# read "Homomorphisms.txt";
# read "G3module.txt";
# read "hypergeometricsols.txt";

# We try to find pullback function of factor (second order) of reducible G3 (a+2b=1) 
# Let b=1/5, let c be from 1 to 44, then every G3(-2b+1,b,x,x+c) gives factorization of [1,2]
# then use Erdal's algorithm to find 2f1-type solution of the second order factor.
# Now we have 44 rational pullback functions. (all of them have degree 6)
# Then set several unknowns to solve for pullback function using these data

# M1:=subs(a=-2*b+1,M)
# M2:=changeVmoduleF1(M1,x,x+c)
# L:=CycVec(M2,x)[2]
# subs(b=1/5,%)
# DFactor(%)
# DFactorLCLM(subs(c=0,%%))
# L2:=collect(%%[2],Dx,factor)
# for i to 44 do 
#         HP:=indets(find_2f1(subs(c=i,L2))[1],function)[1];
#         if op(1,HP)<>[7/30,11/15] or op(2,HP)<>[13/10] then error "unexpected"
#                else FC[i]:=op(3,HP);
#         fi;
# od:

# dataSet:={seq([x+i,FC[i]],i=1..44)}

dataSet:=
{[x+1, 27*(x+1)^2*(27*x^4+54*x^3+9*x^2-26*x-5)/(-11+18*x+54*x^2+27*x^3)^2], [x
+2, 27*(x+2)^2*(27*x^4+108*x^3+90*x^2-44*x-9)/(-20+99*x+108*x^2+27*x^3)^2], [x
+3, 27*(x+3)^2*(27*x^4+162*x^3+225*x^2-62*x-13)/(-29+234*x+162*x^2+27*x^3)^2],
[x+4, 27*(x+4)^2*(27*x^4+216*x^3+414*x^2-80*x-17)/(-38+423*x+216*x^2+27*x^3)^2
], [x+5, 27*(x+5)^2*(27*x^4+270*x^3+657*x^2-98*x-21)/(-47+666*x+270*x^2+27*x^3
)^2], [x+6, 27*(x+6)^2*(27*x^4+324*x^3+954*x^2-116*x-25)/(-56+963*x+324*x^2+27
*x^3)^2], [x+7, 27*(x+7)^2*(27*x^4+378*x^3+1305*x^2-134*x-29)/(-65+1314*x+378*
x^2+27*x^3)^2], [x+8, 27*(x+8)^2*(27*x^4+432*x^3+1710*x^2-152*x-33)/(-74+1719*
x+432*x^2+27*x^3)^2], [x+9, 27*(x+9)^2*(27*x^4+486*x^3+2169*x^2-170*x-37)/(-83
+2178*x+486*x^2+27*x^3)^2], [x+10, 27*(x+10)^2*(27*x^4+540*x^3+2682*x^2-188*x-\
41)/(-92+2691*x+540*x^2+27*x^3)^2], [x+11, 27*(x+11)^2*(27*x^4+594*x^3+3249*x^
2-206*x-45)/(-101+3258*x+594*x^2+27*x^3)^2], [x+12, 27*(x+12)^2*(27*x^4+648*x^
3+3870*x^2-224*x-49)/(-110+3879*x+648*x^2+27*x^3)^2], [x+13, 27*(x+13)^2*(27*x
^4+702*x^3+4545*x^2-242*x-53)/(-119+4554*x+702*x^2+27*x^3)^2], [x+14, 27*(x+14
)^2*(27*x^4+756*x^3+5274*x^2-260*x-57)/(-128+5283*x+756*x^2+27*x^3)^2], [x+15,
27*(x+15)^2*(27*x^4+810*x^3+6057*x^2-278*x-61)/(-137+6066*x+810*x^2+27*x^3)^2]
, [x+16, 27*(x+16)^2*(27*x^4+864*x^3+6894*x^2-296*x-65)/(-146+6903*x+864*x^2+
27*x^3)^2], [x+17, 27*(x+17)^2*(27*x^4+918*x^3+7785*x^2-314*x-69)/(-155+7794*x
+918*x^2+27*x^3)^2], [x+18, 27*(x+18)^2*(27*x^4+972*x^3+8730*x^2-332*x-73)/(-\
164+8739*x+972*x^2+27*x^3)^2], [x+19, 27*(x+19)^2*(27*x^4+1026*x^3+9729*x^2-\
350*x-77)/(-173+9738*x+1026*x^2+27*x^3)^2], [x+20, 27*(x+20)^2*(27*x^4+1080*x^
3+10782*x^2-368*x-81)/(-182+10791*x+1080*x^2+27*x^3)^2], [x+21, 27*(x+21)^2*(
27*x^4+1134*x^3+11889*x^2-386*x-85)/(-191+11898*x+1134*x^2+27*x^3)^2], [x+22,
27*(x+22)^2*(27*x^4+1188*x^3+13050*x^2-404*x-89)/(-200+13059*x+1188*x^2+27*x^3
)^2], [x+23, 27*(x+23)^2*(27*x^4+1242*x^3+14265*x^2-422*x-93)/(-209+14274*x+
1242*x^2+27*x^3)^2], [x+24, 27*(x+24)^2*(27*x^4+1296*x^3+15534*x^2-440*x-97)/(
-218+15543*x+1296*x^2+27*x^3)^2], [x+25, 27*(x+25)^2*(27*x^4+1350*x^3+16857*x^
2-458*x-101)/(-227+16866*x+1350*x^2+27*x^3)^2], [x+26, 27*(x+26)^2*(27*x^4+
1404*x^3+18234*x^2-476*x-105)/(-236+18243*x+1404*x^2+27*x^3)^2], [x+27, 27*(x+
27)^2*(27*x^4+1458*x^3+19665*x^2-494*x-109)/(-245+19674*x+1458*x^2+27*x^3)^2],
[x+28, 27*(x+28)^2*(27*x^4+1512*x^3+21150*x^2-512*x-113)/(-254+21159*x+1512*x^
2+27*x^3)^2], [x+29, 27*(x+29)^2*(27*x^4+1566*x^3+22689*x^2-530*x-117)/(-263+
22698*x+1566*x^2+27*x^3)^2], [x+30, 27*(x+30)^2*(27*x^4+1620*x^3+24282*x^2-548
*x-121)/(-272+24291*x+1620*x^2+27*x^3)^2], [x+31, 27*(x+31)^2*(27*x^4+1674*x^3
+25929*x^2-566*x-125)/(-281+25938*x+1674*x^2+27*x^3)^2], [x+32, 27*(x+32)^2*(
27*x^4+1728*x^3+27630*x^2-584*x-129)/(-290+27639*x+1728*x^2+27*x^3)^2], [x+33,
27*(x+33)^2*(27*x^4+1782*x^3+29385*x^2-602*x-133)/(-299+29394*x+1782*x^2+27*x^
3)^2], [x+34, 27*(x+34)^2*(27*x^4+1836*x^3+31194*x^2-620*x-137)/(-308+31203*x+
1836*x^2+27*x^3)^2], [x+35, 27*(x+35)^2*(27*x^4+1890*x^3+33057*x^2-638*x-141)/
(-317+33066*x+1890*x^2+27*x^3)^2], [x+36, 27*(x+36)^2*(27*x^4+1944*x^3+34974*x
^2-656*x-145)/(-326+34983*x+1944*x^2+27*x^3)^2], [x+37, 27*(x+37)^2*(27*x^4+
1998*x^3+36945*x^2-674*x-149)/(-335+36954*x+1998*x^2+27*x^3)^2], [x+38, 27*(x+
38)^2*(27*x^4+2052*x^3+38970*x^2-692*x-153)/(-344+38979*x+2052*x^2+27*x^3)^2],
[x+39, 27*(x+39)^2*(27*x^4+2106*x^3+41049*x^2-710*x-157)/(-353+41058*x+2106*x^
2+27*x^3)^2], [x+40, 27*(x+40)^2*(27*x^4+2160*x^3+43182*x^2-728*x-161)/(-362+
43191*x+2160*x^2+27*x^3)^2], [x+41, 27*(x+41)^2*(27*x^4+2214*x^3+45369*x^2-746
*x-165)/(-371+45378*x+2214*x^2+27*x^3)^2], [x+42, 27*(x+42)^2*(27*x^4+2268*x^3
+47610*x^2-764*x-169)/(-380+47619*x+2268*x^2+27*x^3)^2], [x+43, 27*(x+43)^2*(
27*x^4+2322*x^3+49905*x^2-782*x-173)/(-389+49914*x+2322*x^2+27*x^3)^2], [x+44,
27*(x+44)^2*(27*x^4+2376*x^3+52254*x^2-800*x-177)/(-398+52263*x+2376*x^2+27*x^
3)^2]}:



pullback:=proc(A)
       local a, c0,d0,r,s,NU,DN, SS; SS:={};
       NU:=add(add(c[i,j]*x^i*y^j,i=0..4),j=0..4); 
       DN:=add(add(d[i,j]*x^i*y^j,i=0..6),j=0..6);
       for a in A do       
             SS:=SS union {collect(subs(y=a[1],y^2*NU-a[2]*DN),x,factor)};
       od;
       s:=solve(SS,{seq(seq(c[i,j],i=0..4),j=0..4),seq(seq(d[i,j],i=0..6),j=0..6)});
       if s<>NULL then return simplify(subs(s,y^2*NU/DN)); fi;
end:

# pullback(dataSet);

# (27*(-1-4*y-4*x-18*y*x+27*y^2*x^2))*y^2/(4+36*y+81*y^2-108*y^2*x-486*y^3*x+729*x^2*y^4)