alias( om = RootOf(Z_^2 + Z_ + 1 = 0) );

X2o := 1/2*T*(-2047*u^3+6174*T-4116*u+6174)*(506689*o*u^3-203616*o*u^2+59297*u^
3+301644*o*u-33768*u^2-166698*T-333396*o+127008*u+166698)*(506689*o*u^3-203616*
o*u^2+447392*u^3+301644*o*u-169848*u^2+166698*T-333396*o+174636*u-500094)*(42*T
-37*u-21)*(22378*o*u^3+8064*o*u^2-92149*u^3+4032*u^2+83349*T-83349*o-83349)*(
22378*o*u^3+8064*o*u^2+114527*u^3+4032*u^2-83349*T-83349*o)/(-2047*u^3+6174*T+
2058*u+6174)/(630961*o*u^3-203616*o*u^2+260699*u^3+301644*o*u-33768*u^2-166698*
T-333396*o-39690*u+166698)/(630961*o*u^3-203616*o*u^2+370262*u^3+301644*o*u-\
169848*u^2+166698*T-333396*o+341334*u-500094)/(42*T-16*u-21)/(22378*o*u^3+8064*
o*u^2-92149*u^3+4032*u^2+83349*T-83349*o-83349*u-83349)/(22378*o*u^3+8064*o*u^2
+114527*u^3+4032*u^2-83349*T-83349*o+83349*u):
Y2o := -1/2*(T-1)*(-2047*u^3+6174*T-4116*u-12348)*(670481*o*u^3+203616*o*u^2+
572140*u^3+301644*o*u+169848*u^2+166698*T+333396*o+174636*u+333396)*(670481*o*u
^3+203616*o*u^2+98341*u^3+301644*o*u+33768*u^2-166698*T+333396*o+127008*u)*(42*
T-37*u-21)*(22378*o*u^3+8064*o*u^2-92149*u^3+4032*u^2+83349*T-83349*o-83349)*(
22378*o*u^3+8064*o*u^2+114527*u^3+4032*u^2-83349*T-83349*o)/(-2047*u^3+6174*T+
2058*u-12348)/(670481*o*u^3+203616*o*u^2+572140*u^3+301644*o*u+169848*u^2+
166698*T+333396*o+341334*u+333396)/(670481*o*u^3+203616*o*u^2+98341*u^3+301644*
o*u+33768*u^2-166698*T+333396*o-39690*u)/(42*T-16*u-21)/(22378*o*u^3+8064*o*u^2
-92149*u^3+4032*u^2+83349*T-83349*o-83349*u-83349)/(22378*o*u^3+8064*o*u^2+
114527*u^3+4032*u^2-83349*T-83349*o+83349*u):
T2o := [[-1/9*u^2+42898/83349*u^3-8335/27783*u^4-1, 1/9*u^2+42898/83349*u^3+256
/3087*u^4+2, -29/1701*u^3+820/9261*u^4+3/14*u+1/2], [(132887/500094*o-43777/
83349)*u^4+(204725/55566*o+43280/83349)*u^3+(-1427/1323*o-730/1323)*u^2+(8/7*o+
2/21)*u-2*o+1, (-96605/500094*o+598379/500094)*u^4+(-91957/23814*o-569515/
166698)*u^3+(-1427/1323*o-697/1323)*u^2+(-8/7*o-22/21)*u-2-2*o, (3919/11907*o+
104677/500094)*u^4+(10280/9261-22378/83349*o)*u^3+u^2*(-128/1323*o-64/1323)+2/3
*u+o+1], [(-132887/500094*o-56507/71442)*u^4+(-204725/55566*o-527615/166698)*u^
3+(1427/1323*o+697/1323)*u^2+(-8/7*o-22/21)*u+2*o+3, (96605/500094*o+347492/
250047)*u^4+(91957/23814*o+12364/27783)*u^3+(1427/1323*o+730/1323)*u^2+(8/7*o+2
/21)*u+2*o, (-3919/11907*o-59921/500094)*u^4+(16414/11907+22378/83349*o)*u^3+u^
2*(128/1323*o+64/1323)+2/3*u-o], [-1+(-2205383/500094*o+131020/250047)*u^4+(
14003/7938*o+4448/27783)*u^3+(-o-1/3)*u^2+2/3*u*o, (1737761/250047*o-1230199/
