Z7(z1,t)=1. Again, we verify instead that

-1+b0(t)+b2 z1^2+b4 z1^4+b6 z1^6=-(b1z1+b3z1^3+b5z1^5+b7z1^7)

(in one stroke we also have Z7(z3,t)=Z7(z5,t)=1)

Off[Root::nup]

ω=(Sqrt[((4-11 t+8 (t^2))/t)]);

iep3=(z^3)-((z^2) (-14+81 t-130 (t^2)-134 (t^3)+728 (t^4)-1020 (t^5)+680 (t^6)-216 (t^7)+24 (t^8)+(-1+2 t) (2+5 t-40 (t^2)+64 (t^3)-38 (t^4)+8 (t^5)) ωω))/(8 (((-1+t))^3) (t^2) (-1+5 t-6 (t^2)+(t^3)))+(z (-2-17 t+245 (t^2)-812 (t^3)+98 (t^4)+5437 (t^5)-13045 (t^6)+7456 (t^7)+19604 (t^8)-46768 (t^9)+47296 (t^10)-26752 (t^11)+8608 (t^12)-1440 (t^13)+96 (t^14)+t (-1+2 t) (7-23 t-36 (t^2)+138 (t^3)+575 (t^4)-3055 (t^5)+5670 (t^6)-5392 (t^7)+2736 (t^8)-688 (t^9)+64 (t^10)) ωω))/(32 (((-1+t))^3) (t^5) (((-1+5 t-6 (t^2)+(t^3)))^2))-(-225+2930 t-16685 (t^2)+53762 (t^3)-104265 (t^4)+112187 (t^5)-23793 (t^6)-119612 (t^7)+201884 (t^8)-171608 (t^9)+89696 (t^10)-29616 (t^11)+5984 (t^12)-672 (t^13)+32 (t^14)+(-1+2 t) (25-90 t-629 (t^2)+5374 (t^3)-17553 (t^4)+32643 (t^5)-37939 (t^6)+28146 (t^7)-13104 (t^8)+3632 (t^9)-536 (t^10)+32 (t^11)) ωω)/(32 (((-1+t))^3) (t^2) (((-1+5 t-6 (t^2)+(t^3)))^3));

q=(2t-1)^10;

p61=16t^5(-10+27 t+288 (t^2)-2022 (t^3)+5825 (t^4)-9477 (t^5)+9336 (t^6)-5494 (t^7)+1712 (t^8)-122 (t^9)-80 (t^10)+16 (t^11));
p62=(-16 t^6)(-25+206 t-726 (t^2)+1459 (t^3)-1927 (t^4)+1870 (t^5)-1394 (t^6)+724 (t^7)-210 (t^8)+24 (t^9));
p41=(-4t^3)(50-687 t+4203 (t^2)-15695 (t^3)+41282 (t^4)-82204 (t^5)+125852 (t^6)-144150 (t^7)+117216 (t^8)-62520 (t^9)+18496 (t^10)-1080 (t^11)-960 (t^12)+192 (t^13));
p42=4t^3(10-121 t+677 (t^2)-2521 (t^3)+7132 (t^4)-15412 (t^5)+24800 (t^6)-29742 (t^7)+27184 (t^8)-18840 (t^9)+9072 (t^10)-2520 (t^11)+288 (t^12));
p21=(-2+40 t-360 (t^2)+2245 (t^3)-11098 (t^4)+42924 (t^5)-126314 (t^6)+282304 (t^7)-483968 (t^8)+638896 (t^9)-640864 (t^10)+470400 (t^11)-234144 (t^12)+65792 (t^13)-2784 (t^14)-3840 (t^15)+768 (t^16));
p22=-t^3(65-774 t+4344 (t^2)-15694 (t^3)+41224 (t^4)-82096 (t^5)+124784 (t^6)-144768 (t^7)+127840 (t^8)-83808 (t^9)+37824 (t^10)-10080 (t^11)+1152 (t^12));
p01=(1-t) (1-19 t+161 (t^2)-924 (t^3)+4066 (t^4)-13822 (t^5)+35840 (t^6)-70584 (t^7)+105400 (t^8)-118168 (t^9)+96704 (t^10)-54208 (t^11)+17760 (t^12)-1440 (t^13)-1024 (t^14)+256 (t^15));
p02=(t^3) (25-290 t+1636 (t^2)-6010 (t^3)+15992 (t^4)-32064 (t^5)+48928 (t^6)-56632 (t^7)+49024 (t^8)-30752 (t^9)+13120 (t^10)-3360 (t^11)+384 (t^12));
p11=8t^3(t-1)(-125+4005 t-60169 (t^2)+570158 (t^3)-3862166 (t^4)+20045945 (t^5)-83223823 (t^6)+284217439 (t^7)-813613881 (t^8)+1978202656 (t^9)-4124892528 (t^10)+7432364016 (t^11)-11645707296 (t^12)+15960414880 (t^13)-19242695952 (t^14)+20530254512 (t^15)-19487643408 (t^16)+16504985216 (t^17)-12441863808 (t^18)+8263236224 (t^19)-4749495424 (t^20)+2309017088 (t^21)-926839040 (t^22)+300215040 (t^23)-76503296 (t^24)+14585856 (t^25)-1794048 (t^26)+73728 (t^27)+8192 (t^28));
p12=8t^3(1-t)(-25+765 t-11171 (t^2)+105634 (t^3)-736638 (t^4)+4055851 (t^5)-18297419 (t^6)+69029409 (t^7)-220386161 (t^8)+599953820 (t^9)-1399432680 (t^10)+2804818368 (t^11)-4835265904 (t^12)+7165985152 (t^13)-9113381872 (t^14)+9916520688 (t^15)-9197207856 (t^16)+7238388416 (t^17)-4811313152 (t^18)+2688147584 (t^19)-1256283520 (t^20)+488377856 (t^21)-156990208 (t^22)+41689856 (t^23)-9309952 (t^24)+1799168 (t^25)-280576 (t^26)+24576 (t^27));
p31=8 t^3(t-1)(-125+4005 t-59369 (t^2)+553998 (t^3)-3730470 (t^4)+19565545 (t^5)-83537119 (t^6)+297885767 (t^7)-901975729 (t^8)+2350136384 (t^9)-5333296784 (t^10)+10652922784 (t^11)-18880321184 (t^12)+29852879392 (t^13)-42255474560 (t^14)+53630116864 (t^15)-60971031168 (t^16)+61759492352 (t^17)-55128623872 (t^18)+42657332224 (t^19)-28037451008 (t^20)+15314942464 (t^21)-6803466496 (t^22)+2405327616 (t^23)-657957120 (t^24)+131469312 (t^25)-16441344 (t^26)+663552 (t^27)+73728 (t^28));
p32=8 t^3(1-t)(-25+765 t-11171 (t^2)+107634 (t^3)-787518 (t^4)+4665067 (t^5)-22973819 (t^6)+95297577 (t^7)-336259225 (t^8)+1016942748 (t^9)-2647457192 (t^10)+5940364352 (t^11)-11479425312 (t^12)+19070848736 (t^13)-27171006848 (t^14)+33097766976 (t^15)-34341234112 (t^16)+30212895232 (t^17)-22415990528 (t^18)+13930544128 (t^19)-7190272256 (t^20)+3053491200 (t^21)-1060255488 (t^22)+303176448 (t^23)-73578240 (t^24)+15455232 (t^25)-2525184 (t^26)+221184 (t^27));
p51=128t^8(t-1)(-1500+28960 t-257939 (t^2)+1404604 (t^3)-5211955 (t^4)+13795890 (t^5)-25855137 (t^6)+29614248 (t^7)+3063112 (t^8)-113012293 (t^9)+333031055 (t^10)-652968101 (t^11)+993241608 (t^12)-1221269400 (t^13)+1224164696 (t^14)-994694744 (t^15)+647101024 (t^16)-331790224 (t^17)+131533872 (t^18)-39147024 (t^19)+8229120 (t^20)-1046016 (t^21)+41472 (t^22)+4608 (t^23));
p52=128 t^8(1-t)(-100+80 t+20039 (t^2)-260728 (t^3)+1786957 (t^4)-8269540 (t^5)+28531279 (t^6)-77414974 (t^7)+170435054 (t^8)-309998835 (t^9)+470441223 (t^10)-598385907 (t^11)+638090092 (t^12)-567729488 (t^13)+417123064 (t^14)-249429768 (t^15)+119783328 (t^16)-46124976 (t^17)+14587824 (t^18)-3960432 (t^19)+919872 (t^20)-157824 (t^21)+13824 (t^22));
p71=2048 t^13(t-1)(-25+56 t+2520 (t^2)-25565 (t^3)+120032 (t^4)-330197 (t^5)+533239 (t^6)-339915 (t^7)-575358 (t^8)+1899107 (t^9)-2748394 (t^10)+2566194 (t^11)-1675233 (t^12)+781067 (t^13)-257689 (t^14)+57232 (t^15)-7392 (t^16)+288 (t^17)+32 (t^18));
p72=2048 t^13(1-t)(-1-50 t+626 (t^2)-2319 (t^3)-1948 (t^4)+47711 (t^5)-201331 (t^6)+485563 (t^7)-779658 (t^8)+878711 (t^9)-712736 (t^10)+425072 (t^11)-193639 (t^12)+72003 (t^13)-23071 (t^14)+6068 (t^15)-1096 (t^16)+96 (t^17));

ContourPlot[Evaluate[{y==1/q(-q+{p61+p62 ωω,p41+p42 ωω,p21+p22 ωω,p01+p02 ωω}.Table[(-Sqrt[Root[iep3,1]])^j,{j,6,0,-2}])/.ωω->ω,y==1/q{-Sqrt[p71+p72 ωω],Sqrt[p51+p52 ωω],-Sqrt[p31+p32 ωω],Sqrt[p11+p12 ωω]}.Table[(-Sqrt[Root[iep3,1]])^j,{j,7,1,-2}]/.ωω->ω}],{t,(NumericalApproximation[Root, (RowDefault["0.777", FEExpressionsNumericalApproximationElider]), Root[-1+12 #1-20 ((#1)^2)+8 ((#1)^3)&,2], ]),.999},{y,-5,5},WorkingPrecision->80,PlotPoints->80]

ContourPlot[Evaluate[{y==1/q(-q+{p61+p62 ωω,p41+p42 ωω,p21+p22 ωω,p01+p02 ωω}.Table[(Sqrt[Root[iep3,2]])^j,{j,6,0,-2}])/.ωω->ω,y==1/q{-Sqrt[p71+p72 ωω],Sqrt[p51+p52 ωω],-Sqrt[p31+p32 ωω],Sqrt[p11+p12 ωω]}.Table[(Sqrt[Root[iep3,2]])^j,{j,7,1,-2}]/.ωω->ω}],{t,(NumericalApproximation[Root, (RowDefault["0.777", FEExpressionsNumericalApproximationElider]), Root[-1+12 #1-20 ((#1)^2)+8 ((#1)^3)&,2], ]),.999},{y,-5,5},WorkingPrecision->80,PlotPoints->80]

ContourPlot[Evaluate[{y==1/q(-q+{p61+p62 ωω,p41+p42 ωω,p21+p22 ωω,p01+p02 ωω}.Table[(-Sqrt[Root[iep3,3]])^j,{j,6,0,-2}])/.ωω->ω,y==1/q{-Sqrt[p71+p72 ωω],Sqrt[p51+p52 ωω],-Sqrt[p31+p32 ωω],Sqrt[p11+p12 ωω]}.Table[(-Sqrt[Root[iep3,3]])^j,{j,7,1,-2}]/.ωω->ω}],{t,(NumericalApproximation[Root, (RowDefault["0.777", FEExpressionsNumericalApproximationElider]), Root[-1+12 #1-20 ((#1)^2)+8 ((#1)^3)&,2], ]),.999},{y,-5,5},WorkingPrecision->80,PlotPoints->80]

Canonical bivariate poly for both lhs and rhs

qqq9=-(((-1+2 t))^33) (40960000 (t^5)-1943552000 (t^6)+45285478400 (t^7)-690435860480 (t^8)+7743927232256 (t^9)-68108919866432 (t^10)+488951367704432 (t^11)-2944750063979824 (t^12)+15175970060479888 (t^13)-67929601432109680 (t^14)+267153306116315968 (t^15)-931564200668344928 (t^16)+2901234873070618208 (t^17)-8117718281903826272 (t^18)+20504594864656612848 (t^19)-46938766854057281136 (t^20)+97689962025293254736 (t^21)-185313014207268621360 (t^22)+321040887724083234272 (t^23)-508698685285134280192 (t^24)+738004829056194882752 (t^25)-980903075179142614464 (t^26)+1194692400632446871424 (t^27)-1333127378402104336384 (t^28)+1362138681532595858944 (t^29)-1273134239034149536256 (t^30)+1086950187535630619648 (t^31)-846062083308585042944 (t^32)+598964136457371922432 (t^33)-384503711839106015232 (t^34)+223000519703744294912 (t^35)-116327416959585554432 (t^36)+54286377662854893568 (t^37)-22516675294486294528 (t^38)+8235452639907504128 (t^39)-2630483836098461696 (t^40)+725003938985705472 (t^41)-169841565680795648 (t^42)+33166940610494464 (t^43)-5261745685790720 (t^44)+654409453273088 (t^45)-60564139147264 (t^46)+3837553278976 (t^47)-142807662592 (t^48)+2147483648 (t^49))-(((-1+2 t))^33) (-2048000 (t^3)+98897920 (t^4)-2301060096 (t^5)+34284058112 (t^6)-366100346048 (t^7)+2964161793936 (t^8)-18691860011004 (t^9)+91956453666440 (t^10)-339027331297076 (t^11)+773368922964576 (t^12)+485970715647896 (t^13)-16256778555818192 (t^14)+103527969481922696 (t^15)-460126867212055792 (t^16)+1652565858521336564 (t^17)-5048648935221121816 (t^18)+13450699414800590588 (t^19)-31703027459360917632 (t^20)+66713310049532305456 (t^21)-126113618703213864096 (t^22)+215088122456108451248 (t^23)-331958151583050360640 (t^24)+464579532249156713728 (t^25)-590376046514544427072 (t^26)+681745680377918777344 (t^27)-715612717391552582400 (t^28)+682789851101715750656 (t^29)-592042736709950430720 (t^30)+466409561118141895680 (t^31)-333820399437898950656 (t^32)+217171088645076603904 (t^33)-128601163772556646400 (t^34)+69502708363299216384 (t^35)-34419756121710096384 (t^36)+15691832007097171968 (t^37)-6607933276682190848 (t^38)+2568139456612139008 (t^39)-913878080313819136 (t^40)+292975978453532672 (t^41)-82638299907751936 (t^42)+19910855360512000 (t^43)-3955441272881152 (t^44)+620075182718976 (t^45)-72266448633856 (t^46)+5709353713664 (t^47)-257698037760 (t^48)+4294967296 (t^49)) y-(((-1+2 t))^33) (-4096+226304 t-6104320 (t^2)+106074496 (t^3)-1324324336 (t^4)+12568448724 (t^5)-93608240131 (t^6)+556354788196 (t^7)-2649553376841 (t^8)+10004029859306 (t^9)-28951916123562 (t^10)+59415590450564 (t^11)-77505287576674 (t^12)+176745320100456 (t^13)-1858385126219215 (t^14)+14082876445044376 (t^15)-73553759124965153 (t^16)+297320808330105826 (t^17)-990278962499888772 (t^18)+2814306694136055344 (t^19)-6972020462350010132 (t^20)+15272227070460526008 (t^21)-29882305705517548848 (t^22)+52630072431602463600 (t^23)-83949745932641885408 (t^24)+121888599331638854656 (t^25)-161772260127566524672 (t^26)+196952719105805828224 (t^27)-220553400610110662912 (t^28)+227577950035522799104 (t^29)-216519086804267177216 (t^30)+189819896193008328704 (t^31)-153040708719458708736 (t^32)+113097452158906555904 (t^33)-76262504269633038336 (t^34)+46661088994199511040 (t^35)-25737575460889006080 (t^36)+12705578545489403904 (t^37)-5568947040026361856 (t^38)+2148594644742111232 (t^39)-722991853128122368 (t^40)+210141777697439744 (t^41)-52239057422385152 (t^42)+10994821218959360 (t^43)-1936636476653568 (t^44)+280320846856192 (t^45)-32093069377536 (t^46)+2666906255360 (t^47)-132070244352 (t^48)+2147483648 (t^49)) (y^2)-(((-1+2 t))^33) (-9728 (t^3)+319616 (t^4)-4262560 (t^5)+21231272 (t^6)+174017144 (t^7)-4358709897 (t^8)+46221876998 (t^9)-346041528932 (t^10)+2115496257784 (t^11)-11537827945399 (t^12)+59243824444198 (t^13)-289966450760380 (t^14)+1328416837044532 (t^15)-5559254885446588 (t^16)+20912000519681028 (t^17)-70248423450837684 (t^18)+210661040704675080 (t^19)-565417341215564252 (t^20)+1363135099498485384 (t^21)-2962035232396486768 (t^22)+5817500497339138992 (t^23)-10346745192826359104 (t^24)+16679998567994197376 (t^25)-24372090387510151232 (t^26)+32244610031834865792 (t^27)-38550755854067812864 (t^28)+41525171375735604480 (t^29)-40127474227170270208 (t^30)+34581308273774562304 (t^31)-26352234840173588480 (t^32)+17530083121667678208 (t^33)-9964650758102171648 (t^34)+4644220093518381056 (t^35)-1598290785896235008 (t^36)+240060898049327104 (t^37)+164502136755585024 (t^38)-177224368158670848 (t^39)+100248003193864192 (t^40)-41072046192459776 (t^41)+12939930401505280 (t^42)-3147949555056640 (t^43)+576997851922432 (t^44)-75406338162688 (t^45)+6282463412224 (t^46)-251255586816 (t^47)) (y^3)-(((-1+2 t))^33) (128 (t^5)-6672 (t^6)+187232 (t^7)-3746757 (t^8)+58668800 (t^9)-745763428 (t^10)+7849951402 (t^11)-69467928703 (t^12)+523633324918 (t^13)-3399714007506 (t^14)+19188108219416 (t^15)-94847507984346 (t^16)+413060552367952 (t^17)-1592556998134424 (t^18)+5457724439694136 (t^19)-16682213545947196 (t^20)+45618158313068376 (t^21)-111908238505973808 (t^22)+246912778434890576 (t^23)-491178722971778912 (t^24)+882986026840374848 (t^25)-1437557480495007040 (t^26)+2123705498248846976 (t^27)-2851318733313996032 (t^28)+3482789737286860288 (t^29)-3871253606239122432 (t^30)+3912883543987812352 (t^31)-3589394351929233408 (t^32)+2978555229037125632 (t^33)-2225509571674685440 (t^34)+1488208430438318080 (t^35)-884019654558154752 (t^36)+462290661732909056 (t^37)-210542393341771776 (t^38)+82424345549864960 (t^39)-27289732007329792 (t^40)+7482714041090048 (t^41)-1651598350614528 (t^42)+281679574859776 (t^43)-34798294794240 (t^44)+2767837986816 (t^45)-106300440576 (t^46)) (y^4)-(((-1+2 t))^33) (152 (t^8)-7398 (t^9)+165929 (t^10)-2211278 (t^11)+18132711 (t^12)-68726802 (t^13)-413256616 (t^14)+9534928924 (t^15)-93166858152 (t^16)+642663579152 (t^17)-3486379980012 (t^18)+15547377177468 (t^19)-58329088678740 (t^20)+186582400399240 (t^21)-512787829217984 (t^22)+1215033657475072 (t^23)-2480957459206208 (t^24)+4342539884621440 (t^25)-6432194348111360 (t^26)+7834870970099712 (t^27)-7297962682238976 (t^28)+3901717429653504 (t^29)+2127573466480640 (t^30)-9119737862225920 (t^31)+14631841735737344 (t^32)-16781407917899776 (t^33)+15285877524660224 (t^34)-11431014044008448 (t^35)+7090339839737856 (t^36)-3645066435887104 (t^37)+1540208050831360 (t^38)-526658570289152 (t^39)+142198158917632 (t^40)-29182012686336 (t^41)+4276210368512 (t^42)-398358216704 (t^43)+17716740096 (t^44)) (y^5)-(((-1+2 t))^33) (-(t^10)+60 (t^11)-1743 (t^12)+32654 (t^13)-443355 (t^14)+4648056 (t^15)-39146185 (t^16)+272061558 (t^17)-1590536844 (t^18)+7934058600 (t^19)-34135077600 (t^20)+127717012800 (t^21)-418214409600 (t^22)+1204361107200 (t^23)-3061184889600 (t^24)+6885051486720 (t^25)-13725049881600 (t^26)+24268379535360 (t^27)-38059283988480 (t^28)+52891781529600 (t^29)-65022693212160 (t^30)+70522375372800 (t^31)-67228313518080 (t^32)+56050744688640 (t^33)-40606924800000 (t^34)+25349271846912 (t^35)-13487906488320 (t^36)+6029853065216 (t^37)-2221563445248 (t^38)+656576348160 (t^39)-149602435072 (t^40)+24662507520 (t^41)-2617245696 (t^42)+134217728 (t^43)) (y^6);

Solve[qqq9==0/.t->4/5,Reals]

ContourPlot[Evaluate[{qqq9==0}],{t,(NumericalApproximation[Root, (RowDefault["0.777", FEExpressionsNumericalApproximationElider]), Root[-1+12 #1-20 ((#1)^2)+8 ((#1)^3)&,2], ]),.999},{y,-1,1},WorkingPrecision->100,PlotPoints->120]

ContourPlot[Evaluate[{qqq9==0,y==1/q(-q+{p61+p62 ωω,p41+p42 ωω,p21+p22 ωω,p01+p02 ωω}.Table[(-Sqrt[Root[iep3,1]])^j,{j,6,0,-2}])/.ωω->ω}],{t,(NumericalApproximation[Root, (RowDefault["0.777", FEExpressionsNumericalApproximationElider]), Root[-1+12 #1-20 ((#1)^2)+8 ((#1)^3)&,2], ]),.999},{y,-1,1},WorkingPrecision->100,PlotPoints->120]

RootReduce[{Root[qqq9/.t->4/5,2],Root[qqq9/.t->4/5,3],Root[qqq9/.t->4/5,4],1/q(-q+{p61+p62 ωω,p41+p42 ωω,p21+p22 ωω,p01+p02 ωω}.Table[(-Sqrt[Root[iep3,1]])^j,{j,6,0,-2}])/.ωω->ω/.t->4/5,1/q(-q+{p61+p62 ωω,p41+p42 ωω,p21+p22 ωω,p01+p02 ωω}.Table[(Sqrt[Root[iep3,2]])^j,{j,6,0,-2}])/.ωω->ω/.t->4/5,1/q(-q+{p61+p62 ωω,p41+p42 ωω,p21+p22 ωω,p01+p02 ωω}.Table[(-Sqrt[Root[iep3,3]])^j,{j,6,0,-2}])/.ωω->ω/.t->4/5}]

ContourPlot[Evaluate[{qqq9==0,y==1/q(-q+{p61+p62 ωω,p41+p42 ωω,p21+p22 ωω,p01+p02 ωω}.Table[(Sqrt[Root[iep3,2]])^j,{j,6,0,-2}])/.ωω->ω}],{t,(NumericalApproximation[Root, (RowDefault["0.777", FEExpressionsNumericalApproximationElider]), Root[-1+12 #1-20 ((#1)^2)+8 ((#1)^3)&,2], ]),.999},{y,-1,1},WorkingPrecision->100,PlotPoints->120]

ContourPlot[Evaluate[{qqq9==0,y==1/q(-q+{p61+p62 ωω,p41+p42 ωω,p21+p22 ωω,p01+p02 ωω}.Table[(-Sqrt[Root[iep3,3]])^j,{j,6,0,-2}])/.ωω->ω}],{t,(NumericalApproximation[Root, (RowDefault["0.777", FEExpressionsNumericalApproximationElider]), Root[-1+12 #1-20 ((#1)^2)+8 ((#1)^3)&,2], ]),.999},{y,-1,1},WorkingPrecision->100,PlotPoints->120]