Radical Solution for solvable quintics ver2  July  2025   by  (HyperlinkURL["mixedmoss", https://mixedmoss.com])

Finds the radical solution of a solvable and irreducible quintic equation.
The initial value is  “(x^5)-2=0”. 
You can find examples of solvable and irreducible quintics below or  at (HyperlinkURL["LMFDB", https://www.lmfdb.org/GaloisGroup/]) (math data base)．
(HyperlinkURL["https://mixedmoss.com/mathematica/Galois/En/solvableQuintics.txt", https://mixedmoss.com/mathematica/Galois/En/solvableQuintics.txt])
　
For detailed information, please check this web site written in Japanese.  
If you can’t read Japanese,  please use DeepL or Google translation. (^^)/
    (HyperlinkTemplate["https://mixedmoss.com/mathematica/Galois/quintic/", https://mixedmoss.com/mathematica/Galois/quintic/, https://mixedmoss.com/mathematica/Galois/quintic/, HyperlinkActionRecycled, HyperlinkActive, , ])

Manipulate[
solveQuintic[x^5+ a x^4 +b x^3+ c x^2 +d x+e],
Text[Style["Select coefficients of a solvable and irreducible quintic: (x^5)+(ax^4)+(bx^3)+(cx^2)+dx+e=0. \nAt the output, ζ=cos(2π/5)+ι sin(2π/5)." ,Bold]],Text[Style[TraditionalForm[f:x^5+Dynamic[a] x^4+Dynamic[b] x^3 +Dynamic[c] x^2 +Dynamic[d] x+Dynamic[e]],Bold,Italic,Larger]],
Delimiter,{{a,0},Range[-15,15]},{{b,0},Range[-15,15]},{{c,0},Range[-15,15]},{{d,0},Range[-15,15]},
{{e,-2},{Join[ Range[-15,-1],{"  "},Range[1,15]]}//Flatten},
ControlType->{SetterBar,RadioButton,SetterBar},
Initialization:>(ClearAll["`*"];
pb=15625 (a2^12)-250000 a1 (a2^10) a3+1500000 (a1^2) (a2^8) (a3^2)+500000 (a2^9) (a3^2)-4000000 (a1^3) (a2^6) (a3^3)-6000000 a1 (a2^7) (a3^3)+4000000 (a1^4) (a2^4) (a3^4)+24000000 (a1^2) (a2^5) (a3^4)+6000000 (a2^6) (a3^4)-32000000 (a1^3) (a2^3) (a3^5)-48000000 a1 (a2^4) (a3^5)+96000000 (a1^2) (a2^2) (a3^6)+32000000 (a2^3) (a3^6)-128000000 a1 a2 (a3^7)+64000000 (a3^8)+500000 (a1^2) (a2^9) a4-875000 (a2^10) a4-6000000 (a1^3) (a2^7) a3 a4+10500000 a1 (a2^8) a3 a4+24000000 (a1^4) (a2^5) (a3^2) a4-30000000 (a1^2) (a2^6) (a3^2) a4-21000000 (a2^7) (a3^2) a4-32000000 (a1^5) (a2^3) (a3^3) a4-40000000 (a1^3) (a2^4) (a3^3) a4+168000000 a1 (a2^5) (a3^3) a4+192000000 (a1^4) (a2^2) (a3^4) a4-240000000 (a1^2) (a2^3) (a3^4) a4-168000000 (a2^4) (a3^4) a4-384000000 (a1^3) a2 (a3^5) a4+672000000 a1 (a2^2) (a3^5) a4+256000000 (a1^2) (a3^6) a4-448000000 a2 (a3^6) a4+6000000 (a1^4) (a2^6) (a4^2)-21000000 (a1^2) (a2^7) (a4^2)+17750000 (a2^8) (a4^2)-48000000 (a1^5) (a2^4) a3 (a4^2)+168000000 (a1^3) (a2^5) a3 (a4^2)-140000000 a1 (a2^6) a3 (a4^2)+96000000 (a1^6) (a2^2) (a3^2) (a4^2)-240000000 (a1^4) (a2^3) (a3^2) (a4^2)-68000000 (a1^2) (a2^4) (a3^2) (a4^2)+284000000 (a2^5) (a3^2) (a4^2)-384000000 (a1^5) a2 (a3^3) (a4^2)+1376000000 (a1^3) (a2^2) (a3^3) (a4^2)-1104000000 a1 (a2^3) (a3^3) (a4^2)+384000000 (a1^4) (a3^4) (a4^2)-1472000000 (a1^2) a2 (a3^4) (a4^2)+1136000000 (a2^2) (a3^4) (a4^2)+128000000 a1 (a3^5) (a4^2)+32000000 (a1^6) (a2^3) (a4^3)-168000000 (a1^4) (a2^4) (a4^3)+284000000 (a1^2) (a2^5) (a4^3)-164000000 (a2^6) (a4^3)-128000000 (a1^7) a2 a3 (a4^3)+672000000 (a1^5) (a2^2) a3 (a4^3)-1104000000 (a1^3) (a2^3) a3 (a4^3)+640000000 a1 (a2^4) a3 (a4^3)+256000000 (a1^6) (a3^2) (a4^3)-1472000000 (a1^4) a2 (a3^2) (a4^3)+2336000000 (a1^2) (a2^2) (a3^2) (a4^3)-1392000000 (a2^3) (a3^2) (a4^3)+256000000 (a1^3) (a3^3) (a4^3)+192000000 a1 a2 (a3^3) (a4^3)-640000000 (a3^4) (a4^3)+64000000 (a1^8) (a4^4)-448000000 (a1^6) a2 (a4^4)+1136000000 (a1^4) (a2^2) (a4^4)-1392000000 (a1^2) (a2^3) (a4^4)+764000000 (a2^4) (a4^4)+128000000 (a1^5) a3 (a4^4)+192000000 (a1^3) a2 a3 (a4^4)-768000000 a1 (a2^2) a3 (a4^4)-1216000000 (a1^2) (a3^2) (a4^4)+2240000000 a2 (a3^2) (a4^4)-640000000 (a1^4) (a4^5)+2240000000 (a1^2) a2 (a4^5)-1760000000 (a2^2) (a4^5)-640000000 a1 a3 (a4^5)+1600000000 (a4^6)-1500000 a1 (a2^9) a5+18000000 (a1^2) (a2^7) a3 a5+2500000 (a2^8) a3 a5-72000000 (a1^3) (a2^5) (a3^2) a5-64000000 a1 (a2^6) (a3^2) a5+96000000 (a1^4) (a2^3) (a3^3) a5+392000000 (a1^2) (a2^4) (a3^3) a5+40000000 (a2^5) (a3^3) a5-704000000 (a1^3) (a2^2) (a3^4) a5-576000000 a1 (a2^3) (a3^4) a5+1664000000 (a1^2) a2 (a3^5) a5+160000000 (a2^2) (a3^5) a5-1280000000 a1 (a3^6) a5-36000000 (a1^3) (a2^6) a4 a5+62000000 a1 (a2^7) a4 a5+288000000 (a1^4) (a2^4) a3 a4 a5-456000000 (a1^2) (a2^5) a3 a4 a5-20000000 (a2^6) a3 a4 a5-576000000 (a1^5) (a2^2) (a3^2) a4 a5+128000000 (a1^3) (a2^3) (a3^2) a4 a5+1096000000 a1 (a2^4) (a3^2) a4 a5+2816000000 (a1^4) a2 (a3^3) a4 a5-3744000000 (a1^2) (a2^2) (a3^3) a4 a5+240000000 (a2^3) (a3^3) a4 a5-3328000000 (a1^3) (a3^4) a4 a5+2560000000 a1 a2 (a3^4) a4 a5+3200000000 (a3^5) a4 a5-288000000 (a1^5) (a2^3) (a4^2) a5+992000000 (a1^3) (a2^4) (a4^2) a5-824000000 a1 (a2^5) (a4^2) a5+1152000000 (a1^6) a2 a3 (a4^2) a5-3808000000 (a1^4) (a2^2) a3 (a4^2) a5+3440000000 (a1^2) (a2^3) a3 (a4^2) a5-960000000 (a2^4) a3 (a4^2) a5-2816000000 (a1^5) (a3^2) (a4^2) a5+6912000000 (a1^3) a2 (a3^2) (a4^2) a5-2240000000 a1 (a2^2) (a3^2) (a4^2) a5+5120000000 (a1^2) (a3^3) (a4^2) a5-11200000000 a2 (a3^3) (a4^2) a5-768000000 (a1^7) (a4^3) a5+3968000000 (a1^5) a2 (a4^3) a5-6592000000 (a1^3) (a2^2) (a4^3) a5+4000000000 a1 (a2^3) (a4^3) a5+2432000000 (a1^4) a3 (a4^3) a5-11840000000 (a1^2) a2 a3 (a4^3) a5+8000000000 (a2^2) a3 (a4^3) a5+9600000000 a1 (a3^2) (a4^3) a5+3840000000 (a1^3) (a4^4) a5-6400000000 a1 a2 (a4^4) a5-16000000000 a3 (a4^4) a5+86000000 (a1^2) (a2^6) (a5^2)-125000000 (a2^7) (a5^2)-688000000 (a1^3) (a2^4) a3 (a5^2)+880000000 a1 (a2^5) a3 (a5^2)+1376000000 (a1^4) (a2^2) (a3^2) (a5^2)+240000000 (a1^2) (a2^3) (a3^2) (a5^2)-1900000000 (a2^4) (a3^2) (a5^2)-7040000000 (a1^3) a2 (a3^3) (a5^2)+6400000000 a1 (a2^2) (a3^3) (a5^2)+9600000000 (a1^2) (a3^4) (a5^2)-8000000000 a2 (a3^4) (a5^2)+1376000000 (a1^4) (a2^3) a4 (a5^2)-4360000000 (a1^2) (a2^4) a4 (a5^2)+3500000000 (a2^5) a4 (a5^2)-5504000000 (a1^5) a2 a3 a4 (a5^2)+16480000000 (a1^3) (a2^2) a3 a4 (a5^2)-14800000000 a1 (a2^3) a3 a4 (a5^2)+14080000000 (a1^4) (a3^2) a4 (a5^2)-30400000000 (a1^2) a2 (a3^2) a4 (a5^2)+32000000000 (a2^2) (a3^2) a4 (a5^2)-32000000000 a1 (a3^3) a4 (a5^2)+5504000000 (a1^6) (a4^2) (a5^2)-26880000000 (a1^4) a2 (a4^2) (a5^2)+43200000000 (a1^2) (a2^2) (a4^2) (a5^2)-22000000000 (a2^3) (a4^2) (a5^2)-16000000000 (a1^3) a3 (a4^2) (a5^2)+24000000000 a1 a2 a3 (a4^2) (a5^2)+40000000000 (a3^2) (a4^2) (a5^2)-16000000000 (a1^2) (a4^3) (a5^2)+40000000000 a2 (a4^3) (a5^2)-2400000000 (a1^3) (a2^3) (a5^3)+6000000000 a1 (a2^4) (a5^3)+9600000000 (a1^4) a2 a3 (a5^3)-20000000000 (a1^2) (a2^2) a3 (a5^3)-10000000000 (a2^3) a3 (a5^3)-32000000000 (a1^3) (a3^2) (a5^3)+80000000000 a1 a2 (a3^2) (a5^3)-19200000000 (a1^5) a4 (a5^3)+80000000000 (a1^3) a2 a4 (a5^3)-80000000000 a1 (a2^2) a4 (a5^3)+80000000000 (a1^2) a3 a4 (a5^3)-200000000000 a2 a3 a4 (a5^3)+40000000000 (a1^4) (a5^4)-200000000000 (a1^2) a2 (a5^4)+250000000000 (a2^2) (a5^4)-(18750 (a2^10)-250000 a1 (a2^8) a3+1200000 (a1^2) (a2^6) (a3^2)+400000 (a2^7) (a3^2)-2400000 (a1^3) (a2^4) (a3^3)-3600000 a1 (a2^5) (a3^3)+1600000 (a1^4) (a2^2) (a3^4)+9600000 (a1^2) (a2^3) (a3^4)+2800000 (a2^4) (a3^4)-6400000 (a1^3) a2 (a3^5)-12800000 a1 (a2^2) (a3^5)+6400000 (a1^2) (a3^6)+6400000 a2 (a3^6)+400000 (a1^2) (a2^7) a4-575000 (a2^8) a4-3600000 (a1^3) (a2^5) a3 a4+4600000 a1 (a2^6) a3 a4+9600000 (a1^4) (a2^3) (a3^2) a4-3600000 (a1^2) (a2^4) (a3^2) a4-7800000 (a2^5) (a3^2) a4-6400000 (a1^5) a2 (a3^3) a4-25600000 (a1^3) (a2^2) (a3^3) a4+25600000 a1 (a2^3) (a3^3) a4+12800000 (a1^4) (a3^4) a4+41600000 (a1^2) a2 (a3^4) a4-25600000 (a2^2) (a3^4) a4-44800000 a1 (a3^5) a4+2800000 (a1^4) (a2^4) (a4^2)-7800000 (a1^2) (a2^5) (a4^2)+6200000 (a2^6) (a4^2)-12800000 (a1^5) (a2^2) a3 (a4^2)+25600000 (a1^3) (a2^3) a3 (a4^2)-14400000 a1 (a2^4) a3 (a4^2)+6400000 (a1^6) (a3^2) (a4^2)+41600000 (a1^4) a2 (a3^2) (a4^2)-91200000 (a1^2) (a2^2) (a3^2) (a4^2)+40000000 (a2^3) (a3^2) (a4^2)-96000000 (a1^3) (a3^3) (a4^2)+124800000 a1 a2 (a3^3) (a4^2)+9600000 (a3^4) (a4^2)+6400000 (a1^6) a2 (a4^3)-25600000 (a1^4) (a2^2) (a4^3)+40000000 (a1^2) (a2^3) (a4^3)-5600000 (a2^4) (a4^3)-44800000 (a1^5) a3 (a4^3)+124800000 (a1^3) a2 a3 (a4^3)-182400000 a1 (a2^2) a3 (a4^3)+96000000 (a1^2) (a3^2) (a4^3)+92800000 a2 (a3^2) (a4^3)+9600000 (a1^4) (a4^4)+92800000 (a1^2) a2 (a4^4)-129600000 (a2^2) (a4^4)-294400000 a1 a3 (a4^4)+339200000 (a4^5)-400000 (a1^3) (a2^6) a5+600000 a1 (a2^7) a5+3200000 (a1^4) (a2^4) a3 a5-3600000 (a1^2) (a2^5) a3 a5-1000000 (a2^6) a3 a5-6400000 (a1^5) (a2^2) (a3^2) a5-6400000 (a1^3) (a2^3) (a3^2) a5+19200000 a1 (a2^4) (a3^2) a5+44800000 (a1^4) a2 (a3^3) a5-57600000 (a1^2) (a2^2) (a3^3) a5+19200000 (a2^3) (a3^3) a5-38400000 (a1^3) (a3^4) a5-38400000 a1 a2 (a3^4) a5+185600000 (a3^5) a5-6400000 (a1^5) (a2^3) a4 a5+23200000 (a1^3) (a2^4) a4 a5-27600000 a1 (a2^5) a4 a5+25600000 (a1^6) a2 a3 a4 a5-83200000 (a1^4) (a2^2) a3 a4 a5+153600000 (a1^2) (a2^3) a3 a4 a5-134400000 (a2^4) a3 a4 a5-89600000 (a1^5) (a3^2) a4 a5+12800000 (a1^3) a2 (a3^2) a4 a5+403200000 a1 (a2^2) (a3^2) a4 a5+364800000 (a1^2) (a3^3) a4 a5-1296000000 a2 (a3^3) a4 a5-25600000 (a1^7) (a4^2) a5+147200000 (a1^5) a2 (a4^2) a5-342400000 (a1^3) (a2^2) (a4^2) a5+236800000 a1 (a2^3) (a4^2) a5+544000000 (a1^4) a3 (a4^2) a5-1731200000 (a1^2) a2 a3 (a4^2) a5+1392000000 (a2^2) a3 (a4^2) a5+1024000000 a1 (a3^2) (a4^2) a5-563200000 (a1^3) (a4^3) a5+992000000 a1 a2 (a4^3) a5-1920000000 a3 (a4^3) a5+19200000 (a1^4) (a2^3) (a5^2)-128400000 (a1^2) (a2^4) (a5^2)+270600000 (a2^5) (a5^2)-76800000 (a1^5) a2 a3 (a5^2)+556800000 (a1^3) (a2^2) a3 (a5^2)-1376000000 a1 (a2^3) a3 (a5^2)+6400000 (a1^4) (a3^2) (a5^2)+112000000 (a1^2) a2 (a3^2) (a5^2)+2040000000 (a2^2) (a3^2) (a5^2)-1280000000 a1 (a3^3) (a5^2)+153600000 (a1^6) a4 (a5^2)-1286400000 (a1^4) a2 a4 (a5^2)+3744000000 (a1^2) (a2^2) a4 (a5^2)-2680000000 (a2^3) a4 (a5^2)-288000000 (a1^3) a3 a4 (a5^2)-2960000000 a1 a2 a3 a4 (a5^2)+3200000000 (a3^2) a4 (a5^2)+2240000000 (a1^2) (a4^2) (a5^2)+400000000 a2 (a4^2) (a5^2)+179200000 (a1^5) (a5^3)-1120000000 (a1^3) a2 (a5^3)+1200000000 a1 (a2^2) (a5^3)+2400000000 (a1^2) a3 (a5^3)-2000000000 a2 a3 (a5^3)-8000000000 a1 a4 (a5^3)+10000000000 (a5^4)) x+(9375 (a2^8)-100000 a1 (a2^6) a3+360000 (a1^2) (a2^4) (a3^2)+120000 (a2^5) (a3^2)-480000 (a1^3) (a2^2) (a3^3)-720000 a1 (a2^3) (a3^3)+160000 (a1^4) (a3^4)+960000 (a1^2) a2 (a3^4)+400000 (a2^2) (a3^4)-640000 a1 (a3^5)+120000 (a1^2) (a2^5) a4-110000 (a2^6) a4-720000 (a1^3) (a2^3) a3 a4+240000 a1 (a2^4) a3 a4+960000 (a1^4) a2 (a3^2) a4+2160000 (a1^2) (a2^2) (a3^2) a4-600000 (a2^3) (a3^2) a4-3520000 (a1^3) (a3^3) a4-1600000 a1 a2 (a3^3) a4+1600000 (a3^4) a4+400000 (a1^4) (a2^2) (a4^2)-600000 (a1^2) (a2^3) (a4^2)+900000 (a2^4) (a4^2)-640000 (a1^5) a3 (a4^2)-1600000 (a1^3) a2 a3 (a4^2)-2400000 a1 (a2^2) a3 (a4^2)+15200000 (a1^2) (a3^2) (a4^2)-800000 a2 (a3^2) (a4^2)+1600000 (a1^4) (a4^3)-800000 (a1^2) a2 (a4^3)+800000 (a2^2) (a4^3)-28800000 a1 a3 (a4^3)+28000000 (a4^4)-240000 (a1^3) (a2^4) a5+720000 a1 (a2^5) a5+1280000 (a1^4) (a2^2) a3 a5-3600000 (a1^2) (a2^3) a3 a5-1200000 (a2^4) a3 a5-1280000 (a1^5) (a3^2) a5+1600000 (a1^3) a2 (a3^2) a5+9600000 a1 (a2^2) (a3^2) a5-1600000 (a1^2) (a3^3) a5-8000000 a2 (a3^3) a5-1280000 (a1^5) a2 a4 a5+4000000 (a1^3) (a2^2) a4 a5-1200000 a1 (a2^3) a4 a5+12800000 (a1^4) a3 a4 a5-43200000 (a1^2) a2 a3 a4 a5+12000000 (a2^2) a3 a4 a5+24000000 a1 (a3^2) a4 a5-28800000 (a1^3) (a4^2) a5+88000000 a1 a2 (a4^2) a5-80000000 a3 (a4^2) a5+2560000 (a1^6) (a5^2)-19200000 (a1^4) a2 (a5^2)+42000000 (a1^2) (a2^2) (a5^2)-15000000 (a2^3) (a5^2)+16000000 (a1^3) a3 (a5^2)-80000000 a1 a2 a3 (a5^2)+100000000 (a3^2) (a5^2)+40000000 (a1^2) a4 (a5^2)-100000000 a2 a4 (a5^2)) (x^2)-(2500 (a2^6)-20000 a1 (a2^4) a3+48000 (a1^2) (a2^2) (a3^2)+16000 (a2^3) (a3^2)-32000 (a1^3) (a3^3)-48000 a1 a2 (a3^3)+16000 (a3^4)+16000 (a1^2) (a2^3) a4+2000 (a2^4) a4-48000 (a1^3) a2 a3 a4-120000 a1 (a2^2) a3 a4+336000 (a1^2) (a3^2) a4+24000 a2 (a3^2) a4+16000 (a1^4) (a4^2)+24000 (a1^2) a2 (a4^2)+184000 (a2^2) (a4^2)-1024000 a1 a3 (a4^2)+1120000 (a4^3)-48000 (a1^3) (a2^2) a5+168000 a1 (a2^3) a5+128000 (a1^4) a3 a5-432000 (a1^2) a2 a3 a5-280000 (a2^2) a3 a5+640000 a1 (a3^2) a5-416000 (a1^3) a4 a5+1360000 a1 a2 a4 a5-1600000 a3 a4 a5+400000 (a1^2) (a5^2)-1000000 a2 (a5^2)) (x^3)+(375 (a2^4)-2000 a1 (a2^2) a3+2400 (a1^2) (a3^2)+800 a2 (a3^2)+800 (a1^2) a2 a4+2600 (a2^2) a4-13600 a1 a3 a4+22000 (a4^2)-3200 (a1^3) a5+12000 a1 a2 a5-20000 a3 a5) (x^4)-(30 (a2^2)-80 a1 a3+200 a4) (x^5)+(x^6);
pc=15625 (a1^18) (a2^6)-93750 (a1^16) (a2^7)+234375 (a1^14) (a2^8)-312500 (a1^12) (a2^9)+234375 (a1^10) (a2^10)-93750 (a1^8) (a2^11)+15625 (a1^6) (a2^12)-218750 (a1^17) (a2^5) a3+1250000 (a1^15) (a2^6) a3-2968750 (a1^13) (a2^7) a3+3750000 (a1^11) (a2^8) a3-2656250 (a1^9) (a2^9) a3+1000000 (a1^7) (a2^10) a3-156250 (a1^5) (a2^11) a3+1109375 (a1^16) (a2^4) (a3^2)-5343750 (a1^14) (a2^5) (a3^2)+9890625 (a1^12) (a2^6) (a3^2)-8312500 (a1^10) (a2^7) (a3^2)+2390625 (a1^8) (a2^8) (a3^2)+656250 (a1^6) (a2^9) (a3^2)-390625 (a1^4) (a2^10) (a3^2)-2562500 (a1^15) (a2^3) (a3^3)+7125000 (a1^13) (a2^4) (a3^3)+2437500 (a1^11) (a2^5) (a3^3)-32250000 (a1^9) (a2^6) (a3^3)+49312500 (a1^7) (a2^7) (a3^3)-31875000 (a1^5) (a2^8) (a3^3)+7812500 (a1^3) (a2^9) (a3^3)+2984375 (a1^14) (a2^2) (a3^4)-1781250 (a1^12) (a2^3) (a3^4)-6734375 (a1^10) (a2^4) (a3^4)+17812500 (a1^8) (a2^5) (a3^4)-32984375 (a1^6) (a2^6) (a3^4)+30468750 (a1^4) (a2^7) (a3^4)-9765625 (a1^2) (a2^8) (a3^4)-1718750 (a1^13) a2 (a3^5)+17250000 (a1^11) (a2^2) (a3^5)-225968750 (a1^9) (a2^3) (a3^5)+548250000 (a1^7) (a2^4) (a3^5)-552656250 (a1^5) (a2^5) (a3^5)+312500000 (a1^3) (a2^6) (a3^5)-97656250 a1 (a2^7) (a3^5)+390625 (a1^12) (a3^6)-37781250 (a1^10) a2 (a3^6)+483609375 (a1^8) (a2^2) (a3^6)-626437500 (a1^6) (a2^3) (a3^6)-56640625 (a1^4) (a2^4) (a3^6)+136718750 (a1^2) (a2^5) (a3^6)+244140625 (a2^6) (a3^6)+19375000 (a1^9) (a3^7)-333000000 (a1^7) a2 (a3^7)-773750000 (a1^5) (a2^2) (a3^7)+2875000000 (a1^3) (a2^3) (a3^7)-3515625000 a1 (a2^4) (a3^7)+396500000 (a1^6) (a3^8)-1743750000 (a1^4) a2 (a3^8)+5625000000 (a1^2) (a2^2) (a3^8)+3906250000 (a2^3) (a3^8)+3875000000 (a1^3) (a3^9)-21875000000 a1 a2 (a3^9)+15625000000 (a3^10)+1281250 (a1^16) (a2^5) a4-8656250 (a1^14) (a2^6) a4+24937500 (a1^12) (a2^7) a4-38812500 (a1^10) (a2^8) a4+34156250 (a1^8) (a2^9) a4-16031250 (a1^6) (a2^10) a4+3125000 (a1^4) (a2^11) a4-13281250 (a1^15) (a2^4) a3 a4+81531250 (a1^13) (a2^5) a3 a4-212562500 (a1^11) (a2^6) a3 a4+312562500 (a1^9) (a2^7) a3 a4-278406250 (a1^7) (a2^8) a3 a4+141406250 (a1^5) (a2^9) a3 a4-31250000 (a1^3) (a2^10) a3 a4+47312500 (a1^14) (a2^3) (a3^2) a4-199500000 (a1^12) (a2^4) (a3^2) a4+205125000 (a1^10) (a2^5) (a3^2) a4+74000000 (a1^8) (a2^6) (a3^2) a4-201937500 (a1^6) (a2^7) (a3^2) a4+75000000 (a1^4) (a2^8) (a3^2) a4-75062500 (a1^13) (a2^2) (a3^3) a4-8687500 (a1^11) (a2^3) (a3^3) a4+1660125000 (a1^9) (a2^4) (a3^3) a4-4117625000 (a1^7) (a2^5) (a3^3) a4+4252187500 (a1^5) (a2^6) (a3^3) a4-2492187500 (a1^3) (a2^7) (a3^3) a4+781250000 a1 (a2^8) (a3^3) a4+54906250 (a1^12) a2 (a3^4) a4+282593750 (a1^10) (a2^2) (a3^4) a4-2126062500 (a1^8) (a2^3) (a3^4) a4+765437500 (a1^6) (a2^4) (a3^4) a4+3863281250 (a1^4) (a2^5) (a3^4) a4-2207031250 (a1^2) (a2^6) (a3^4) a4-1953125000 (a2^7) (a3^4) a4-15156250 (a1^11) (a3^5) a4+155156250 (a1^9) a2 (a3^5) a4-3542062500 (a1^7) (a2^2) (a3^5) a4+17661562500 (a1^5) (a2^3) (a3^5) a4-30988281250 (a1^3) (a2^4) (a3^5) a4+33300781250 a1 (a2^5) (a3^5) a4-302437500 (a1^8) (a3^6) a4+2914500000 (a1^6) a2 (a3^6) a4+14909375000 (a1^4) (a2^2) (a3^6) a4-60937500000 (a1^2) (a2^3) (a3^6) a4-38085937500 (a2^4) (a3^6) a4-1210000000 (a1^5) (a3^7) a4-57375000000 (a1^3) a2 (a3^7) a4+243750000000 a1 (a2^2) (a3^7) a4+14687500000 (a1^2) (a3^8) a4-179687500000 a2 (a3^8) a4-2000000 (a1^16) (a2^3) (a4^2)+63171875 (a1^14) (a2^4) (a4^2)-387937500 (a1^12) (a2^5) (a4^2)+1085781250 (a1^10) (a2^6) (a4^2)-1675437500 (a1^8) (a2^7) (a4^2)+1497671875 (a1^6) (a2^8) (a4^2)-737500000 (a1^4) (a2^9) (a4^2)+156250000 (a1^2) (a2^10) (a4^2)+14000000 (a1^15) (a2^2) a3 (a4^2)-508937500 (a1^13) (a2^3) a3 (a4^2)+2923375000 (a1^11) (a2^4) a3 (a4^2)-7068750000 (a1^9) (a2^5) a3 (a4^2)+9280625000 (a1^7) (a2^6) a3 (a4^2)-7765312500 (a1^5) (a2^7) a3 (a4^2)+4687500000 (a1^3) (a2^8) a3 (a4^2)-1562500000 a1 (a2^9) a3 (a4^2)-22000000 (a1^14) a2 (a3^2) (a4^2)+1316281250 (a1^12) (a2^2) (a3^2) (a4^2)-6117750000 (a1^10) (a2^3) (a3^2) (a4^2)+5224187500 (a1^8) (a2^4) (a3^2) (a4^2)+10187750000 (a1^6) (a2^5) (a3^2) (a4^2)-20386718750 (a1^4) (a2^6) (a3^2) (a4^2)+10000000000 (a1^2) (a2^7) (a3^2) (a4^2)+3906250000 (a2^8) (a3^2) (a4^2)+10000000 (a1^13) (a3^3) (a4^2)-1222437500 (a1^11) a2 (a3^3) (a4^2)+1225125000 (a1^9) (a2^2) (a3^3) (a4^2)+33082250000 (a1^7) (a2^3) (a3^3) (a4^2)-86225625000 (a1^5) (a2^4) (a3^3) (a4^2)+94617187500 (a1^3) (a2^5) (a3^3) (a4^2)-95312500000 a1 (a2^6) (a3^3) (a4^2)+63921875 (a1^10) (a3^4) (a4^2)+7072187500 (a1^8) a2 (a3^4) (a4^2)-53506968750 (a1^6) (a2^2) (a3^4) (a4^2)-20139062500 (a1^4) (a2^3) (a3^4) (a4^2)+220732421875 (a1^2) (a2^4) (a3^4) (a4^2)+117187500000 (a2^5) (a3^4) (a4^2)-1257000000 (a1^7) (a3^5) (a4^2)-475000000 (a1^5) a2 (a3^5) (a4^2)+307062500000 (a1^3) (a2^2) (a3^5) (a4^2)-918750000000 a1 (a2^3) (a3^5) (a4^2)+38218750000 (a1^4) (a3^6) (a4^2)-285468750000 (a1^2) a2 (a3^6) (a4^2)+730468750000 (a2^2) (a3^6) (a4^2)+126562500000 a1 (a3^7) (a4^2)-82000000 (a1^14) (a2^2) (a4^3)+1649687500 (a1^12) (a2^3) (a4^3)-9270812500 (a1^10) (a2^4) (a4^3)+23499812500 (a1^8) (a2^5) (a4^3)-30909937500 (a1^6) (a2^6) (a4^3)+23593750000 (a1^4) (a2^7) (a4^3)-12812500000 (a1^2) (a2^8) (a4^3)+276000000 (a1^13) a2 a3 (a4^3)-9299812500 (a1^11) (a2^2) a3 (a4^3)+58172687500 (a1^9) (a2^3) a3 (a4^3)-140834687500 (a1^7) (a2^4) a3 (a4^3)+146251562500 (a1^5) (a2^5) a3 (a4^3)-68656250000 (a1^3) (a2^6) a3 (a4^3)+73437500000 a1 (a2^7) a3 (a4^3)-194000000 (a1^12) (a3^2) (a4^3)+16419562500 (a1^10) a2 (a3^2) (a4^3)-115629562500 (a1^8) (a2^2) (a3^2) (a4^3)+226842687500 (a1^6) (a2^3) (a3^2) (a4^3)-28354687500 (a1^4) (a2^4) (a3^2) (a4^3)-286250000000 (a1^2) (a2^5) (a3^2) (a4^3)-109375000000 (a2^6) (a3^2) (a4^3)-6857437500 (a1^9) (a3^3) (a4^3)+46774812500 (a1^7) a2 (a3^3) (a4^3)+171299687500 (a1^5) (a2^2) (a3^3) (a4^3)-932257812500 (a1^3) (a2^3) (a3^3) (a4^3)+1295312500000 a1 (a2^4) (a3^3) (a4^3)+7441875000 (a1^6) (a3^4) (a4^3)-510687500000 (a1^4) a2 (a3^4) (a4^3)+1688359375000 (a1^2) (a2^2) (a3^4) (a4^3)-1218750000000 (a2^3) (a3^4) (a4^3)+130625000000 (a1^3) (a3^5) (a4^3)-945312500000 a1 a2 (a3^5) (a4^3)-78125000000 (a3^6) (a4^3)+64000000 (a1^14) (a4^4)-1006000000 (a1^12) a2 (a4^4)+14016609375 (a1^10) (a2^2) (a4^4)-79824468750 (a1^8) (a2^3) (a4^4)+208062359375 (a1^6) (a2^4) (a4^4)-247587500000 (a1^4) (a2^5) (a4^4)+80156250000 (a1^2) (a2^6) (a4^4)+118000000 (a1^11) a3 (a4^4)-39360843750 (a1^9) a2 a3 (a4^4)+347420125000 (a1^7) (a2^2) a3 (a4^4)-1124197031250 (a1^5) (a2^3) a3 (a4^4)+1600062500000 (a1^3) (a2^4) a3 (a4^4)-481250000000 a1 (a2^5) a3 (a4^4)+35535484375 (a1^8) (a3^2) (a4^4)-459506406250 (a1^6) a2 (a3^2) (a4^4)+1723271484375 (a1^4) (a2^2) (a3^2) (a4^4)-3116562500000 (a1^2) (a2^3) (a3^2) (a4^4)+687500000000 (a2^4) (a3^2) (a4^4)+218128125000 (a1^5) (a3^3) (a4^4)-640937500000 (a1^3) a2 (a3^3) (a4^4)+1575000000000 a1 (a2^2) (a3^3) (a4^4)+80468750000 (a1^2) (a3^4) (a4^4)+625000000000 a2 (a3^4) (a4^4)+3330000000 (a1^10) (a4^5)-21301468750 (a1^8) a2 (a4^5)+60400468750 (a1^6) (a2^2) (a4^5)-82234375000 (a1^4) (a2^3) (a4^5)-81875000000 (a1^2) (a2^4) (a4^5)-10468906250 (a1^7) a3 (a4^5)+42444531250 (a1^5) a2 a3 (a4^5)+181250000000 (a1^3) (a2^2) a3 (a4^5)+725000000000 a1 (a2^3) a3 (a4^5)-268085937500 (a1^4) (a3^2) (a4^5)-218750000000 (a1^2) a2 (a3^2) (a4^5)-1250000000000 (a2^2) (a3^2) (a4^5)-125000000000 a1 (a3^3) (a4^5)+38445703125 (a1^6) (a4^6)-159062500000 (a1^4) a2 (a4^6)-112500000000 (a1^2) (a2^2) (a4^6)+400000000000 (a1^3) a3 (a4^6)+500000000000 a1 a2 a3 (a4^6)-250000000000 (a1^2) (a4^7)-2000000 (a1^17) (a2^4) a5+14593750 (a1^15) (a2^5) a5-37156250 (a1^13) (a2^6) a5+30937500 (a1^11) (a2^7) a5-17062500 (a1^9) (a2^8) a5+60218750 (a1^7) (a2^9) a5-80781250 (a1^5) (a2^10) a5+31250000 (a1^3) (a2^11) a5+24000000 (a1^16) (a2^3) a3 a5-205093750 (a1^14) (a2^4) a3 a5+600718750 (a1^12) (a2^5) a3 a5-471687500 (a1^10) (a2^6) a3 a5-256062500 (a1^8) (a2^7) a3 a5+249531250 (a1^6) (a2^8) a3 a5+214843750 (a1^4) (a2^9) a3 a5-156250000 (a1^2) (a2^10) a3 a5-92000000 (a1^15) (a2^2) (a3^2) a5+975437500 (a1^13) (a2^3) (a3^2) a5-3443312500 (a1^11) (a2^4) (a3^2) a5+3137625000 (a1^9) 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(a5^3)-468750000000 a2 (a4^2) (a5^3)-6250000000 (a1^5) (a5^4)-95214843750 (a1^3) a2 (a5^4)+95214843750 a1 (a2^2) (a5^4)+222167968750 (a1^2) a3 (a5^4)-158691406250 a2 a3 (a5^4)-793457031250 a1 a4 (a5^4)+793457031250 (a5^5)) x+(9375 (a1^12) (a2^4)-37500 (a1^10) (a2^5)+56250 (a1^8) (a2^6)-37500 (a1^6) (a2^7)+9375 (a1^4) (a2^8)-87500 (a1^11) (a2^3) a3+325000 (a1^9) (a2^4) a3-450000 (a1^7) (a2^5) a3+275000 (a1^5) (a2^6) a3-62500 (a1^3) (a2^7) a3+266250 (a1^10) (a2^2) (a3^2)-750000 (a1^8) (a2^3) (a3^2)+607500 (a1^6) (a2^4) (a3^2)-30000 (a1^4) (a2^5) (a3^2)-93750 (a1^2) (a2^6) (a3^2)-307500 (a1^9) a2 (a3^3)+165000 (a1^7) (a2^2) (a3^3)+1230000 (a1^5) (a2^3) (a3^3)-2025000 (a1^3) (a2^4) (a3^3)+937500 a1 (a2^5) (a3^3)+119375 (a1^8) (a3^4)+207500 (a1^6) a2 (a3^4)+1406250 (a1^4) (a2^2) (a3^4)-1062500 (a1^2) (a2^3) (a3^4)-390625 (a2^4) (a3^4)+810000 (a1^5) (a3^5)-10800000 (a1^3) a2 (a3^5)+8750000 a1 (a2^2) (a3^5)+9000000 (a1^2) (a3^6)-6250000 a2 (a3^6)+432500 (a1^10) (a2^3) a4-1837500 (a1^8) (a2^4) a4+3127500 (a1^6) (a2^5) a4-2472500 (a1^4) (a2^6) a4+750000 (a1^2) (a2^7) a4-2707500 (a1^9) (a2^2) a3 a4+9892500 (a1^7) (a2^3) a3 a4-13972500 (a1^5) (a2^4) a3 a4+10537500 (a1^3) (a2^5) a3 a4-3750000 a1 (a2^6) a3 a4+4957500 (a1^8) a2 (a3^2) a4-9037500 (a1^6) (a2^2) (a3^2) a4-9487500 (a1^4) (a2^3) (a3^2) a4+12187500 (a1^2) (a2^4) (a3^2) a4-2682500 (a1^7) (a3^3) a4-10312500 (a1^5) a2 (a3^3) a4+73312500 (a1^3) (a2^2) (a3^3) a4-57187500 a1 (a2^3) (a3^3) a4+3975000 (a1^4) (a3^4) a4-35000000 (a1^2) a2 (a3^4) a4+34375000 (a2^2) (a3^4) a4-32500000 a1 (a3^5) a4-240000 (a1^10) a2 (a4^2)+9021250 (a1^8) (a2^2) (a4^2)-34572500 (a1^6) (a2^3) (a4^2)+50531250 (a1^4) (a2^4) (a4^2)-30500000 (a1^2) (a2^5) (a4^2)+6250000 (a2^6) (a4^2)+560000 (a1^9) a3 (a4^2)-37752500 (a1^7) a2 a3 (a4^2)+141000000 (a1^5) (a2^2) a3 (a4^2)-166937500 (a1^3) (a2^3) a3 (a4^2)+77500000 a1 (a2^4) a3 (a4^2)+38031250 (a1^6) (a3^2) (a4^2)-141737500 (a1^4) a2 (a3^2) (a4^2)+29531250 (a1^2) (a2^2) (a3^2) (a4^2)-12500000 (a2^3) (a3^2) (a4^2)+59500000 (a1^3) (a3^3) (a4^2)+112500000 a1 a2 (a3^3) (a4^2)+43750000 (a3^4) (a4^2)-2000000 (a1^8) (a4^3)+63362500 (a1^6) a2 (a4^3)-211512500 (a1^4) (a2^2) (a4^3)+211250000 (a1^2) (a2^3) (a4^3)-75000000 (a2^4) (a4^3)-142112500 (a1^5) a3 (a4^3)+559562500 (a1^3) a2 a3 (a4^3)-250000000 a1 (a2^2) a3 (a4^3)-408125000 (a1^2) (a3^2) (a4^3)-150000000 a2 (a3^2) (a4^3)+167484375 (a1^4) (a4^4)-605000000 (a1^2) a2 (a4^4)+300000000 (a2^2) (a4^4)+800000000 a1 a3 (a4^4)-400000000 (a4^5)-480000 (a1^11) (a2^2) a5+2157500 (a1^9) (a2^3) a5-3697500 (a1^7) (a2^4) a5+1432500 (a1^5) (a2^5) a5-3162500 (a1^3) (a2^6) a5+3750000 a1 (a2^7) a5+2880000 (a1^10) a2 a3 a5-16502500 (a1^8) (a2^2) a3 a5+34137500 (a1^6) (a2^3) a3 a5+8362500 (a1^4) (a2^4) a3 a5-19187500 (a1^2) (a2^5) a3 a5-6250000 (a2^6) a3 a5-3680000 (a1^9) (a3^2) a5+39612500 (a1^7) a2 (a3^2) a5-142462500 (a1^5) (a2^2) (a3^2) a5+2687500 (a1^3) (a2^3) (a3^2) a5+82812500 a1 (a2^4) (a3^2) a5-36787500 (a1^6) (a3^3) a5+280562500 (a1^4) a2 (a3^3) a5-111562500 (a1^2) (a2^2) (a3^3) a5-76562500 (a2^3) (a3^3) a5-177500000 (a1^3) (a3^4) a5+112500000 a1 a2 (a3^4) a5-12500000 (a3^5) a5-9440000 (a1^9) a2 a4 a5+40587500 (a1^7) (a2^2) a4 a5-104325000 (a1^5) (a2^3) a4 a5+100187500 (a1^3) (a2^4) a4 a5-77500000 a1 (a2^5) a4 a5+24800000 (a1^8) a3 a4 a5-134075000 (a1^6) a2 a3 a4 a5+477750000 (a1^4) (a2^2) a3 a4 a5-180625000 (a1^2) (a2^3) a3 a4 a5+137500000 (a2^4) a3 a4 a5+155687500 (a1^5) (a3^2) a4 a5-1294375000 (a1^3) a2 (a3^2) a4 a5+342187500 a1 (a2^2) (a3^2) a4 a5+1106250000 (a1^2) (a3^3) a4 a5-187500000 a2 (a3^3) a4 a5-30800000 (a1^7) (a4^2) a5-46437500 (a1^5) a2 (a4^2) a5+16562500 (a1^3) (a2^2) (a4^2) a5-162500000 a1 (a2^3) (a4^2) a5+214687500 (a1^4) a3 (a4^2) a5+895312500 (a1^2) a2 a3 (a4^2) a5-250000000 (a2^2) a3 (a4^2) a5-1718750000 a1 (a3^2) (a4^2) a5-929687500 (a1^3) (a4^3) a5+1250000000 a1 a2 (a4^3) a5+2560000 (a1^10) (a5^2)-14000000 (a1^8) a2 (a5^2)+105531250 (a1^6) (a2^2) (a5^2)-259062500 (a1^4) (a2^3) (a5^2)+294531250 (a1^2) (a2^4) (a5^2)-12500000 (a2^5) (a5^2)-26000000 (a1^7) a3 (a5^2)-37812500 (a1^5) a2 a3 (a5^2)+156250000 (a1^3) (a2^2) a3 (a5^2)-898437500 a1 (a2^3) a3 (a5^2)-199218750 (a1^4) (a3^2) (a5^2)+1570312500 (a1^2) a2 (a3^2) (a5^2)+488281250 (a2^2) (a3^2) (a5^2)-1718750000 a1 (a3^3) (a5^2)+90000000 (a1^6) a4 (a5^2)+492187500 (a1^4) a2 a4 (a5^2)-492187500 (a1^2) (a2^2) a4 (a5^2)+781250000 (a2^3) a4 (a5^2)-1148437500 (a1^3) a3 a4 (a5^2)-2070312500 a1 a2 a3 a4 (a5^2)+4296875000 (a3^2) a4 (a5^2)+4472656250 (a1^2) (a4^2) (a5^2)-3125000000 a2 (a4^2) (a5^2)-150000000 (a1^5) (a5^3)-820312500 (a1^3) a2 (a5^3)+585937500 a1 (a2^2) (a5^3)+2539062500 (a1^2) a3 (a5^3)-976562500 a2 a3 (a5^3)-10742187500 a1 a4 (a5^3)+13427734375 (a5^4)) (x^2)+(-2500 (a1^9) (a2^3)+7500 (a1^7) (a2^4)-7500 (a1^5) (a2^5)+2500 (a1^3) (a2^6)+17500 (a1^8) (a2^2) a3-47500 (a1^6) (a2^3) a3+42500 (a1^4) (a2^4) a3-12500 (a1^2) (a2^5) a3-35500 (a1^7) a2 (a3^2)+64500 (a1^5) (a2^2) (a3^2)-16500 (a1^3) (a2^3) (a3^2)-12500 a1 (a2^4) (a3^2)+20500 (a1^6) (a3^3)+19500 (a1^4) a2 (a3^3)-82500 (a1^2) (a2^2) (a3^3)+62500 (a2^3) (a3^3)-4000 (a1^3) (a3^4)-150000 a1 a2 (a3^4)+250000 (a3^5)-78500 (a1^7) (a2^2) a4+229000 (a1^5) (a2^3) a4-250500 (a1^3) (a2^4) a4+100000 a1 (a2^5) a4+329000 (a1^6) a2 a3 a4-770000 (a1^4) (a2^2) a3 a4+595000 (a1^2) (a2^3) a3 a4-250000 (a2^4) a3 a4-306500 (a1^5) (a3^2) a4+159000 (a1^3) a2 (a3^2) a4+937500 a1 (a2^2) (a3^2) a4+330000 (a1^2) (a3^3) a4-1500000 a2 (a3^3) a4+16000 (a1^7) (a4^2)-873500 (a1^5) a2 (a4^2)+2111500 (a1^3) (a2^2) (a4^2)-1500000 a1 (a2^3) (a4^2)+1883500 (a1^4) a3 (a4^2)-4682500 (a1^2) a2 a3 (a4^2)+2000000 (a2^2) a3 (a4^2)+2050000 a1 (a3^2) (a4^2)-2377500 (a1^3) (a4^3)+4400000 a1 a2 (a4^3)-4000000 a3 (a4^3)+64000 (a1^8) a2 a5-135500 (a1^6) (a2^2) a5+193000 (a1^4) (a2^3) a5-27500 (a1^2) (a2^4) a5+250000 (a2^5) a5-192000 (a1^7) a3 a5+483000 (a1^5) a2 a3 a5-1140000 (a1^3) (a2^2) a3 a5-1575000 a1 (a2^3) a3 a5-707500 (a1^4) (a3^2) a5+5875000 (a1^2) a2 (a3^2) a5+812500 (a2^2) (a3^2) a5-5500000 a1 (a3^3) a5+544000 (a1^6) a4 a5+895000 (a1^4) a2 a4 a5+375000 (a1^2) (a2^2) a4 a5+1500000 (a2^3) a4 a5-4675000 (a1^3) a3 a4 a5-7875000 a1 a2 a3 a4 a5+13750000 (a3^2) a4 a5+18062500 (a1^2) (a4^2) a5-10000000 a2 (a4^2) a5-1200000 (a1^5) (a5^2)-3437500 (a1^3) a2 (a5^2)+937500 a1 (a2^2) (a5^2)+14687500 (a1^2) a3 (a5^2)-1562500 a2 a3 (a5^2)-70312500 a1 a4 (a5^2)+117187500 (a5^3)) (x^3)+(375 (a1^6) (a2^2)-750 (a1^4) (a2^3)+375 (a1^2) (a2^4)-1750 (a1^5) a2 a3+3000 (a1^3) (a2^2) a3-1250 a1 (a2^3) a3+1775 (a1^4) (a3^2)-1450 (a1^2) a2 (a3^2)-625 (a2^2) (a3^2)-1000 a1 (a3^3)+7050 (a1^4) a2 a4-10650 (a1^2) (a2^2) a4+5000 (a2^3) a4-14850 (a1^3) a3 a4+14250 a1 a2 a3 a4+2500 (a3^2) a4+29875 (a1^2) (a4^2)-20000 a2 (a4^2)-3200 (a1^5) a5-11250 (a1^3) a2 a5+3750 a1 (a2^2) a5+46250 (a1^2) a3 a5-6250 a2 a3 a5-218750 a1 a4 a5+546875 (a5^2)) (x^4)+(-30 (a1^3) a2+30 a1 (a2^2)+70 (a1^2) a3-50 a2 a3-250 a1 a4+1250 a5) (x^5)+(x^6);
pd=(((-15625 (a1^18) (a2^6)+468750 (a1^16) (a2^7)-5859375 (a1^14) (a2^8)+39062500 (a1^12) (a2^9)-146484375 (a1^10) (a2^10)+292968750 (a1^8) (a2^11)-244140625 (a1^6) (a2^12)+125000 (a1^19) (a2^4) a3-3906250 (a1^17) (a2^5) a3+50000000 (a1^15) (a2^6) a3-332031250 (a1^13) (a2^7) a3+1171875000 (a1^11) (a2^8) a3-1855468750 (a1^9) (a2^9) a3+2441406250 (a1^5) (a2^11) a3-250000 (a1^20) (a2^2) (a3^2)+8500000 (a1^18) (a2^3) (a3^2)-111484375 (a1^16) (a2^4) (a3^2)+683593750 (a1^14) (a2^5) (a3^2)-1594140625 (a1^12) (a2^6) (a3^2)-2492187500 (a1^10) (a2^7) (a3^2)+18740234375 (a1^8) (a2^8) (a3^2)-22753906250 (a1^6) (a2^9) (a3^2)-7568359375 (a1^4) (a2^10) (a3^2)-1500000 (a1^19) a2 (a3^3)+21250000 (a1^17) (a2^2) (a3^3)+97812500 (a1^15) (a2^3) (a3^3)-3362500000 (a1^13) (a2^4) (a3^3)+22257812500 (a1^11) (a2^5) (a3^3)-52562500000 (a1^9) (a2^6) (a3^3)-2382812500 (a1^7) (a2^7) (a3^3)+124218750000 (a1^5) (a2^8) (a3^3)+2929687500 (a1^3) (a2^9) (a3^3)-2250000 (a1^18) (a3^4)-47500000 (a1^16) a2 (a3^4)+1066015625 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p20=(a3^8)+a1 (a2^2) (a3^5) a4-13 a2 (a3^6) a4+(a2^5) (a3^2) (a4^2)+2 (a1^3) (a2^2) (a3^3) (a4^2)-15 a1 (a2^3) (a3^3) (a4^2)-2 (a1^4) (a3^4) (a4^2)+5 (a1^2) a2 (a3^4) (a4^2)+65 (a2^2) (a3^4) (a4^2)+5 a1 (a3^5) (a4^2)+(a1^2) (a2^5) (a4^3)-4 (a2^6) (a4^3)+(a1^5) (a2^2) a3 (a4^3)-15 (a1^3) (a2^3) a3 (a4^3)+44 a1 (a2^4) a3 (a4^3)+5 (a1^4) a2 (a3^2) (a4^3)+19 (a1^2) (a2^2) (a3^2) (a4^3)-128 (a2^3) (a3^2) (a4^3)+6 (a1^3) (a3^3) (a4^3)-118 a1 a2 (a3^3) (a4^3)+17 (a3^4) (a4^3)+(a1^8) (a4^4)-13 (a1^6) a2 (a4^4)+65 (a1^4) (a2^2) (a4^4)-128 (a1^2) (a2^3) (a4^4)+48 (a2^4) (a4^4)+5 (a1^5) a3 (a4^4)-118 (a1^3) a2 a3 (a4^4)+384 a1 (a2^2) a3 (a4^4)+82 (a1^2) (a3^2) (a4^4)-16 a2 (a3^2) (a4^4)+17 (a1^4) (a4^5)-16 (a1^2) a2 (a4^5)-192 (a2^2) (a4^5)-192 a1 a3 (a4^5)+256 (a4^6)+(a1^2) (a2^4) (a3^3) a5-4 (a2^5) (a3^3) a5-6 (a1^3) (a2^2) (a3^4) a5+25 a1 (a2^3) (a3^4) a5+8 (a1^4) (a3^5) a5-36 (a1^2) a2 (a3^5) a5-12 (a2^2) (a3^5) a5+38 a1 (a3^6) a5+(a1^4) (a2^4) a3 a4 a5-9 (a1^2) (a2^5) a3 a4 a5+18 (a2^6) a3 a4 a5-4 (a1^5) (a2^2) (a3^2) a4 a5+43 (a1^3) (a2^3) (a3^2) a4 a5-95 a1 (a2^4) (a3^2) a4 a5-28 (a1^4) a2 (a3^3) a4 a5+93 (a1^2) (a2^2) (a3^3) a4 a5+12 (a2^3) (a3^3) a4 a5-12 (a1^3) (a3^4) a4 a5-29 a1 a2 (a3^4) a4 a5-124 (a3^5) a4 a5+2 (a1^7) (a2^2) (a4^2) a5-22 (a1^5) (a2^3) (a4^2) a5+76 (a1^3) (a2^4) (a4^2) a5-78 a1 (a2^5) (a4^2) a5-8 (a1^8) a3 (a4^2) a5+88 (a1^6) a2 a3 (a4^2) a5-287 (a1^4) (a2^2) a3 (a4^2) a5+168 (a1^2) (a2^3) a3 (a4^2) a5+196 (a2^4) a3 (a4^2) a5-54 (a1^5) (a3^2) (a4^2) a5+447 (a1^3) a2 (a3^2) (a4^2) a5-528 a1 (a2^2) (a3^2) (a4^2) a5-172 (a1^2) (a3^3) (a4^2) a5+590 a2 (a3^3) (a4^2) a5+12 (a1^7) (a4^3) a5-148 (a1^5) a2 (a4^3) a5+514 (a1^3) (a2^2) (a4^3) a5-384 a1 (a2^3) (a4^3) a5+180 (a1^4) a3 (a4^3) a5-1174 (a1^2) a2 a3 (a4^3) a5-160 (a2^2) a3 (a4^3) a5+780 a1 (a3^2) (a4^3) a5-276 (a1^3) (a4^4) a5+1760 a1 a2 (a4^4) a5-1600 a3 (a4^4) a5+(a1^6) (a2^4) (a5^2)-9 (a1^4) (a2^5) (a5^2)+27 (a1^2) (a2^6) (a5^2)-27 (a2^7) (a5^2)-8 (a1^7) (a2^2) a3 (a5^2)+72 (a1^5) (a2^3) a3 (a5^2)-210 (a1^3) (a2^4) a3 (a5^2)+198 a1 (a2^5) a3 (a5^2)+16 (a1^8) (a3^2) (a5^2)-144 (a1^6) a2 (a3^2) (a5^2)+370 (a1^4) (a2^2) (a3^2) (a5^2)-185 (a1^2) (a2^3) (a3^2) (a5^2)-150 (a2^4) (a3^2) (a5^2)+152 (a1^5) (a3^3) (a5^2)-700 (a1^3) a2 (a3^3) (a5^2)+525 a1 (a2^2) (a3^3) (a5^2)+375 (a1^2) (a3^4) (a5^2)-125 a2 (a3^4) (a5^2)+12 (a1^6) (a2^2) a4 (a5^2)-76 (a1^4) (a2^3) a4 (a5^2)+141 (a1^2) (a2^4) a4 (a5^2)-99 (a2^5) a4 (a5^2)-48 (a1^7) a3 a4 (a5^2)+304 (a1^5) a2 a3 a4 (a5^2)-397 (a1^3) (a2^2) a3 a4 (a5^2)+15 a1 (a2^3) a3 a4 (a5^2)-676 (a1^4) (a3^2) a4 (a5^2)+1995 (a1^2) a2 (a3^2) a4 (a5^2)-725 (a2^2) (a3^2) a4 (a5^2)-2050 a1 (a3^3) a4 (a5^2)+86 (a1^6) (a4^2) (a5^2)-294 (a1^4) a2 (a4^2) (a5^2)-610 (a1^2) (a2^2) (a4^2) (a5^2)+1200 (a2^3) (a4^2) (a5^2)+1470 (a1^3) a3 (a4^2) (a5^2)-850 a1 a2 a3 (a4^2) (a5^2)+3250 (a3^2) (a4^2) (a5^2)-1450 (a1^2) (a4^3) (a5^2)-2000 a2 (a4^3) (a5^2)-14 (a1^5) (a2^2) (a5^3)+50 (a1^3) (a2^3) (a5^3)+56 (a1^6) a3 (a5^3)-200 (a1^4) a2 a3 (a5^3)-125 (a1^2) (a2^2) a3 (a5^3)+250 (a1^3) (a3^2) (a5^3)+625 a1 a2 (a3^2) (a5^3)-468 (a1^5) a4 (a5^3)+2300 (a1^3) a2 a4 (a5^3)-1750 a1 (a2^2) a4 (a5^3)-3500 (a1^2) a3 a4 (a5^3)-1250 a2 a3 a4 (a5^3)+7500 a1 (a4^2) (a5^3)+625 (a1^4) (a5^4)-3125 (a1^2) a2 (a5^4)+3125 (a2^2) (a5^4)+3125 a1 a3 (a5^4)-9375 a4 (a5^4)+(-2 a2 (a3^6)-a1 (a2^3) (a3^3) a4-2 (a1^2) a2 (a3^4) a4+19 (a2^2) (a3^4) a4+6 a1 (a3^5) a4-(a1^3) (a2^3) a3 (a4^2)+3 a1 (a2^4) a3 (a4^2)-2 (a1^4) a2 (a3^2) (a4^2)+28 (a1^2) (a2^2) (a3^2) (a4^2)-51 (a2^3) (a3^2) (a4^2)+4 (a1^3) (a3^3) (a4^2)-89 a1 a2 (a3^3) (a4^2)+3 (a3^4) (a4^2)-2 (a1^6) a2 (a4^3)+19 (a1^4) (a2^2) (a4^3)-51 (a1^2) (a2^3) (a4^3)+32 (a2^4) (a4^3)+6 (a1^5) a3 (a4^3)-89 (a1^3) a2 a3 (a4^3)+212 a1 (a2^2) a3 (a4^3)+102 (a1^2) (a3^2) (a4^3)+76 a2 (a3^2) (a4^3)+3 (a1^4) (a4^4)+76 (a1^2) a2 (a4^4)-256 (a2^2) (a4^4)-384 a1 a3 (a4^4)+512 (a4^5)-2 (a1^3) (a2^3) (a3^2) a5+9 a1 (a2^4) (a3^2) a5+8 (a1^4) a2 (a3^3) a5-32 (a1^2) (a2^2) (a3^3) a5-31 (a2^3) (a3^3) a5-8 (a1^3) (a3^4) a5+112 a1 a2 (a3^4) a5-58 (a3^5) a5-2 (a1^5) (a2^3) a4 a5+15 (a1^3) (a2^4) a4 a5-27 a1 (a2^5) a4 a5+8 (a1^6) a2 a3 a4 a5-50 (a1^4) (a2^2) a3 a4 a5+33 (a1^2) (a2^3) a3 a4 a5+117 (a2^4) a3 a4 a5-24 (a1^5) (a3^2) a4 a5+194 (a1^3) a2 (a3^2) a4 a5-231 a1 (a2^2) (a3^2) a4 a5-234 (a1^2) (a3^3) a4 a5+105 a2 (a3^3) a4 a5+8 (a1^7) (a4^2) a5-94 (a1^5) a2 (a4^2) a5+347 (a1^3) (a2^2) (a4^2) a5-366 a1 (a2^3) (a4^2) a5+166 (a1^4) a3 (a4^2) a5-901 (a1^2) a2 a3 (a4^2) a5+260 (a2^2) a3 (a4^2) a5+1220 a1 (a3^2) (a4^2) a5-364 (a1^3) (a4^3) a5+1640 a1 a2 (a4^3) a5-2400 a3 (a4^3) a5+8 (a1^6) (a2^2) (a5^2)-66 (a1^4) (a2^3) (a5^2)+162 (a1^2) (a2^4) (a5^2)-108 (a2^5) (a5^2)-32 (a1^7) a3 (a5^2)+264 (a1^5) a2 a3 (a5^2)-574 (a1^3) (a2^2) a3 (a5^2)+180 a1 (a2^3) a3 (a5^2)-352 (a1^4) (a3^2) (a5^2)+1340 (a1^2) a2 (a3^2) (a5^2)-325 (a2^2) (a3^2) (a5^2)-850 a1 (a3^3) (a5^2)+48 (a1^6) a4 (a5^2)-318 (a1^4) a2 a4 (a5^2)+305 (a1^2) (a2^2) a4 (a5^2)+525 (a2^3) a4 (a5^2)+1090 (a1^3) a3 a4 (a5^2)-3075 a1 a2 a3 a4 (a5^2)+2750 (a3^2) a4 (a5^2)-550 (a1^2) (a4^2) (a5^2)-500 a2 (a4^2) (a5^2)-56 (a1^5) (a5^3)+350 (a1^3) a2 (a5^3)-375 a1 (a2^2) (a5^3)-750 (a1^2) a3 (a5^3)+625 a2 a3 (a5^3)+2500 a1 a4 (a5^3)-3125 (a5^4)) x+((a2^2) (a3^4)+2 a1 (a3^5)+3 (a1^2) (a2^2) (a3^2) a4-6 (a2^3) (a3^2) a4-19 a1 a2 (a3^3) a4-8 (a3^4) a4+(a1^4) (a2^2) (a4^2)-6 (a1^2) (a2^3) (a4^2)+9 (a2^4) (a4^2)+2 (a1^5) a3 (a4^2)-19 (a1^3) a2 a3 (a4^2)+30 a1 (a2^2) a3 (a4^2)+41 (a1^2) (a3^2) (a4^2)+76 a2 (a3^2) (a4^2)-8 (a1^4) (a4^3)+76 (a1^2) a2 (a4^3)-136 (a2^2) (a4^3)-264 a1 a3 (a4^3)+400 (a4^4)+2 (a1^4) (a2^2) a3 a5-9 (a1^2) (a2^3) a3 a5-8 (a1^5) (a3^2) a5+28 (a1^3) a2 (a3^2) a5+45 a1 (a2^2) (a3^2) a5-70 (a1^2) (a3^3) a5-50 a2 (a3^3) a5-8 (a1^5) a2 a4 a5+46 (a1^3) (a2^2) a4 a5-54 a1 (a2^3) a4 a5+68 (a1^4) a3 a4 a5-354 (a1^2) a2 a3 a4 a5+90 (a2^2) a3 a4 a5+630 a1 (a3^2) a4 a5-144 (a1^3) (a4^2) a5+640 a1 a2 (a4^2) a5-1400 a3 (a4^2) a5+16 (a1^6) (a5^2)-120 (a1^4) a2 (a5^2)+225 (a1^2) (a2^2) (a5^2)+250 (a1^3) a3 (a5^2)-875 a1 a2 a3 (a5^2)+625 (a3^2) (a5^2)-200 (a1^2) a4 (a5^2)+500 a2 a4 (a5^2)) (x^2)+(-2 a1 a2 (a3^3)-2 (a3^4)-2 (a1^3) a2 a3 a4+5 a1 (a2^2) a3 a4+6 (a1^2) (a3^2) a4+21 a2 (a3^2) a4-2 (a1^4) (a4^2)+21 (a1^2) a2 (a4^2)-40 (a2^2) (a4^2)-80 a1 a3 (a4^2)+160 (a4^3)-2 (a1^3) (a2^2) a5+9 a1 (a2^3) a5+16 (a1^4) a3 a5-66 (a1^2) a2 a3 a5-15 (a2^2) a3 a5+120 a1 (a3^2) a5-44 (a1^3) a4 a5+190 a1 a2 a4 a5-400 a3 a4 a5-50 (a1^2) (a5^2)+125 a2 (a5^2)) (x^3)+((a1^2) (a3^2)+2 a2 (a3^2)+2 (a1^2) a2 a4-6 (a2^2) a4-14 a1 a3 a4+40 (a4^2)-8 (a1^3) a5+30 a1 a2 a5-50 a3 a5) (x^4)+(-2 a1 a3+8 a4) (x^5)+(x^6);

(*gの有理数解で重複度がmultiplicityのいずれかとなるものを取り出す*)
getTheta[g_,multiplicity_List]:=Module[{ichiji},ichiji = Select[FactorList[g], Exponent[#[[1]], x] == 1&&MemberQ[multiplicity,#[[2]]]& ];
If[ichiji!={},x/.Solve[ichiji[[1,1]]==0,x][[1]],{}]];

calculateQuintic[f_]:=Module[{(*alist,fb,pc,pd,θa,θb,θc,θd,wa1,wa2,seki1,seki2,perms,ζ,Ra,Rb,Rc,Rd,ΔRa,ΔRb,ΔRc,ΔRd,ΔRabcd,$R1234,r1234,$r1234Table,$R1234Table,pos,$r1234,power,$sol*)},
(*Part1 (R1,R2,R3,R4)の4つの候補Ra,Rb,Rc,Rdを求める*)
alist=(CoefficientList[f,x]//Drop[#,-1]&//Reverse){-1,1,-1,1,-1};(*{a1,a2,a3,a4,a5}*)
fb:=pb/. AssociationThread[{a1,a2,a3,a4,a5}->alist];
fc:=pc/. AssociationThread[{a1,a2,a3,a4,a5}->alist];
fd:=pd/. AssociationThread[{a1,a2,a3,a4,a5}->alist];
θa=2 ((alist[[1]])^2)-5 alist[[2]];(*θa=r1r4+r2r3*)
θb=getTheta[fb,{1,6}];(*(r1 r4-r2 r3)^2*)
θc=getTheta[fc,{1,6}];(*θc=l1 l4+l2 l3*)
θd=getTheta[fd,{1,6}];(*(Subscript[θ, d])=(l1+l4-l2-((l3))^2)*)
{wa1,wa2}=(1/2){(4 ((alist[[1]])^5)-5θc+(Sqrt[θd])(Sqrt[5])),(4 ((alist[[1]])^5)-5θc-(Sqrt[θd])(Sqrt[5]))};(*{R1+R4,R2+R3}={wa1,wa2}または{wa2,wa1}*)
{seki1,seki2}=(1/2{θa+Sqrt[θb],θa-Sqrt[θb]})^5;(*{R1R4,R2R3}={seki1,seki2}または{seki2,seki1}*)
(*case a*)Ra=Flatten[{t/.Solve[t^2-wa1 t +seki1==0],t/.Solve[t^2-wa2 t+seki2==0,t]},1][[{1,3,4,2}]]//Simplify;
(*case b*)Rb=Ra[[{1,3,2,4}]];
(*case c*)Rc=Flatten[{t/.Solve[t^2-wa1 t +seki2==0],t/.Solve[t^2-wa2 t+seki1==0,t]},1][[{1,3,4,2}]]//Simplify;
(*case d*)Rd=Rc[[{1,3,2,4}]];
(*Part2 Ra,Rb,Rc,Rdのうち正しいものを 数値解から求める.数値解は先頭に$を付ける*)
perms={{1,2,3,4,5},{1,3,5,2,4},{1,5,4,3,2},{1,4,2,5,3},{2,1,5,3,4},{2,5,4,1,3},{2,4,3,5,1},{2,3,1,4,5},{3,2,5,1,4},{3,5,4,2,1},{3,4,1,5,2},{3,1,2,4,5},
{2,3,5,1,4},{2,5,4,3,1},{2,4,1,5,3},{2,1,3,4,5},{3,1,5,2,4},{3,5,4,1,2},{3,4,2,5,1},{3,2,1,4,5},{1,2,5,3,4},{1,5,4,2,3},{1,4,3,5,2},{1,3,2,4,5}};(*解の置換*)
$sol=x/.NSolve[f==0,x];(*数値解*)
ζ=Cos[2Pi/5]+I Sin[2Pi/5];(*最後のr1234の値はζの選択に依存する事に注意*)
$r1234Table=Table[{{1,ζ,ζ^2,ζ^3,ζ^4},{1,ζ^2,ζ^4,ζ,ζ^3},{1,ζ^3,ζ,ζ^4,ζ^2},{1,ζ^4,ζ^3,ζ^2,ζ}}.$sol[[perms[[i]]]],{i,1,24}];
$R1234Table=$r1234Table^5//Quiet;(*$solsに対する{R1,R2,R3,R4}*)
ΔRa=Norm/@($R1234Table-Table[N[Ra],24]);(*{R1,R2,R3,R4}=Raのとき，$R1234Talbeとの差*)
ΔRb=Norm/@($R1234Table-Table[N[Rb],24]);
ΔRc=Norm/@($R1234Table-Table[N[Rc],24]);
ΔRd=Norm/@($R1234Table-Table[N[Rd],24]);
ΔRabcd={ΔRa,ΔRb,ΔRc,ΔRd};
pos=Position[ΔRabcd,Min[ΔRabcd]][[1]];(*厳密解{R1,R2,R3,R4}と$R1234Talbeとの差が最小のposition*)
R1234={Ra,Rb,Rc,Rd}[[pos[[1]]]];(*{R1,R2,R3,R4}の厳密解*)
r1234=N@R1234^(1/5);(*{R1,R2,R3,R4}の偏角が最小の5乗根*)
$r1234=$r1234Table[[pos[[2]]]];(*正しい数値解{r1,r2,r3,r4}*)
(*Part3 正しい(R1,R2,R3,R4)の組から，(R1,R2,R3,R4)^1/5 に掛けるζの指数を求める*)
power=Mod[Round[(Arg[$r1234]-Arg[r1234])/Arg[ζ]],5];
Clear[ζ];(*解をζを使って表すために「記号(偏角は2/5piで固定)」に戻す*)
{r1,r2,r3,r4}=R1234^(1/5)* ζ^power;(*{r1,r2,r3,r4}の厳密解*)
{x1,x2,x3,x4,x5}=1/5{{1,1,1,1,1},{1,ζ^4,ζ^3,ζ^2,ζ},{1,ζ^3,ζ,ζ^4,ζ^2},{1,ζ^2,ζ^4,ζ,ζ^3},{1,ζ,ζ^2,ζ^3,ζ^4}}.{alist[[1]],r1,r2,r3,r4}//PolynomialMod[#,ζ^5-1]&//Simplify//Sort;
Style[{TraditionalForm[x1],TraditionalForm[x2],TraditionalForm[x3],TraditionalForm[x4],TraditionalForm[x5]},Small]
];

solveQuintic[f_]:=Module[{alist,f20,θ20},alist=(CoefficientList[f,x]//Drop[#,-1]&//Reverse){-1,1,-1,1,-1};f20:=p20/. AssociationThread[{a1,a2,a3,a4,a5}->alist];
θ20=getTheta[f20,{1}];
If[IrreduciblePolynomialQ[f]&&θ20=!={},calculateQuintic[f], "可解で既約な5次方程式でありません(Not an irreducible solvable quintic)"
]]
)]