500094)*u^4+(-88681/11907*o-515167/166698)*u^3+(1513/1323*o-256/1323)*u^2+(-22/
7*o-12/7)*u-2-2*o, (-1223729/500094*o+1175513/500094)*u^4+(156400/83349*o+12776
/11907)*u^3+(-1048/1323*o-524/1323)*u^2+(4/3+2/3*o)*u-o], [(138375245/600362847
*o+155960500/600362847)*u^4+(13237793/4084101*o+3069079/4084101)*u^3+(-10784/
9261*o-4177/9261)*u^2+(40/21*o+8/21)*u-2*o+1, (-27111484/600362847*o+116256109/
600362847)*u^4+(16270795/4084101*o+1855937/4084101)*u^3+(10784/9261*o+4177/9261
)*u^2+(40/21*o+8/21)*u+2*o, 1/2+(-145933/12252303*o-126173/12252303)*u^4+u^3*(-\
48446/1361367*o-130561/1361367)+u*(1/2+2/21*o)], [(-158128/35721*o+2473853/
500094)*u^4+(-446027/83349*o-173303/166698)*u^3+(-1513/1323*o+256/1323)*u^2+(-\
22/7*o-12/7)*u+2*o+3, 2+(1873045/500094*o-290525/250047)*u^4+(14003/7938*o+4448
/27783)*u^3+(o+1/3)*u^2+2/3*u*o, (782833/500094*o-1168849/500094)*u^4+(111644/
83349*o+67054/83349)*u^3+(1048/1323*o+524/1323)*u^2+(4/3+2/3*o)*u+o+1], [(
158128/35721*o+4687645/500094)*u^4+(446027/83349*o+718751/166698)*u^3+(1513/
1323*o+1769/1323)*u^2+(22/7*o+10/7)*u-2*o+1, 2+(-1873045/500094*o-350585/71442)
*u^4+(-14003/7938*o-89125/55566)*u^3+(-o-2/3)*u^2+u*(-2/3-2/3*o), (-782833/
500094*o-975841/250047)*u^4+(-111644/83349*o-130/243)*u^3+(-1048/1323*o-524/
1323)*u^2+(-2/3*o+2/3)*u-o], [-1+(2205383/500094*o+352489/71442)*u^4+(-14003/
7938*o-89125/55566)*u^3+(o+2/3)*u^2+u*(-2/3-2/3*o), (-1737761/250047*o-4705721/
500094)*u^4+(88681/11907*o+726367/166698)*u^3+(-1513/1323*o-1769/1323)*u^2+(22/
7*o+10/7)*u+2*o, (1223729/500094*o+1199621/250047)*u^4+(-156400/83349*o-66968/
83349)*u^3+(1048/1323*o+524/1323)*u^2+(-2/3*o+2/3)*u+o+1], [(-138375245/
600362847*o+17585255/600362847)*u^4+(-13237793/4084101*o-10168714/4084101)*u^3+
(10784/9261*o+6607/9261)*u^2+(-40/21*o-32/21)*u+2*o+3, (27111484/600362847*o+
143367593/600362847)*u^4+(-16270795/4084101*o-14414858/4084101)*u^3+(-10784/
9261*o-6607/9261)*u^2+(-40/21*o-32/21)*u-2-2*o, 1/2+(145933/12252303*o+19760/
12252303)*u^4+u^3*(48446/1361367*o-82115/1361367)+u*(-2/21*o+17/42)]]:

X2 := subs(o=om,X2o); Y2 := subs(o=om,Y2o); T2 := subs(o=om,T2o);

#
#  Check that [X2,Y2]  is a solution up to $O(u^4)$.
#

for k from 1 to 9 do
  for j from 1 to 3 do
    xx[j] := convert(series( subs( T = T2[k,j], X2 ), u,5),polynom);
    yy[j] := convert(series( subs( T = T2[k,j], Y2 ), u,5),polynom);
  od;
  print( factor(xx[1]-xx[2]), factor(xx[2]-xx[3]), factor(yy[1]-yy[2]), factor(yy[2]-yy[3]) );
od: