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

X3o := 1/2*T*(49985*u^4-902727*u^3+2722734*T-1815156*u+2722734)*(294531115*o*u^
4-223449849*o*u^3+66788105*u^4+89794656*o*u^2-26149977*u^3-133025004*o*u+
14891688*u^2+73513818*T+147027636*o-56010528*u-73513818)*(294531115*o*u^4-\
223449849*o*u^3+227743010*u^4+89794656*o*u^2-197299872*u^3-133025004*o*u+
74902968*u^2-73513818*T+147027636*o-77014476*u+220541454)*(42*T-37*u-21)*(
40318628*o*u^4+9868698*o*u^3+41291650*u^4+3556224*o*u^2-40637709*u^3+1778112*u^
2+36756909*T-36756909*o-36756909)*(40318628*o*u^4+9868698*o*u^3-973022*u^4+
3556224*o*u^2+50506407*u^3+1778112*u^2-36756909*T-36756909*o)/(49985*u^4-902727
*u^3+2722734*T+907578*u+2722734)/(398978311*o*u^4-278253801*o*u^3+249494657*u^4
+89794656*o*u^2-114968259*u^3-133025004*o*u+14891688*u^2+73513818*T+147027636*o
+17503290*u-73513818)/(398978311*o*u^4-278253801*o*u^3+149483654*u^4+89794656*o
*u^2-163285542*u^3-133025004*o*u+74902968*u^2-73513818*T+147027636*o-150528294*
u+220541454)/(42*T-16*u-21)/(40318628*o*u^4+9868698*o*u^3+41291650*u^4+3556224*
o*u^2-40637709*u^3+1778112*u^2+36756909*T-36756909*o-36756909*u-36756909)/(
40318628*o*u^4+9868698*o*u^3-973022*u^4+3556224*o*u^2+50506407*u^3+1778112*u^2-\
36756909*T-36756909*o+36756909*u):
Y3o := -1/2*(T-1)*(49985*u^4-902727*u^3+2722734*T-1815156*u-5445468)*(196805101
*o*u^4+295682121*o*u^3+171411410*u^4+89794656*o*u^2+252313740*u^3+133025004*o*u
+74902968*u^2+73513818*T+147027636*o+77014476*u+147027636)*(196805101*o*u^4+
295682121*o*u^3+25393691*u^4+89794656*o*u^2+43368381*u^3+133025004*o*u+14891688
*u^2-73513818*T+147027636*o+56010528*u)*(42*T-37*u-21)*(40318628*o*u^4+9868698*
o*u^3+41291650*u^4+3556224*o*u^2-40637709*u^3+1778112*u^2+36756909*T-36756909*o
-36756909)*(40318628*o*u^4+9868698*o*u^3-973022*u^4+3556224*o*u^2+50506407*u^3+
1778112*u^2-36756909*T-36756909*o)/(49985*u^4-902727*u^3+2722734*T+907578*u-\
5445468)/(196805101*o*u^4+295682121*o*u^3+171411410*u^4+89794656*o*u^2+
252313740*u^3+133025004*o*u+74902968*u^2+73513818*T+147027636*o+150528294*u+
147027636)/(196805101*o*u^4+295682121*o*u^3+25393691*u^4+89794656*o*u^2+
43368381*u^3+133025004*o*u+14891688*u^2-73513818*T+147027636*o-17503290*u)/(42*
T-16*u-21)/(40318628*o*u^4+9868698*o*u^3+41291650*u^4+3556224*o*u^2-40637709*u^
3+1778112*u^2+36756909*T-36756909*o-36756909*u-36756909)/(40318628*o*u^4+
9868698*o*u^3-973022*u^4+3556224*o*u^2+50506407*u^3+1778112*u^2-36756909*T-\
36756909*o+36756909*u):
T3o := [[-1/9*u^2+42898/83349*u^3-866815/2722734*u^4+109917565/220541454*u^5-1,
1/9*u^2+42898/83349*u^3+175807/2722734*u^4+105673363/220541454*u^5+2, -29/1701*
u^3+820/9261*u^4+10763899/31505922*u^5+3/14*u+1/2], [(3053557/147027636*o-\
50763845/73513818)*u^5+(-198400547/36756909*o-157572323/73513818)*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, (-\
110486497/147027636*o-23569949/21003948)*u^5+(-105503018/36756909*o-83449697/
73513818)*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, (-1398713/36756909*o+14425897/36756909)*u^5+(-4031525/
5250987*o-67195781/73513818)*u^4+(10280/9261-22378/83349*o)*u^3+u^2*(-128/1323*
o-64/1323)+2/3*u+o+1], [(-3053557/147027636*o-104581247/147027636)*u^5+(
198400547/36756909*o+239228771/73513818)*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, (110486497/147027636*o-\
27251573/73513818)*u^5+(105503018/36756909*o+127556339/73513818)*u^4+(91957/
23814*o+12364/27783)*u^3+(1427/1323*o+730/1323)*u^2+(8/7*o+2/21)*u+2*o, (
1398713/36756909*o+1758290/4084101)*u^5+(4031525/5250987*o-10754431/73513818)*u
^4+(16414/11907+22378/83349*o)*u^3+u^2*(128/1323*o+64/1323)+2/3*u-o], [-1+(
1062267769/110270727*o-1549359499/441082908)*u^5+(-2205383/500094*o+12390095/
24504606)*u^4+(14003/7938*o+4448/27783)*u^3+(-o-1/3)*u^2+2/3*u*o, (-617195657/
63011844*o+6025140797/441082908)*u^5+(314096633/73513818*o-352250663/73513818)*
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, (-47355629/110270727*o-1196405824/110270727)*u^5+(-14178701/10501974
*o+170854367/73513818)*u^4+(156400/83349*o+12776/11907)*u^3+(-1048/1323*o-524/
1323)*u^2+(4/3+2/3*o)*u-o], [(-157137536455/529520031054*o-455330951381/
529520031054)*u^5+(-15186243829/3602177082*o-6234796697/3602177082)*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, (43556837495/529520031054*o-112770096143/529520031054)*u^5+(
9480781045/3602177082*o+1941827513/3602177082)*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+(-\
182265733789/529520031054*o-275604653531/529520031054)*u^5+(-145933/12252303*o-\
126173/12252303)*u^4+u^3*(-48446/1361367*o-130561/1361367)+u*(1/2+2/21*o)], [(-\
3407731595/441082908*o+5451063713/441082908)*u^5+(-204873581/73513818*o+
399963569/73513818)*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+(832935583/110270727*o-2287100695/441082908
)*u^5+(1873045/500094*o-28921315/24504606)*u^4+(14003/7938*o+4448/27783)*u^3+(o
+1/3)*u^2+2/3*u*o, (-256819718/110270727*o-1141694899/110270727)*u^5+(4919885/
10501974*o-254404103/73513818)*u^4+(111644/83349*o+67054/83349)*u^3+(1048/1323*
o+524/1323)*u^2+(4/3+2/3*o)*u+o+1], [(3407731595/441082908*o+2214698827/
110270727)*u^5+(204873581/73513818*o+302418575/36756909)*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+(-\
832935583/110270727*o-802691861/63011844)*u^5+(-1873045/500094*o-60350260/
12252303)*u^4+(-14003/7938*o-89125/55566)*u^3+(-o-2/3)*u^2+u*(-2/3-2/3*o), (
256819718/110270727*o-884875181/110270727)*u^5+(-4919885/10501974*o-144421649/
36756909)*u^4+(-111644/83349*o-130/243)*u^3+(-1048/1323*o-524/1323)*u^2+(-2/3*o
+2/3)*u-o], [-1+(-1062267769/110270727*o-828347225/63011844)*u^5+(2205383/
500094*o+60226931/12252303)*u^4+(-14003/7938*o-89125/55566)*u^3+(o+2/3)*u^2+u*(
-2/3-2/3*o), (617195657/63011844*o+2586377599/110270727)*u^5+(-314096633/
73513818*o-333173648/36756909)*u^4+(88681/11907*o+726367/166698)*u^3+(-1513/
1323*o-1769/1323)*u^2+(22/7*o+10/7)*u+2*o, (47355629/110270727*o-1149050195/
110270727)*u^5+(14178701/10501974*o+135052637/36756909)*u^4+(-156400/83349*o-\
66968/83349)*u^3+(1048/1323*o+524/1323)*u^2+(-2/3*o+2/3)*u+o+1], [(157137536455
/529520031054*o-149096707463/264760015527)*u^5+(15186243829/3602177082*o+
4475723566/1801088541)*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, (-43556837495/529520031054*o-\
78163466819/264760015527)*u^5+(-9480781045/3602177082*o-3769476766/1801088541)*
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+(182265733789/529520031054*o-46669459871/
264760015527)*u^5+(145933/12252303*o+19760/12252303)*u^4+u^3*(48446/1361367*o-\
82115/1361367)+u*(-2/21*o+17/42)]]:

X3 := subs(o=om,X3o); Y3 := subs(o=om,Y3o); T3 := subs(o=om,T3o);

#
#  Check that [X3,Y3]  is a solution up to $O(u^5)$.
#

for k from 1 to 9 do
  for j from 1 to 3 do
    xx[j] := convert(series( subs( T = T3[k,j], X3 ), u,6),polynom);
    yy[j] := convert(series( subs( T = T3[k,j], Y3 ), u,6),polynom);
  od;
  print( factor(xx[1]-xx[2]), factor(xx[2]-xx[3]), factor(yy[1]-yy[2]), factor(yy[2]-yy[3]) );
od: