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

X5o := 1/8*T*(7151422046*u^6+48004128018*u^5+1388733255*u^4-25080464241*u^3+
75645718722*T-50430479148*u+75645718722)*(81331937469206*o*u^6-36987304330872*o
*u^5+18520179855967*u^6+16365915936090*o*u^4+3096383503626*u^5-12416214309534*o
*u^3+3711147842430*u^4+4989529855296*o*u^2-1453049621982*u^3-7391667372264*o*u+
827471535408*u^2+4084868810988*T+8169737621976*o-3112280998848*u-4084868810988)
*(81331937469206*o*u^6-36987304330872*o*u^5+62811757613239*u^6+16365915936090*o
*u^4-40083687834498*u^5-12416214309534*o*u^3+12654768093660*u^4+4989529855296*o
*u^2-10963164687552*u^3-7391667372264*o*u+4162058319888*u^2-4084868810988*T+
8169737621976*o-4279386373416*u+12254606432964)*(42*T-37*u-21)*(3384095161496*o
*u^6+2130651563040*o*u^5+8205192949132*u^6+1120172441724*o*u^4-2597618287341*u^
5+274182036534*o*u^3+1147205911950*u^4+98802571392*o*u^2-1129037469147*u^3+
49401285696*u^2+1021217202747*T-1021217202747*o-1021217202747)*(3384095161496*o
*u^6+2130651563040*o*u^5-4821097787636*u^6+1120172441724*o*u^4+4728269850381*u^
5+274182036534*o*u^3-27033470226*u^4+98802571392*o*u^2+1403219505681*u^3+
49401285696*u^2-1021217202747*T-1021217202747*o)/(7151422046*u^6+48004128018*u^
5+1388733255*u^4-25080464241*u^3+75645718722*T+25215239574*u+75645718722)/(
70769677201582*o*u^6-32889384562356*o*u^5+68143590963122*u^6+11084814414513*o*u
^4-25232475273579*u^5-7730725353183*o*u^3+6931710055431*u^4+2494764927648*o*u^2
-3194163139797*u^3-3695833686132*o*u+413735767704*u^2+2042434405494*T+
4084868810988*o+486293906070*u-2042434405494)/(70769677201582*o*u^6-\
32889384562356*o*u^5+2626086238460*u^6+11084814414513*o*u^4-7656909288777*u^5-\
7730725353183*o*u^3+4153104359082*u^4+2494764927648*o*u^2-4536562213386*u^3-\
3695833686132*o*u+2081029159944*u^2-2042434405494*T+4084868810988*o-\
4182127592202*u+6127303216482)/(42*T-16*u-21)/(3384095161496*o*u^6+
2130651563040*o*u^5+8205192949132*u^6+1120172441724*o*u^4-2597618287341*u^5+
274182036534*o*u^3+1147205911950*u^4+98802571392*o*u^2-1129037469147*u^3+
49401285696*u^2+1021217202747*T-1021217202747*o-1021217202747*u-1021217202747)/
(3384095161496*o*u^6+2130651563040*o*u^5-4821097787636*u^6+1120172441724*o*u^4+
4728269850381*u^5+274182036534*o*u^3-27033470226*u^4+98802571392*o*u^2+
1403219505681*u^3+49401285696*u^2-1021217202747*T-1021217202747*o+1021217202747
*u):
Y5o := -1/2*(T-1)*(7151422046*u^6+48004128018*u^5+1388733255*u^4-25080464241*u^
3+75645718722*T-50430479148*u-151291437444)*(2088655377434*o*u^6+55199567797416
*o*u^5+8574188295199*u^6+10935672242166*o*u^4+53301604432734*u^5+16429872735486
*o*u^3+9524646408060*u^4+4989529855296*o*u^2+14020065276840*u^3+7391667372264*o
*u+4162058319888*u^2+4084868810988*T+8169737621976*o+4279386373416*u+
8169737621976)*(2088655377434*o*u^6+55199567797416*o*u^5-6485532917765*u^6+
10935672242166*o*u^4+1897963364682*u^5+16429872735486*o*u^3+1411025834106*u^4+
4989529855296*o*u^2+2409807458646*u^3+7391667372264*o*u+827471535408*u^2-\
4084868810988*T+8169737621976*o+3112280998848*u)*(42*T-37*u-21)*(3384095161496*
o*u^6+2130651563040*o*u^5+8205192949132*u^6+1120172441724*o*u^4-2597618287341*u
^5+274182036534*o*u^3+1147205911950*u^4+98802571392*o*u^2-1129037469147*u^3+
49401285696*u^2+1021217202747*T-1021217202747*o-1021217202747)*(3384095161496*o
*u^6+2130651563040*o*u^5-4821097787636*u^6+1120172441724*o*u^4+4728269850381*u^
5+274182036534*o*u^3-27033470226*u^4+98802571392*o*u^2+1403219505681*u^3+
49401285696*u^2-1021217202747*T-1021217202747*o)/(7151422046*u^6+48004128018*u^
5+1388733255*u^4-25080464241*u^3+75645718722*T+25215239574*u-151291437444)/(
2088655377434*o*u^6+55199567797416*o*u^5+8574188295199*u^6+10935672242166*o*u^4
+53301604432734*u^5+16429872735486*o*u^3+9524646408060*u^4+4989529855296*o*u^2+
14020065276840*u^3+7391667372264*o*u+4162058319888*u^2+4084868810988*T+
8169737621976*o+8364255184404*u+8169737621976)/(2088655377434*o*u^6+
55199567797416*o*u^5-6485532917765*u^6+10935672242166*o*u^4+1897963364682*u^5+
16429872735486*o*u^3+1411025834106*u^4+4989529855296*o*u^2+2409807458646*u^3+
7391667372264*o*u+827471535408*u^2-4084868810988*T+8169737621976*o-972587812140
*u)/(42*T-16*u-21)/(3384095161496*o*u^6+2130651563040*o*u^5+8205192949132*u^6+
1120172441724*o*u^4-2597618287341*u^5+274182036534*o*u^3+1147205911950*u^4+
98802571392*o*u^2-1129037469147*u^3+49401285696*u^2+1021217202747*T-\
1021217202747*o-1021217202747*u-1021217202747)/(3384095161496*o*u^6+
2130651563040*o*u^5-4821097787636*u^6+1120172441724*o*u^4+4728269850381*u^5+
274182036534*o*u^3-27033470226*u^4+98802571392*o*u^2+1403219505681*u^3+
49401285696*u^2-1021217202747*T-1021217202747*o+1021217202747*u):
T5o := [[-1/9*u^2+42898/83349*u^3-866815/2722734*u^4-30036161/220541454*u^5-\
594361069658/340405734249*u^6+5083200846715/2042434405494*u^7-1, 1/9*u^2+42898/
83349*u^3+175807/2722734*u^4-34280363/220541454*u^5-164472642172/340405734249*u
^6+1927700418869/1021217202747*u^7+2, -29/1701*u^3+820/9261*u^4+10763899/
31505922*u^5+314994503/514596726*u^6+102406736293/291776343642*u^7+3/14*u+1/2],
[(-38198135526937/12254606432964*o-17013102997063/1750658061852)*u^7+(-\
47414291956163/1361622936996*o-12731958522229/1361622936996)*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, (-71331221278411/12254606432964*o-4428260735473/437664515463)*u^7+(-\
1808754035971/1361622936996*o+1523641334203/680811468498)*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, (-2193184979032/3063651608241*o+3197928064703/
3063651608241)*u^7+(-229928016713/97258781214*o-370335580591/50430479148)*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], [(38198135526937/12254606432964*o-6741132121042/1021217202747)*u^7
+(17341166716967/680811468498+47414291956163/1361622936996*o)*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, (71331221278411/12254606432964*o-\
17553359771611/4084868810988)*u^7+(1618678901459/453874312332+1808754035971/
1361622936996*o)*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, (2193184979032/3063651608241*o+
599012560415/340405734249)*u^7+(229928016713/97258781214*o-6780068441975/
1361622936996)*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+(193567492112188/3063651608241*o-\
586181862239569/12254606432964)*u^7+(-177965812849/6947055801*o+4209722164852/
340405734249)*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, (927763315639/250094008836*o+736435507538515/6127303216482)*u^7+(-\
180577886664835/4084868810988+49336517907079/4084868810988*o)*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, (-436618802019301/6127303216482*o-\
945341896706455/12254606432964)*u^7+(27329672779720/1021217202747+9895540065863
/583552687284*o)*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], [(-123023389210490462/
51490792579706487*o-1682361620606031685/205963170318825948)*u^7+(-\
4369606275457876/350277500542221-7709192371046534/350277500542221*o)*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, (119796859748121995/
102981585159412974*o-130585664934341671/205963170318825948)*u^7+(-\
438189309385718/350277500542221-10902197410786/350277500542221*o)*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+(-77946308194613407/
102981585159412974*o-121783767885834959/102981585159412974)*u^7+(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)], [(290480684979733/
12254606432964*o+9927899526841/125047004418)*u^7+(203886761998771/4084868810988
+52658517325649/4084868810988*o)*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+(
142174568830949/6127303216482*o-839895048189007/12254606432964)*u^7+(
92151454478/6947055801*o-7021874315252/340405734249)*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, (-479930605751647/6127303216482*o-\
575460604566439/12254606432964)*u^7+(-33362123770231/1021217202747-\
15102565244435/583552687284*o)*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], [(-\
290480684979733/12254606432964*o+682453468650685/12254606432964)*u^7+(
75614122336561/2042434405494-52658517325649/4084868810988*o)*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+(-142174568830949/6127303216482*o-160606312264415/
1750658061852)*u^7+(-92151454478/6947055801*o-11537295584674/340405734249)*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), (
479930605751647/6127303216482*o+384400606936855/12254606432964)*u^7+(-\
27730538369879/4084868810988+15102565244435/583552687284*o)*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+(-193567492112188/3063651608241*o-194350261526903/1750658061852)*u^7+(
177965812849/6947055801*o+12930046994453/340405734249)*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), (-927763315639/
250094008836*o+1427410612610719/12254606432964)*u^7+(-114957202285957/
2042434405494-49336517907079/4084868810988*o)*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, (436618802019301/6127303216482*o-72104292667853/12254606432964)*u^7+(
40049910657839/4084868810988-9895540065863/583552687284*o)*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], [(123023389210490462/51490792579706487*o-1190268063764069837/
205963170318825948)*u^7+(3339586095588658/350277500542221+7709192371046534/
350277500542221*o)*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, (-119796859748121995/102981585159412974*o-370179384430585661/
205963170318825948)*u^7+(-427287111974932/350277500542221+10902197410786/
350277500542221*o)*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+(77946308194613407/102981585159412974*o-21918729845610776
/51490792579706487)*u^7+(-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)]]:

X5 := subs(o=om,X5o); Y5 := subs(o=om,Y5o); T5 := subs(o=om,T5o);

#
#  Check that [X5,Y5]  is a solution up to $O(u^7)$.
#

for k from 1 to 9 do
  for j from 1 to 3 do
    xx[j] := convert(series( subs( T = T5[k,j], X5 ), u,8),polynom);
    yy[j] := convert(series( subs( T = T5[k,j], Y5 ), u,8),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: