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

X4o := 1/2*T*(15550414*u^5+449865*u^4-8124543*u^3+24504606*T-16336404*u+
24504606)*(41935719196*o*u^5-18555460245*o*u^4-3510638893*u^5+14077340487*o*u^3
-4207650615*u^4-5657063328*o*u^2+1647448551*u^3+8380575252*o*u-938176344*u^2-\
4631370534*T-9262741068*o+3528663264*u+4631370534)*(41935719196*o*u^5-\
18555460245*o*u^4+45446358089*u^5+14077340487*o*u^3-14347809630*u^4-5657063328*
o*u^2+12429891936*u^3+8380575252*o*u-4718886984*u^2+4631370534*T-9262741068*o+
4851911988*u-13894111602)*(42*T-37*u-21)*(4831409440*o*u^5+2540073564*o*u^4-\
5890290901*u^5+621727974*o*u^3+2601373950*u^4+224042112*o*u^2-2560175667*u^3+
112021056*u^2+2315685267*T-2315685267*o-2315685267)*(4831409440*o*u^5+
2540073564*o*u^4+10721700341*u^5+621727974*o*u^3-61300386*u^4+224042112*o*u^2+
3181903641*u^3+112021056*u^2-2315685267*T-2315685267*o)/(15550414*u^5+449865*u^
4-8124543*u^3+24504606*T+8168202*u+24504606)/(74579103316*o*u^5-25135633593*o*u
^4+57216497219*u^5+17529989463*o*u^3-15718163391*u^4-5657063328*o*u^2+
7243000317*u^3+8380575252*o*u-938176344*u^2-4631370534*T-9262741068*o-\
1102707270*u+4631370534)/(74579103316*o*u^5-25135633593*o*u^4+17362606097*u^5+
17529989463*o*u^3-9417470202*u^4-5657063328*o*u^2+10286989146*u^3+8380575252*o*
u-4718886984*u^2+4631370534*T-9262741068*o+9483282522*u-13894111602)/(42*T-16*u
-21)/(4831409440*o*u^5+2540073564*o*u^4-5890290901*u^5+621727974*o*u^3+
2601373950*u^4+224042112*o*u^2-2560175667*u^3+112021056*u^2+2315685267*T-\
2315685267*o-2315685267*u-2315685267)/(4831409440*o*u^5+2540073564*o*u^4+
10721700341*u^5+621727974*o*u^3-61300386*u^4+224042112*o*u^2+3181903641*u^3+
112021056*u^2-2315685267*T-2315685267*o+2315685267*u):
Y4o := -1/2*(T-1)*(15550414*u^5+449865*u^4-8124543*u^3+24504606*T-16336404*u-\
49009212)*(62584543988*o*u^5+12398721363*o*u^4+60432658087*u^5+18627973623*o*u^
3+10798918830*u^4+5657063328*o*u^2+15895765620*u^3+8380575252*o*u+4718886984*u^
2+4631370534*T+9262741068*o+4851911988*u+9262741068)*(62584543988*o*u^5+
12398721363*o*u^4+2151885901*u^5+18627973623*o*u^3+1599802533*u^4+5657063328*o*
u^2+2732208003*u^3+8380575252*o*u+938176344*u^2-4631370534*T+9262741068*o+
3528663264*u)*(42*T-37*u-21)*(4831409440*o*u^5+2540073564*o*u^4-5890290901*u^5+
621727974*o*u^3+2601373950*u^4+224042112*o*u^2-2560175667*u^3+112021056*u^2+
2315685267*T-2315685267*o-2315685267)*(4831409440*o*u^5+2540073564*o*u^4+
10721700341*u^5+621727974*o*u^3-61300386*u^4+224042112*o*u^2+3181903641*u^3+
112021056*u^2-2315685267*T-2315685267*o)/(15550414*u^5+449865*u^4-8124543*u^3+
24504606*T+8168202*u-49009212)/(62584543988*o*u^5+12398721363*o*u^4+60432658087
*u^5+18627973623*o*u^3+10798918830*u^4+5657063328*o*u^2+15895765620*u^3+
8380575252*o*u+4718886984*u^2+4631370534*T+9262741068*o+9483282522*u+9262741068
)/(62584543988*o*u^5+12398721363*o*u^4+2151885901*u^5+18627973623*o*u^3+
1599802533*u^4+5657063328*o*u^2+2732208003*u^3+8380575252*o*u+938176344*u^2-\
4631370534*T+9262741068*o-1102707270*u)/(42*T-16*u-21)/(4831409440*o*u^5+
2540073564*o*u^4-5890290901*u^5+621727974*o*u^3+2601373950*u^4+224042112*o*u^2-\
2560175667*u^3+112021056*u^2+2315685267*T-2315685267*o-2315685267*u-2315685267)
/(4831409440*o*u^5+2540073564*o*u^4+10721700341*u^5+621727974*o*u^3-61300386*u^
4+224042112*o*u^2+3181903641*u^3+112021056*u^2-2315685267*T-2315685267*o+
2315685267*u):
T4o := [[-1/9*u^2+42898/83349*u^3-866815/2722734*u^4-30036161/220541454*u^5-\
11473054499/6947055801*u^6-1, 1/9*u^2+42898/83349*u^3+175807/2722734*u^4-\
34280363/220541454*u^5-385688755/992436543*u^6+2, -29/1701*u^3+820/9261*u^4+
10763899/31505922*u^5+314994503/514596726*u^6+3/14*u+1/2], [(359191761515/
83364669612*o+190789398437/41682334806)*u^6+(165033327299/9262741068*o+
858124300/330812181)*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, (-\
68114423071/83364669612*o+361551761233/83364669612)*u^6+(-132129737287/
9262741068*o-131259663683/9262741068)*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, (39585754531/41682334806*o+57624280993/83364669612)*u^6+(
-4919528359/2315685267*o+6799122412/2315685267)*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], [(-359191761515/83364669612*o+22387035359/83364669612)*u^6+(-\
165033327299/9262741068*o-141005846899/9262741068)*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, (68114423071/83364669612*o+15345220868/
2977309629)*u^6+(132129737287/9262741068*o+217518401/2315685267)*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, (-39585754531/41682334806*o-21547228069/
83364669612)*u^6+(4919528359/2315685267*o+11718650771/2315685267)*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+(-177965812849/6947055801*o+86569460491/6947055801)*u^
6+(1062267769/110270727*o-1829266951/441082908)*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, (
349831110779/27788223204*o-292523296547/6947055801)*u^6+(-215896849555/
9262741068*o+5662640563/9262741068)*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, (379131971531/27788223204*o+218712724948/6947055801)*u^6+
(3836941231/2315685267*o-14402821963/2315685267)*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], [(253908980500190/116759166847407*o+2076331270819111/
467036667389628)*u^6+(17260538701471/1588560093162*o+3215610502723/794280046581
)*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, (-\
126669597350501/233518333694814*o+157260184416841/467036667389628)*u^6+(
21597169100369/1588560093162*o+199893287807/794280046581)*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+(168999154465/
4765680279486*o+367104455233/2382840139743)*u^6+(-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)], [(600074634881/27788223204*o+
883417064813/13894111602)*u^6+(-37019284415/9262741068*o+148073648771/
9262741068)*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+(
92151454478/6947055801*o-142646794205/6947055801)*u^6+(832935583/110270727*o-\
2567008147/441082908)*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, (-89583650209/3969746172*o-171135583817/
6947055801)*u^6+(-10224623518/2315685267*o-18085301978/2315685267)*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], [(-600074634881/27788223204*o+
1166759494745/27788223204)*u^6+(37019284415/9262741068*o+13220923799/661624362)
*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+(-92151454478/
6947055801*o-234798248683/6947055801)*u^6+(-832935583/110270727*o-5898750479/
441082908)*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), (89583650209/3969746172*o-57456783805/
27788223204)*u^6+(10224623518/2315685267*o-7860678460/2315685267)*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+(177965812849/6947055801*o+264535273340/
6947055801)*u^6+(-1062267769/110270727*o-6078338027/441082908)*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), (-349831110779/27788223204*o-1519924296967/27788223204)*u^6+(
215896849555/9262741068*o+110779745059/4631370534)*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, (-379131971531/27788223204*o+495718928261/
27788223204)*u^6+(-3836941231/2315685267*o-18239763194/2315685267)*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], [(-253908980500190/
116759166847407*o+1060695348818351/467036667389628)*u^6+(-17260538701471/
1588560093162*o-10829317696025/1588560093162)*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, (126669597350501/
233518333694814*o+410599379117843/467036667389628)*u^6+(-21597169100369/
1588560093162*o-21197382524755/1588560093162)*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+(-168999154465/
4765680279486*o+565209756001/4765680279486)*u^6+(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)]]:

X4 := subs(o=om,X4o); Y4 := subs(o=om,Y4o); T4 := subs(o=om,T4o);

#
#  Check that [X4,Y4]  is a solution up to $O(u^6)$.
#

for k from 1 to 9 do
  for j from 1 to 3 do
    xx[j] := convert(series( subs( T = T4[k,j], X4 ), u,7),polynom);
    yy[j] := convert(series( subs( T = T4[k,j], Y4 ), u,7),polynom);
  od;
  print( [[k]], factor(xx[1]-xx[2]), factor(xx[2]-xx[3]), factor(yy[1]-yy[2]), factor(yy[2]-yy[3]) );
od: