theta1^6 + 
1/4728427520*(-101918460*theta0^11 + 21582815*theta0^10 - 1290310084*theta0^9 - 
7475077839*theta0^8 + 3437526768*theta0^7 - 7289160714*theta0^6 + 
12632818560*theta0^5 + 9821960850*theta0^4 + 44230303980*theta0^3 + 
24903512395*theta0^2 + 10818659812*theta0 + 20982213405)*theta1^5 + 
1/4728427520*(-143864665*theta0^11 + 95473940*theta0^10 - 1801045691*theta0^9 - 
9769265814*theta0^8 + 10088881222*theta0^7 - 10490711432*theta0^6 + 
19277269418*theta0^5 + 10701432292*theta0^4 + 42683754547*theta0^3 + 
14403255076*theta0^2 - 15892179599*theta0 + 20640543330)*theta1^4 + 
1/2364213760*(-134851580*theta0^11 + 22607485*theta0^10 - 1695789852*theta0^9 - 
9946233821*theta0^8 + 4240049056*theta0^7 - 8468698342*theta0^6 + 
17626478656*theta0^5 + 13385930926*theta0^4 + 56089770780*theta0^3 + 
36032275433*theta0^2 + 4442385692*theta0 + 20879110479)*theta1^3 + 
1/2364213760*(-80695375*theta0^11 + 25263820*theta0^10 - 1028575805*theta0^9 - 
5848601202*theta0^8 + 3279135258*theta0^7 - 6859639128*theta0^6 + 
8378663334*theta0^5 + 8970587316*theta0^4 + 32202498549*theta0^3 + 
18605126444*theta0^2 + 13177551511*theta0 + 23410523582)*theta1^2 + 
1/9456855040*(-179353640*theta0^11 + 79724425*theta0^10 - 2385946776*theta0^9 - 
12701703545*theta0^8 + 7916333864*theta0^7 - 23417140950*theta0^6 + 
23200093224*theta0^5 + 16293312366*theta0^4 + 66868812704*theta0^3 + 
54784093485*theta0^2 + 49046459728*theta0 + 81836137323)*theta1 + 
1/9456855040*(-127819975*theta0^11 - 116687080*theta0^10 - 1652501405*theta0^9 -
11080614662*theta0^8 - 7070272166*theta0^7 - 7811091888*theta0^6 + 
15112535462*theta0^5 + 21844069188*theta0^4 + 80900298189*theta0^3 + 
86655092808*theta0^2 + 71193703831*theta0 + 23951200690))*t + 
1/1891371008*(-1039395*theta0^11 + 822960*theta0^10 - 13458513*theta0^9 - 
68482206*theta0^8 + 76963394*theta0^7 - 106196736*theta0^6 + 182641630*theta0^5 
+ 1950388*theta0^4 + 428390017*theta0^3 + 14658688*theta0^2 + 18502355*theta0 + 
143833562)*theta1^9 + 1/1891371008*(2598480*theta0^11 - 2775535*theta0^10 + 
34083192*theta0^9 + 161553231*theta0^8 - 239956104*theta0^7 + 304442074*theta0^6
- 538135704*theta0^5 + 228026558*theta0^4 - 1105846696*theta0^3 + 
241804469*theta0^2 - 219839104*theta0 - 166354413)*theta1^8 + 
1/2364213760*(-32586785*theta0^11 + 23670105*theta0^10 - 421980779*theta0^9 - 
2165374203*theta0^8 + 2248801942*theta0^7 - 3196182678*theta0^6 + 
6208042730*theta0^5 - 109203046*theta0^4 + 13368060651*theta0^3 + 
87647309*theta0^2 + 1680989825*theta0 + 3365225841)*theta1^7 + 
1/2364213760*(-14592135*theta0^11 - 2237905*theta0^10 - 155110429*theta0^9 - 
1160785977*theta0^8 + 462655458*theta0^7 + 988985782*theta0^6 - 
803344602*theta0^5 + 4194269694*theta0^4 + 792072917*theta0^3 + 
7324306347*theta0^2 - 7178395577*theta0 + 7652280331)*theta1^6 + 
1/4728427520*(-190378205*theta0^11 + 105835030*theta0^10 - 2446237047*theta0^9 -
13094459448*theta0^8 + 10909982558*theta0^7 - 17032912036*theta0^6 + 
31255457842*theta0^5 + 5155990592*theta0^4 + 81172519775*theta0^3 + 
21328603166*theta0^2 + 18284898853*theta0 + 11527800264)*theta1^5 + 
1/4728427520*(-89633690*theta0^11 - 21014125*theta0^10 - 1065511926*theta0^9 - 
7182146855*theta0^8 + 936451252*theta0^7 - 1756642226*theta0^6 + 
559560508*theta0^5 + 24851606098*theta0^4 + 27110133222*theta0^3 + 
58164276959*theta0^2 - 22843365062*theta0 + 36329733909)*theta1^4 + 
1/2364213760*(-132111225*theta0^11 + 76211735*theta0^10 - 1696134155*theta0^9 - 
9048776481*theta0^8 + 7826582854*theta0^7 - 11685942058*theta0^6 + 
22322013514*theta0^5 + 5730846926*theta0^4 + 57324220307*theta0^3 + 
4810556451*theta0^2 + 2494312833*theta0 + 8038614243)*theta1^3 + 
1/2364213760*(-54530665*theta0^11 - 73638845*theta0^10 - 620566651*theta0^9 - 
5111700193*theta0^8 - 3634516322*theta0^7 + 2359838750*theta0^6 - 
1381019958*theta0^5 + 27485047726*theta0^4 + 11662192859*theta0^3 + 
53367955023*theta0^2 - 7196741471*theta0 + 16256922531)*theta1^2 + 
1/9456855040*(-1785375*theta0^11 + 45205860*theta0^10 - 246641525*theta0^9 + 
762652694*theta0^8 + 448251690*theta0^7 - 13988055608*theta0^6 + 
29714864342*theta0^5 - 41655975268*theta0^4 + 33343654165*theta0^3 - 
34544855228*theta0^2 + 64225303039*theta0 - 29715780994)*theta1 + 
1/9456855040*(-88334140*theta0^11 - 71832115*theta0^10 - 891584556*theta0^9 - 
7739660261*theta0^8 - 1034588288*theta0^7 + 11005598962*theta0^6 - 
7916868864*theta0^5 + 35137055062*theta0^4 + 12037591068*theta0^3 + 
52012639393*theta0^2 - 41874432180*theta0 + 6569854063)

p_{4} = t^5 + 1/15252992*(-2250*theta0^9 + 23625*theta0^8 - 11400*theta0^7 + 
65700*theta0^6 + 48900*theta0^5 + 12750*theta0^4 + 47000*theta0^3 - 
152700*theta0^2 - 15850*theta0 + 2225)*theta1^9 + 1/15252992*(-3750*theta0^10 + 
34875*theta0^9 - 18000*theta0^8 + 38700*theta0^7 + 191400*theta0^6 + 
38250*theta0^5 + 4500*theta0^4 - 543300*theta0^3 + 244350*theta0^2 - 
64125*theta0 - 118500)*theta1^8 + 1/3813248*(-1500*theta0^11 + 11250*theta0^10 -
16950*theta0^9 + 63675*theta0^8 + 160860*theta0^7 + 164520*theta0^6 + 
128640*theta0^5 - 480450*theta0^4 + 212400*theta0^3 - 656970*theta0^2 - 
125610*theta0 + 12135)*theta1^7 + 1/7626496*(36000*theta0^11 - 1500*theta0^10 + 
495750*theta0^9 + 2133225*theta0^8 - 187560*theta0^7 + 3089580*theta0^6 - 
2418300*theta0^5 - 1818450*theta0^4 - 5582760*theta0^3 - 5260680*theta0^2 - 
2307450*theta0 - 1222575)*theta1^6 + 1/7626496*(-4200*theta0^11 + 
34500*theta0^10 - 251880*theta0^9 + 207675*theta0^8 + 1421160*theta0^7 - 
1023420*theta0^6 + 5187240*theta0^5 + 6662370*theta0^4 + 5776160*theta0^3 + 
7038720*theta0^2 + 6321920*theta0 + 374315)*theta1^5 + 
1/7626496*(71400*theta0^11 - 77925*theta0^10 + 920685*theta0^9 + 
4662930*theta0^8 - 2531220*theta0^7 + 6276270*theta0^6 - 2709810*theta0^5 - 
4591540*theta0^4 - 13268220*theta0^3 - 12727345*theta0^2 - 3566755*theta0 - 
1456070)*theta1^4 + 1/3813248*(-30000*theta0^11 + 7350*theta0^10 - 
342700*theta0^9 - 1634535*theta0^8 + 1908420*theta0^7 + 1219080*theta0^6 + 
4792420*theta0^5 + 10211850*theta0^4 + 9229580*theta0^3 + 6420370*theta0^2 + 
2742840*theta0 - 616355)*theta1^3 + 1/7626496*(37200*theta0^11 - 43050*theta0^10
+ 714030*theta0^9 + 2258805*theta0^8 - 175800*theta0^7 - 860280*theta0^6 + 
2529060*theta0^5 - 3800770*theta0^4 - 14678920*theta0^3 + 8368730*theta0^2 + 
6156670*theta0 - 350235)*theta1^2 + 1/15252992*(-112800*theta0^11 + 
104400*theta0^10 - 1390270*theta0^9 - 6869415*theta0^8 + 5636280*theta0^7 - 
2530620*theta0^6 + 8360860*theta0^5 + 20272030*theta0^4 + 28520280*theta0^3 + 
18068660*theta0^2 + 4919570*theta0 - 94415)*theta1 + 
1/15252992*(-189600*theta0^11 - 23700*theta0^10 - 2209165*theta0^9 - 
15130080*theta0^8 + 4586460*theta0^7 - 6824140*theta0^6 + 14399242*theta0^5 + 
31242780*theta0^4 + 62389100*theta0^3 + 50870960*theta0^2 - 5609365*theta0 - 
231116)

For each i, theta_{i} is a root of p_{i}.

[K : K0] = 2400

Total time: 1236.829 seconds, Total memory usage: 168.38MB
\end{verbatim}
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