This is a Mathematica notebook that supports Section 3 of the paper. The first two lines need to be modified according to your system.

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SetDirectory["DeskTop/EnricoFPP/paperfiles"];

Consider the symplectic form x_{16}+x_{25}+x_{43}. The dim 14 subspace is the kernel of the contraction map.

Weight 0: e124, e356
Weight 1: e456, e125-e143
Weight 2: e135, e243-e216
Weight 4: e263, e416-e425
Weight 3: e415, e316-e325
Weight 5: e246, e543-e516
Weight 6: e123, e625-e643

GrPt=Transpose[{{1,0,x1, 0,x2,x3},{0,1,x4,0,x5,x6},{0,0,x7,1,x8,x9}}];
GrPt//MatrixForm
e124=Det[{GrPt[[1]],GrPt[[2]],GrPt[[4]]}];
e356=Det[{GrPt[[3]],GrPt[[5]],GrPt[[6]]}];
e456=Det[{GrPt[[4]],GrPt[[5]],GrPt[[6]]}];
e125=Det[{GrPt[[1]],GrPt[[2]],GrPt[[5]]}];
e143=Det[{GrPt[[1]],GrPt[[4]],GrPt[[3]]}];
e135=Det[{GrPt[[1]],GrPt[[3]],GrPt[[5]]}];
e243=Det[{GrPt[[2]],GrPt[[4]],GrPt[[3]]}];
e216=Det[{GrPt[[2]],GrPt[[1]],GrPt[[6]]}];
e263=Det[{GrPt[[2]],GrPt[[6]],GrPt[[3]]}];
e416=Det[{GrPt[[4]],GrPt[[1]],GrPt[[6]]}];
e425=Det[{GrPt[[4]],GrPt[[2]],GrPt[[5]]}];
e415=Det[{GrPt[[4]],GrPt[[1]],GrPt[[5]]}];
e316=Det[{GrPt[[3]],GrPt[[1]],GrPt[[6]]}];
e325=Det[{GrPt[[3]],GrPt[[2]],GrPt[[5]]}];
e246=Det[{GrPt[[2]],GrPt[[4]],GrPt[[6]]}];
e625=Det[{GrPt[[6]],GrPt[[2]],GrPt[[5]]}];
e643=Det[{GrPt[[6]],GrPt[[4]],GrPt[[3]]}];
e123=Det[{GrPt[[1]],GrPt[[2]],GrPt[[3]]}];
e543=Det[{GrPt[[5]],GrPt[[4]],GrPt[[3]]}];
e516=Det[{GrPt[[5]],GrPt[[1]],GrPt[[6]]}];

(* this is a general form of the C_3 semidirect C_14 invariant complete intersections *)

Rels=Expand[{p0 e124 + e356,p1 e456 + e125-e143,p2 e135 + e243-e216,p4 e263 + e416-e425,
p3 e415 + e316-e325,p5 e246 + e543-e516,p6 e123 + e625-e643}];

VarsX={x1,x2,x3,x4,x5,x6,x7,x8,x9};

SolsP0P3={};
solp={p1->1,p2->1,p4->1,p3->1,p5->1,p6->1};
Relsp=Rels/.solp

(* We look at the minors of the Jacobian matrix to find when the complete intersection is singular. We exclude p0=0 case since it does not have free action of C_7. The computation takes awhile, but under an hour. *)

JacMat = Table[D[Relsp,VarsX[[k]]],{k,1,9}];
Min7=Expand[Minors[JacMat,7]];
Eqs1=Table[Relsp[[k]]==0,{k,1,7}];
Eqs2=Table[Min7[[k]]==0,{k,1,36}];
Print["solving equations"];
maybe=NSolve[Join[Join[Eqs1,Eqs2],{p0≠0}],WorkingPrecision->200];
Print["done solving equations"];

We collect the values of p0, avoiding repetition and erroneous values close to zero. For each value we pick one singular point.

Length[maybe]
count=0;
newmaybe={};
For[k=1,k≤Length[maybe],k++,
If[Abs[(p0/.maybe[[k]])]^2>10^(-30),
For[j=1,j≤count,j++,
If[Abs[(p0/.newmaybe[[j]])-(p0/.maybe[[k]]) ]<10^(-30) ,
j=count+100;
];
];
If[j<count+100,
count++;
AppendTo[newmaybe,maybe[[k]]];
];
];
];

We will now compute the polynomial whose roots are the first eight values of p0 for p3=1. There were two spurious solutions which we were not interested in (but perhaps should look at in more detail!).

RootApproximant[p0/.newmaybe]
PolyForp3equal1 = Product[p0-(p0/.newmaybe[[k]]),{k,1,8}];
PolyForp3equal1=Simplify[999Sum[Rationalize[SeriesCoefficient[PolyForp3equal1,{p0,0,k}]]p0^k,{k,0,8}]]

We will now look for the solutions to a slightly different p3.

prec=5000;
solp3={p1->1,p2->1,p4->1,p5->p3,p6->p3};
Relsp3=Rels/.solp3;
JacMatp3 =Table[D[Relsp3,VarsX[[k]]],{k,1,9}];
Min7p3=Minors[JacMatp3,7];
Eqs1p3=Table[Relsp3[[k]]==0,{k,1,7}];
Eqs2p3={Min7p3[[1]]==0,Min7p3[[7]]==0,Min7p3[[15]]==0};
Eqsp3=Join[Eqs1p3,Eqs2p3];

Eqsp3good=Eqsp3/.p3->1+1/10^20;

eightp0={};
For[j=1,j≤8,j++,
Print[j];
fr=FindRoot[Eqsp3good,{{p0,p0/.newmaybe[[j]]},{x1,x1/.newmaybe[[j]]},{x2,x2/.newmaybe[[j]]},{x3,x3/.newmaybe[[j]]},{x4,x4/.newmaybe[[j]]},{x5,x5/.newmaybe[[j]]},{x6,x6/.newmaybe[[j]]},{x7,x7/.newmaybe[[j]]},{x8,x8/.newmaybe[[j]]},{x9,x9/.newmaybe[[j]]}},AccuracyGoal->prec,PrecisionGoal->prec,WorkingPrecision->prec+500];

AppendTo[eightp0,p0/.fr];
];

We are looking for a nice polynomial in two variables which restricts to Product[A-eightp0[[j]],{j,1,8}] for p3=1+10^(-20). This is slightly tricky. We were helped by a nice pattern of signs of the terms.

poly=Product[A-eightp0[[j]],{j,1,8}];
polynew=Sum[Rationalize[SeriesCoefficient[poly,{A,0,j}]] A^j,{j,0,8}]
polyblah=Expand[10 Numerator[Factor[polynew]]];

runr=0;
For[j=0,j≤8,j++,
c0=(-1)^j SeriesCoefficient[polyblah,{A,0,j}];
runc0=c0;
For[i=0,i≤30,i++,
n=Mod[runc0,10^20];
runr+=n r^i A^j (-1)^j;
runc0=(runc0-n)/10^20;
];
];
runr
polyblah-runr/.r->10^20
Factor[Simplify[runr/.(r->1/s)/.(s->p-1)](p-1)^20/2]

FinalPoly=Factor[Simplify[runr/.(r->1/s)/.(s->p-1)](p-1)^20/2]/.{A->P0,p->P3}
sols=Solve[{D[FinalPoly,P0]==0,D[FinalPoly,P3]==0,FinalPoly==0,P0≠0}];

We should get a symmetry with p3->p3, p0 -> p0^{-1} p3^3. And we do see it in our polynomial.

Simplify[(FinalPoly/.(P0->P3^3/P0))/FinalPoly]

sols=FullSimplify[sols]

The true parameter is P0 + P3^3/ P0. The reasonable guess is that 9-12 give two FPP/C_3 s, conjugate to each other.
Surfaces 9-10 are the same and 11-12 are the same.

Here we are going to see for which of the solutions the group action is free. The C_7 is always ok, but C_2 requires a calculation.

FullSimplify[(P0+P3^3/P0)/.sols]

For[m=1,m≤Length[sols],m++,
solp3={p0->(P0/.sols[[m]]),p1->1,p2->1,p4->1,p3->(P3/.sols[[m]]),p5->(P3/.sols[[m]]),p6->(P3/.sols[[m]])};
Relsp3=Rels/.solp3;
Eqs = Join[Table[Relsp3[[k]]==0,{k,1,7}],{x1==x9,x4==x8,x2==x6}];
blah=NSolve[Eqs];
If[Length[blah]==0,Print[m]];
];

We see that m=3,4 m=7,8 m=9,...,12 are candidates for FPP quotients -- there are three possibilities to explore. We looked at m=12, since the field of definition was consistent with our expectations.

Here is how the singular points were found. I used Magma to find the equations for x9 for singular points. It looks like it is the root of the polynomial below. Then I changed the order of variables and found equations for the others, also below. This should be enough to find the points. The Magma file appears to be lost, but it can be easily reproduced by the reader.

eq1=x1+(1/26621783451473574858136930628130253984787303116897196074752481951744*(-4210824707957225035844576438040052618416616173802226186048476851423*(I (Sqrt[15]))+17972434963131763927469955197608544276914647049639619210601506784519)*(I (Sqrt[7]))+1/26621783451473574858136930628130253984787303116897196074752481951744*(12277510414108135920188173779894064727105936400121217681294623528533*(I (Sqrt[15]))+43148113466523888614585614038840485863489853632071699862714903313907))*x8*x7^20+(1/50777022269198560444139348274479396791052442773622886800293888*(25048511410302101811311208126368856086294368518713341358566995*(I (Sqrt[15]))+18314131705334055755273948203064276471518377321155197159862181)*(I (Sqrt[7]))+1/50777022269198560444139348274479396791052442773622886800293888*(12510933722607662467268957649016350224278690813708165034605887*(I (Sqrt[15]))-256670863238493897927768485955881979346509031205232406292162151))*x8*x7^13+(1/193698968006891481186444657419126116909227152914516017152*(-28491157632055800868292985281728047061698323510797472901*(I (Sqrt[15]))-215847607801371081249638863730391599573907285050916489027)*(I (Sqrt[7]))+1/193698968006891481186444657419126116909227152914516017152*(-147431561918551524595216538757820954887450654663405962415*(I (Sqrt[15]))+291899660606877667890186322536346946415840786975237338199))*x8*x7^6+(1/23570113691595342654033858854518088084332593563096027236401152*(5043258063867711122446027669504993225351160796922969831061211*(I (Sqrt[15]))+20507713011747017400133751671921916852002130627269973232079965)*(I (Sqrt[7]))+1/481022728399904952123139976622818124170052929859102596661248*(285906837488163594677452304292109269401558318855434304814471*(I (Sqrt[15]))-1054653409805726259182779675900383500799975751123960170333039))*x7^19+(1/22478212062449782041581972937124336323101609767051722752*(-1248291380323860954412265927964916596126507438164567727*(I (Sqrt[15]))-60684525138709522581029042340045393066196001242049837481)*(I (Sqrt[7]))+1/401396643972317536456820945305791720055385888697352192*(-740275411865663354140869046903625242381405202704721877*(I (Sqrt[15]))+228413937486701555235282774585118592891516327650760589))*x7^12+(1/1371961185452257204686399715400655293158057236758528*(-318606192644925934422719221975362881598736294396083*(I (Sqrt[15]))+1658327823700693152408124090647372836117790108343227)*(I (Sqrt[7]))+1/195994455064608172098057102200093613308293890965504*(161368094320928555011874513839679905647839824738425*(I (Sqrt[15]))+462430285302904442205732802181812414352742544102767))*x7^5;
eq2=x2+(1/42738193457108508892173814786070875946205972009240303239168*(-255044428480922064091736382460336334028276346545460080051*(I (Sqrt[15]))-1269144816895799429189565946778262591926144652730132854597)*(I (Sqrt[7]))+1/42738193457108508892173814786070875946205972009240303239168*(-866990963573641186491815458148425627691006721217986707343*(I (Sqrt[15]))+2613427701776915405662128394628897404064369177211402705975))*x8*x7^18+(1/81516634859292047294948224613324882404720253008347136*(-17064052107294337250093543504219208430131992182761*(I (Sqrt[15]))-14787254137501560721751833485526683190633735066251135)*(I (Sqrt[7]))+1/81516634859292047294948224613324882404720253008347136*(-10101618345245414590832832527225169492129115858633965*(I (Sqrt[15]))+174854806371193345823791528574269741313266877112549))*x8*x7^11+(1/310961284100692929439347170308398751849061023744*(2247791973908483265807047024257799751529198407*(I (Sqrt[15]))-38254247994586552587848890077190592927430259503)*(I (Sqrt[7]))+1/310961284100692929439347170308398751849061023744*(-26526034075420956241250977463426780204535537291*(I (Sqrt[15]))-27028578032212011690392438108452367493709615149))*x8*x7^4+(1/75678181410621084705373510429777588016712184116543488*(-7598398145869122848488153929420362471304689064300231*(I (Sqrt[15]))+12223996009332416141133579821275510039754201811690159)*(I (Sqrt[7]))+1/75678181410621084705373510429777588016712184116543488*(8350579018060612090245420263988294967064074798000461*(I (Sqrt[15]))+77860411700888284988186844279466746458417098708240827))*x7^17+(1/144344675847284478579279919490389991792129867776*(25128157849720158540425652128302226613685687077*(I (Sqrt[15]))+43368309046005230085373518060368710882555359651)*(I (Sqrt[7]))+1/144344675847284478579279919490389991792129867776*(29626184655067834287400512765430612403039882133*(I (Sqrt[15]))-257486992924051013263010635978419735722044106061))*x7^10+(1/2202524961048652322071531974645843380617216*(75760474780385462352441201990535323885093*(I (Sqrt[15]))-962614282995361316621648743715938494554013)*(I (Sqrt[7]))+1/2202524961048652322071531974645843380617216*(-683681865948997067608570432235709289734153*(I (Sqrt[15]))-1075285639187228292043626912172060189389087))*x7^3;eq3=x3+(1/17152806303404597621344857305026651862355436109824*(-506152598942570599023448990906726611612643358415*(I (Sqrt[15]))-1878212609102487956539597179902342663184863121033)*(I (Sqrt[7]))+1/17152806303404597621344857305026651862355436109824*(-1283063475568660816882691843487235111091789865051*(I (Sqrt[15]))+5186520761415355266645228816808979753776040712483))*x7^16+(1/32716381651696391337098803148320487713538048*(1647375634856657404315304293277054011327947*(I (Sqrt[15]))+14065560363582786121837448329008279643679469)*(I (Sqrt[7]))+1/32716381651696391337098803148320487713538048*(9608612406872301667809978535281028855706919*(I (Sqrt[15]))-16880643051799937526082677573875560573069839))*x7^9+(1/62401545813935072588155370995179152896*(-3001421011246707180220143781201938947*(I (Sqrt[15]))-35197931151681569983649882956528164549)*(I (Sqrt[7]))+1/62401545813935072588155370995179152896*(-10807378505530574465343860652296463005*(I (Sqrt[15]))+29791117511986356893217391913586455205))*x7^2;eq4=x4+x8+(1/831930732858549214316779082129070437024603222403037377336015060992*(-115118881212715722421538583731443616323735045370387489126557129653*(I (Sqrt[15]))+18521820835908783461031320469904315840028399987292211909416783981)*(I (Sqrt[7]))+1/831930732858549214316779082129070437024603222403037377336015060992*(12652812413433169437698293045827465948853704644449408724466485511*(I (Sqrt[15]))+1179617508019054715717647741439480807757910982161639661438843305233))*x7^20+(1/1586781945912455013879354633577481149720388836675715212509184*(689270795310592439982660581090183709583956707961452136832089*(I (Sqrt[15]))+1332646466483896789648562628145165570626651603767154799891599)*(I (Sqrt[7]))+1/1586781945912455013879354633577481149720388836675715212509184*(910370844867762660678007927824763596415483661493893440865421*(I (Sqrt[15]))-7062923880711038427235649923128411086914240775984872623525765))*x7^13+(1/3026546375107679393538197772173845576706674264289312768*(-825729989311280896449135739410208679317752654297970287*(I (Sqrt[15]))-3878806772039759136416367949997293563393211441914558441)*(I (Sqrt[7]))+1/3026546375107679393538197772173845576706674264289312768*(-2655234866142678091710028490338714600517279673934236921*(I (Sqrt[15]))+8454539203883966246523740368474166664989555852153166609))*x7^6;eq5=x5+(1/10684548364277127223043453696517718986551493002310075809792*(-35257791027051122400546474206205787926167396118571190211*(I (Sqrt[15]))-6244533137544671611802117074682249402585219953429397938101)*(I (Sqrt[7]))+1/10684548364277127223043453696517718986551493002310075809792*(-4265828241121450237758284211137025147346597761184754466015*(I (Sqrt[15]))+361284848771417757963141461464642198592820394712955163911))*x7^18+(1/20379158714823011823737056153331220601180063252086784*(-5645131586557643603987336449721517752523383748989361*(I (Sqrt[15]))+27532693465731944983337630445171886446131764272857865)*(I (Sqrt[7]))+1/20379158714823011823737056153331220601180063252086784*(18808409700628617651813829528637714652087924720097563*(I (Sqrt[15]))+57845388139052438976282249502658905660626291272731677))*x7^11+(1/4858770064073327022489799536068730497641578496*(1323734741751922874644754683173073478805827565*(I (Sqrt[15]))-1990334338304848124655657524485199144609480805)*(I (Sqrt[7]))+1/9717540128146654044979599072137460995283156992*(-2884973821562830039555561999888895490582721533*(I (Sqrt[15]))-27483409800261213350327583256909797211169312139))*x7^4;eq6=x6+(1/42738193457108508892173814786070875946205972009240303239168*(255044428480922064091736382460336334028276346545460080051*(I (Sqrt[15]))+1269144816895799429189565946778262591926144652730132854597)*(I (Sqrt[7]))+1/42738193457108508892173814786070875946205972009240303239168*(866990963573641186491815458148425627691006721217986707343*(I (Sqrt[15]))-2613427701776915405662128394628897404064369177211402705975))*x8*x7^18+(1/81516634859292047294948224613324882404720253008347136*(17064052107294337250093543504219208430131992182761*(I (Sqrt[15]))+14787254137501560721751833485526683190633735066251135)*(I (Sqrt[7]))+1/81516634859292047294948224613324882404720253008347136*(10101618345245414590832832527225169492129115858633965*(I (Sqrt[15]))-174854806371193345823791528574269741313266877112549))*x8*x7^11+(1/310961284100692929439347170308398751849061023744*(-2247791973908483265807047024257799751529198407*(I (Sqrt[15]))+38254247994586552587848890077190592927430259503)*(I (Sqrt[7]))+1/310961284100692929439347170308398751849061023744*(26526034075420956241250977463426780204535537291*(I (Sqrt[15]))+27028578032212011690392438108452367493709615149))*x8*x7^4+(1/75678181410621084705373510429777588016712184116543488*(-740108262758202953830717697651213246076388137876705*(I (Sqrt[15]))-45296071075280730875861851920848181004335301022785863)*(I (Sqrt[7]))+1/75678181410621084705373510429777588016712184116543488*(-30943107346659895339571956553822930883166654612129557*(I (Sqrt[15]))+7583852929966002785373042277318816014122629565502669))*x7^17+(1/144344675847284478579279919490389991792129867776*(-17020343630324830206148081319213359783034561613*(I (Sqrt[15]))+171473842529246380209699037886478551392537426245)*(I (Sqrt[7]))+1/144344675847284478579279919490389991792129867776*(117138923959516017462302492475981108232595296515*(I (Sqrt[15]))+174406539493996340141325228537411867952808921205))*x7^10+(1/2202524961048652322071531974645843380617216*(59445362172274732708964592895838605829763*(I (Sqrt[15]))-1052166325842387274073650608354859755060363)*(I (Sqrt[7]))+1/2202524961048652322071531974645843380617216*(-602520882269351803939511816826500255421375*(I (Sqrt[15]))-1027447056041392868129210199890819052692889))*x7^3;eq9=x9+(1/26621783451473574858136930628130253984787303116897196074752481951744*(4210824707957225035844576438040052618416616173802226186048476851423*(I (Sqrt[15]))-17972434963131763927469955197608544276914647049639619210601506784519)*(I (Sqrt[7]))+1/26621783451473574858136930628130253984787303116897196074752481951744*(-12277510414108135920188173779894064727105936400121217681294623528533*(I (Sqrt[15]))-43148113466523888614585614038840485863489853632071699862714903313907))*x8*x7^20+(1/50777022269198560444139348274479396791052442773622886800293888*(-25048511410302101811311208126368856086294368518713341358566995*(I (Sqrt[15]))-18314131705334055755273948203064276471518377321155197159862181)*(I (Sqrt[7]))+1/50777022269198560444139348274479396791052442773622886800293888*(-12510933722607662467268957649016350224278690813708165034605887*(I (Sqrt[15]))+256670863238493897927768485955881979346509031205232406292162151))*x8*x7^13+(1/193698968006891481186444657419126116909227152914516017152*(28491157632055800868292985281728047061698323510797472901*(I (Sqrt[15]))+215847607801371081249638863730391599573907285050916489027)*(I (Sqrt[7]))+1/193698968006891481186444657419126116909227152914516017152*(147431561918551524595216538757820954887450654663405962415*(I (Sqrt[15]))-291899660606877667890186322536346946415840786975237338199))*x8*x7^6+(1/11785056845797671327016929427259044042166296781548013618200576*(773621440695416239879582035019041808492763431455628921646997*(I (Sqrt[15]))+821053728135970012942970935582906067861167322412369455781555)*(I (Sqrt[7]))+1/11785056845797671327016929427259044042166296781548013618200576*(560886475227785342707783615339388852856162059367808155436377*(I (Sqrt[15]))-7927260814296500250403010561859199677682370854471120436522097))*x7^19+(1/44956424124899564083163945874248672646203219534103445504*(-4573677685089714151048208235200157343191176046526545723*(I (Sqrt[15]))-61840606941643610103777887236067397867379361871983917821)*(I (Sqrt[7]))+1/44956424124899564083163945874248672646203219534103445504*(-42245177329148340393892146052110622711637460210257302461*(I (Sqrt[15]))+46866250346777852812186271981712592418837333713529728821))*x7^12+(1/1371961185452257204686399715400655293158057236758528*(-149359665426252530998748458155655286554617706520013*(I (Sqrt[15]))+1460764659761311981069790624300127398463391938864837)*(I (Sqrt[7]))+1/1371961185452257204686399715400655293158057236758528*(995821526378021932539175869510614807819178658386033*(I (Sqrt[15]))+1514023879672033209003188007429278404422535020108855))*x7^5;eq8=x8^2+(1/831930732858549214316779082129070437024603222403037377336015060992*(-115118881212715722421538583731443616323735045370387489126557129653*(I (Sqrt[15]))+18521820835908783461031320469904315840028399987292211909416783981)*(I (Sqrt[7]))+1/831930732858549214316779082129070437024603222403037377336015060992*(12652812413433169437698293045827465948853704644449408724466485511*(I (Sqrt[15]))+1179617508019054715717647741439480807757910982161639661438843305233))*x8*x7^20+(1/1586781945912455013879354633577481149720388836675715212509184*(689270795310592439982660581090183709583956707961452136832089*(I (Sqrt[15]))+1332646466483896789648562628145165570626651603767154799891599)*(I (Sqrt[7]))+1/1586781945912455013879354633577481149720388836675715212509184*(910370844867762660678007927824763596415483661493893440865421*(I (Sqrt[15]))-7062923880711038427235649923128411086914240775984872623525765))*x8*x7^13+(1/3026546375107679393538197772173845576706674264289312768*(-825729989311280896449135739410208679317752654297970287*(I (Sqrt[15]))-3878806772039759136416367949997293563393211441914558441)*(I (Sqrt[7]))+1/3026546375107679393538197772173845576706674264289312768*(-2655234866142678091710028490338714600517279673934236921*(I (Sqrt[15]))+8454539203883966246523740368474166664989555852153166609))*x8*x7^6+(1/23570113691595342654033858854518088084332593563096027236401152*(6668279632264218378220797261641966810501812935671795719960671*(I (Sqrt[15]))+35925260569442503001655163809836771169827460489359963995093881)*(I (Sqrt[7]))+1/23570113691595342654033858854518088084332593563096027236401152*(24541625087289506049105505110760409381435276403825465598810923*(I (Sqrt[15]))-68329533085463334774828915306828191584659315304753068310950771))*x7^19+(1/44956424124899564083163945874248672646203219534103445504*(5382899834208725730523324116955677626622393627414849189*(I (Sqrt[15]))-243946779004467325899078784678207365499938036342008042973)*(I (Sqrt[7]))+1/44956424124899564083163945874248672646203219534103445504*(-166647375257689366597570787245491448380860804845721376663*(I (Sqrt[15]))-55158314889461253383140699667619671064654436137231164673))*x7^12+(1/10718446761345759411612497776567619477797322162176*(-6576142106485823622571545082103379532943733740547*(I (Sqrt[15]))+28758587577347410068286589209541694895673999399643)*(I (Sqrt[7]))+1/21436893522691518823224995553135238955594644324352*(39573598916375756467607967684073249504632631069845*(I (Sqrt[15]))+134863337594256928907749486519461220536772828029075))*x7^5;eq7=x7^21+(1/14155776*(511399974589*(I (Sqrt[15]))+5420204276277)*(I (Sqrt[7]))+1/14155776*(-3702697798751*(I (Sqrt[15]))+5240257947513))*x7^14+(1/3131031158784*(-113671691040691066841929*(I (Sqrt[15]))+304635178280446195198751)*(I (Sqrt[7]))+1/3131031158784*(-208105444896085436684957*(I (Sqrt[15]))-1164788221575442098979605))*x7^7+1/3116402981210161152*(-252265040575458476470499521665557*(I (Sqrt[15]))+923860473808999763070454228930611)*(I (Sqrt[7]))+1/3116402981210161152*(-631116852649128571792369859899609*(I (Sqrt[15]))-2584947450749523543998221211667441);

solssing=NSolve[{eq7==0,eq1==0,eq2==0,eq3==0,eq4==0,eq5==0,eq6==0,eq8==0,eq9==0},1000];

Length[solssing]

prec=3000;
SingPts={};
For[j=1,j≤42,j++,
fr=FindRoot[{eq7==0,eq1==0,eq2==0,eq3==0,eq4==0,eq5==0,eq6==0,eq8==0,eq9==0},{{x1,x1/.solssing[[j]]},{x2,x2/.solssing[[j]]},{x3,x3/.solssing[[j]]},{x4,x4/.solssing[[j]]},{x5,x5/.solssing[[j]]},{x6,x6/.solssing[[j]]},{x7,x7/.solssing[[j]]},{x8,x8/.solssing[[j]]},{x9,x9/.solssing[[j]]}},AccuracyGoal->prec,PrecisionGoal->prec,WorkingPrecision->prec+500];
AppendTo[SingPts,fr];
];
JacMat = Table[D[Relsp3,VarsX[[k]]],{k,1,9}];Table[MatrixRank[JacMat/.SingPts[[k]]],{k,1,42}]

We write equations of the C_7 quotient in its bicanonical map, hopefully an embedding.

For[deg=2,deg≤3,deg++,
maxlength=500;
CoeffList=Table[ToExpression[StringJoin["A",ToString[i]]],{i,1,maxlength}];
AllMons=MonomialList[(U0+U1+U2+U3+U4+U5+U6+U7)^deg];
AllMons=Table[AllMons[[k]]/(AllMons[[k]]/.{U0->1,U1->1,U2->1,U3->1,U4->1,U5->1,U6->1,U7->1}),{k,1,Length[AllMons]}];
count=Length[AllMons];
UnivPoly=Sum[CoeffList[[k]] AllMons[[k]],{k,1,count}];

solp3={p0->(1/256) (-2475+49 I (Sqrt[7])-35 I (Sqrt[15])+231 (Sqrt[105])),p1->1,p2->1,p4->1,p3->(1/16) (-17+7 I (Sqrt[15])),p5->(1/16) (-17+7 I (Sqrt[15])),p6->(1/16) (-17+7 I (Sqrt[15]))};
Relsp3=Rels/.solp3;

prec=3000;
p0point=2;
 
XtoU={U0->(-(8925/4)-((1561 I (Sqrt[7]))/4)+((1115 I (Sqrt[15]))/4)+((833 (Sqrt[105]))/4))e124^2,U1->32 (7- I (Sqrt[15]))e456  e123, U2 ->32 (7- I (Sqrt[15])) e135 e246, U3->32 (7- I (Sqrt[15]))e263 e415, U4->128e456 (e625+ e643),  U5-> 128e135(e543+e516),U6->128 e263(e316+e325),
U7->((136-56 I (Sqrt[15]))(e123 (e125+e143)+e246 (e243+e216)+e415 (e416+e425)) + 128( e456 (e625+ e643)+e135(e543+e516)+e263(e316+e325)))/3};

UnivEq={};

For[numpt=1,numpt≤count+50,numpt++,
If[Mod[numpt,100]==0,Print[numpt]];

rpt1=(RandomInteger[{1,100000}]+I RandomInteger[{1,100000}])/100000;
rpt2=(RandomInteger[{1,100000}]+I RandomInteger[{1,100000}])/100000;
rpt3=(RandomInteger[{1,100000}]+I RandomInteger[{1,100000}])/100000;
rpt4=(RandomInteger[{1,100000}]+I RandomInteger[{1,100000}])/100000;
rpt5=(RandomInteger[{1,100000}]+I RandomInteger[{1,100000}])/100000;
rpt6=(RandomInteger[{1,100000}]+I RandomInteger[{1,100000}])/100000;
rpt7=(RandomInteger[{1,100000}]+I RandomInteger[{1,100000}])/100000;
rpt8=(RandomInteger[{1,100000}]+I RandomInteger[{1,100000}])/100000;
rpt9=(RandomInteger[{1,100000}]+I RandomInteger[{1,100000}])/100000;

EqsToSolve=Table[Relsp3[[i]]==0,{i,1,7}]/.{x1->rpt1,x2->rpt2};
sols=FindRoot[EqsToSolve,{{x3,rpt3},{x4,rpt4},{x5,rpt5},{x6,rpt6},{x7,rpt7},{x8,rpt8},{x9,rpt9}},AccuracyGoal->prec,PrecisionGoal->prec,WorkingPrecision->prec+100];
pt=({U0,U1,U2,U3,U4,U5,U6,U7}/.XtoU)/.sols/.x1->rpt1/.x2->rpt2;
AppendTo[UnivEq,0==UnivPoly/.{U0->pt[[1]],U1->pt[[2]],U2->pt[[3]],U3->pt[[4]],U4->pt[[5]],U5->pt[[6]],U6->pt[[7]],U7->pt[[8]]}];
];


Print["starting big matrix calculation"];
sols=NSolve[UnivEq];
Print["done with matrix calculation"];
sols=Chop[sols[[1]]];

AllEqs={};

For[k=1,k≤count+10,k++,
UEq=Coefficient[UnivPoly/.sols,CoeffList[[k]]];
If[ByteCount[UEq]>100,
Print[CoeffList[[k]]];
Eq=Sum[RootApproximant[Coefficient[UEq, AllMons[[k]]],4] AllMons[[k]],{k,1,count}];
AppendTo[AllEqs,Eq];
];
];
If[deg==2,AllEqs2=AllEqs;AllMons2=AllMons];
If[deg==3,AllEqs3=AllEqs;AllMons3=AllMons];
];

(* This is just a reality check. *)
Length[AllEqs3]
Table[ByteCount[AllEqs3[[k]]],{k,1,Length[AllEqs3]}] 
AllEqs3=Table[Numerator[Factor[AllEqs3[[k]]]],{k,1,Length[AllEqs3]}];

(* Slight reorganization. *)
Coefficient[AllEqs2,U0];
AllEqs2[[2]]=AllEqs2[[2]]/Coefficient[AllEqs2[[2]],U0 U1];
AllEqs2[[3]]=AllEqs2[[3]]/Coefficient[AllEqs2[[3]],U0 U2];
AllEqs2=FullSimplify[Factor[AllEqs2]]

Here we clean up the equations a little bit to reduce the number of degree three equations.

VarU={U0,U1,U2,U3,U4,U5,U6,U7};
OldEqs3={};
For[i=1,i≤8,i++,For[j=1,j≤3,j++,
AppendTo[OldEqs3,VarU[[i]] AllEqs2[[j]]];
];];
CoefOldEqs3=Table[Coefficient[OldEqs3[[j]],AllMons3[[k]]],{j,1,24},{k,1,Length[AllMons3]}];
MatrixRank[CoefOldEqs3]

OldAndNewEqs3=OldEqs3;
rk=24;
For[j=1,j≤Length[AllEqs3],j++,
new=Append[OldAndNewEqs3,AllEqs3[[j]]];
Coefnew=Table[Coefficient[new[[m]],AllMons3[[k]]],{m,1,Length[new]},{k,1,Length[AllMons3]}];
If[MatrixRank[Coefnew]==rk+1,
rk++;
AppendTo[OldAndNewEqs3,AllEqs3[[j]]];
Print[rk, " ",j];
];
];

This is a useful function to have, various versions of it will be used later.

Beautify[x_]:=
Ret=RootApproximant[N[Re[x],3000]]+RootApproximant[N[Im[x]/(Sqrt[15]),3000]] I(Sqrt[15]);

Making the new list and checking the symmetry action. The first two are odd, the last four are even.

For[m=25,m≤30,m++,OldAndNewEqs3[[m]]=Numerator[Factor[Sum[Beautify[Coefficient[OldAndNewEqs3[[m]],AllMons3[[j]]]]AllMons3[[j]],{j,1,Length[AllMons3]}]]];];

Sym2={U4->-U4,U5->-U5,U6->-U6,U7->-U7};
OldAndNewEqs3[[25]]+(OldAndNewEqs3[[25]]/.Sym2)
OldAndNewEqs3[[26]]+(OldAndNewEqs3[[26]]/.Sym2)
OldAndNewEqs3[[27]]-(OldAndNewEqs3[[27]]/.Sym2)
OldAndNewEqs3[[28]]-(OldAndNewEqs3[[28]]/.Sym2)
OldAndNewEqs3[[29]]-(OldAndNewEqs3[[29]]/.Sym2)
OldAndNewEqs3[[30]]-(OldAndNewEqs3[[30]]/.Sym2)

We write the output into a file called “EquationsOfXintermsofU”.

AllEqsU=Join[Table[AllEqs2[[k]]==0,{k,1,3}],Table[OldAndNewEqs3[[k]]==0,{k,25,30}]];
AllEqs23 = Join[Table[AllEqs2[[k]],{k,1,3}],Table[OldAndNewEqs3[[k]],{k,25,30}]];
fname="EquationsOfXintermsofU";
OpenWrite[fname];
Write[fname,AllEqs23];
Close[fname];

We will now compute the coordinates of the singular points, first approximately and then precisely.

Factor[Beautify[RootApproximant[(U1+U2+U3)/U0/.XtoU/.SingPts[[1]]]]]
Factor[Beautify[RootApproximant[(U1^2+U2^2+U3^2)/U0^2/.XtoU/.SingPts[[1]]]]]
Factor[Beautify[RootApproximant[(U1 U2^2+U2 U3^2+U3 U1^2)/U0^3/.XtoU/.SingPts[[1]]]]]
solssingU=NSolve[Join[{U0==1,-(1/9) I (-503 I+21 (Sqrt[15]))U0^3-(U1 U2^2+U2 U3^2+U3 U1^2)==0,17 U0^2 + 3(U1^2+U2^2+U3^2)==0,5U0+U1+U2+U3==0},AllEqsU],400];
Length[solssingU]

SingPtsExact=Table[RootApproximant[VarU/.solssingU[[k]],24],{k,1,Length[solssingU]}];
solssingU=NSolve[Join[{U0==1,-(1/9) I (-503 I+21 (Sqrt[15]))U0^3-(U1 U2^2+U2 U3^2+U3 U1^2)==0,17 U0^2 + 3(U1^2+U2^2+U3^2)==0,5U0+U1+U2+U3==0},AllEqsU],4000];
For[i=1,i≤6,i++,Print[N[solssingU[[i]]]]];
solssingUexact=Table[Table[VarU[[j]]->SingPtsExact[[i]][[j]],{j,1,8}],{i,1,6}];

By Magma calculation with exact coefficients, we have an embedding into P^7.
The next step is to make a change of variables from U to W so that the equations are nicer.
We will make a change of variables U to move the  points to (*:1:0:0:\pm 1:0:0:*) and cyclic.
There is still some wiggle room afterwards. Namely, one can change W0 to any multiple of W0. One can add 
multiple of W0 to W1,W2,W3 and similarly for the other indices. 

There is a unique degree 2 C_3 symmetric equation and we can try to make its coefficients nicer. The C_3 orbits of the monomials will be represented by W0^2, W0 W1, W1^2,  W1 W2, W7^2, W4 W7, W4^2, W4 W5.

We know that the surface contains (0:1:0:0:1:0:0:0), which means that the coefficients at W1^2 and W^4^2 are opposite of each other.
We can avoid W0^2 and W7^2 and we make a couple of reasonable guesses about the relative sizes of the other coefficients.
However, after I did it, the equations were not nice. So I had to modify the goals.

I will look at the tangent space to the points and make sure that these are defined over the correct field.

SymEqU = Simplify[AllEqs2[[1]]+AllEqs2[[2]]+AllEqs2[[3]]]
VarsV={V0,V1,V2,V3,V4,V5,V6,V7};
VarsW={W0,W1,W2,W3,W4,W5,W6,W7};

(* moving the singular points to (0:1:0:0:\pm 1:0:0:0) and cyclic orbit *)
subUV={U0->V0+(V1+V2+V3),
U1-> (U1/.solssingUexact[[1]]) V1 +  (U1/.solssingUexact[[2]]) V2 +  (U1/.solssingUexact[[3]]) V3 ,U2-> (U2/.solssingUexact[[1]]) V1 +  (U2/.solssingUexact[[2]]) V2 +  (U2/.solssingUexact[[3]]) V3 ,U3-> (U3/.solssingUexact[[1]]) V1 +  (U3/.solssingUexact[[2]]) V2 +  (U3/.solssingUexact[[3]]) V3,
U4-> (U4/.solssingUexact[[1]]) V4 +  (U4/.solssingUexact[[2]]) V5 +  (U4/.solssingUexact[[3]]) V6,U5-> (U5/.solssingUexact[[1]]) V4 +  (U5/.solssingUexact[[2]]) V5 +  (U5/.solssingUexact[[3]]) V6 ,U6-> (U6/.solssingUexact[[1]]) V4+  (U6/.solssingUexact[[2]]) V5+  (U6/.solssingUexact[[3]]) V6,
U7-> V7+  (U7/.solssingUexact[[1]]) (V4+V5+V6)
 };
AllEqs23V = Chop[N[Expand[AllEqs23/.N[subUV,10000]],10000]];

pts={{0,1,0,0,1,0,0,0},{0,1,0,0,-1,0,0,0},{0,0,1,0,0,1,0,0},{0,0,1,0,0,-1,0,0},{0,0,0,1,0,0,1,0},{0,0,0,1,0,0,-1,0}};
Table[MatrixRank[Table[D[AllEqs23V[[j]],VarsV[[i]]]/.Table[VarsV[[k]]->pts[[m]][[k]],{k,1,8}],{j,1,9},{i,1,8}]],{m,1,6}]

VarsV={V0,V1,V2,V3,V4,V5,V6,V7};
VarsW={W0,W1,W2,W3,W4,W5,W6,W7};


Mons2W = {};
For[i=1,i≤4,i++,For[j=1,j≤i,j++,AppendTo[Mons2W,VarsW[[i]] VarsW[[j]]]]];
For[i=5,i≤8,i++,For[j=5,j≤i,j++,AppendTo[Mons2W,VarsW[[i]] VarsW[[j]]]]];

subUV={U0->V0+(V1+V2+V3),
U1-> (U1/.solssingUexact[[1]]) V1 +  (U1/.solssingUexact[[2]]) V2 +  (U1/.solssingUexact[[3]]) V3 ,U2-> (U2/.solssingUexact[[1]]) V1 +  (U2/.solssingUexact[[2]]) V2 +  (U2/.solssingUexact[[3]]) V3 ,U3-> (U3/.solssingUexact[[1]]) V1 +  (U3/.solssingUexact[[2]]) V2 +  (U3/.solssingUexact[[3]]) V3,
U4-> (U4/.solssingUexact[[1]]) V4 +  (U4/.solssingUexact[[2]]) V5 +  (U4/.solssingUexact[[3]]) V6,U5-> (U5/.solssingUexact[[1]]) V4 +  (U5/.solssingUexact[[2]]) V5 +  (U5/.solssingUexact[[3]]) V6 ,U6-> (U6/.solssingUexact[[1]]) V4+  (U6/.solssingUexact[[2]]) V5+  (U6/.solssingUexact[[3]]) V6,
U7-> V7+  (U7/.solssingUexact[[1]]) (V4+V5+V6)
 };
subVW={V0->  W0,V1->W1,V2->W2 ,V3->W3,V4->W4+c3 W7,V5->W5+c3 W7, V6-> W6+c3 W7,V7->c4 W7};
subUW= subUV/.subVW;

AllEqs23W = Chop[N[Expand[AllEqs23/.N[subUW,5000]],5000]];

(* compute the tangent space for a variable change *)

pts={{0,1,0,0,1,0,0,0},{0,1,0,0,-1,0,0,0},{0,0,1,0,0,1,0,0},{0,0,1,0,0,-1,0,0},{0,0,0,1,0,0,1,0},{0,0,0,1,0,0,-1,0}};
RR=Chop[RowReduce[Table[D[AllEqs23W[[j]],VarsW[[i]]]/.Table[VarsW[[k]]->pts[[1]][[k]],{k,1,8}],{j,1,9},{i,1,8}]]];

(* require a simpler form of the tangent space at first point and modify accordingly *)
subUW= subUW/.Solve[{RR[[2]][[8]]==1,RR[[1]][[8]]==0}][[1]];

Beautify[x_]:=
Ret=Simplify[Expand[RootApproximant[N[Re[x],3000],2]+RootApproximant[N[Im[x]/(Sqrt[15]),3000],2] I(Sqrt[15])]];

We are looking to find nicer degree two equations.

AllEqs23W=Chop[N[Expand[AllEqs23/.N[subUW,4500]],4000]];

(* computing degree 2 equations *)
Mons2W = {};
For[i=1,i≤4,i++,For[j=1,j≤i,j++,AppendTo[Mons2W,VarsW[[i]] VarsW[[j]]]]];
For[i=5,i≤8,i++,For[j=5,j≤i,j++,AppendTo[Mons2W,VarsW[[i]] VarsW[[j]]]]];

Eqs2W={};
UnivPolyW2 = c1 AllEqs23W[[1]] + c2  AllEqs23W[[2]] + c3  AllEqs23W[[3]];
For[j1=1,j1≤20,j1++,
Print["j1=",j1];
For[j2=1,j2≤20,j2++,
For[j3=1,j3≤20,j3++,

If[(j1-j2)(j2-j3)(j3-j1)≠0,
IndicesToTry={j1,j2,j3};
EqsToSolve= {1==Coefficient[UnivPolyW2,Mons2W[[IndicesToTry[[1]]]]]};
For[j=2,j≤3,j++,
AppendTo[EqsToSolve,0==Coefficient[UnivPolyW2,Mons2W[[IndicesToTry[[j]]]]]];
];

sols=Chop[NSolve[EqsToSolve]]; 
(* need to make sure there is a unique solution *)
If[Length[sols]>0,
sols=sols[[1]];
If[Length[sols]==3,

Polysolve=UnivPolyW2/.sols;
goodeq=Numerator[Factor[Sum[RootApproximant[Coefficient[Polysolve,Mons2W[[k]]]] Mons2W[[k]],{k,1,Length[Mons2W]}]]];

If[ByteCount[goodeq]≤5072, (* found experimentally *)
new=Append[Eqs2W,goodeq];
If[Length[Eqs2W]==0,Eqs2W=new; Print[ByteCount[goodeq]," ","Length of Eqs2W is ",Length[Eqs2W]]];
If[MatrixRank[Table[Coefficient[Eqs2W[[i]],Mons2W[[j]]],{i,1,Length[Eqs2W]},{j,1,20}]]<MatrixRank[Table[Coefficient[new[[i]],Mons2W[[j]]],{i,1,Length[new]},{j,1,20}]],
Eqs2W=new;
Print[ByteCount[goodeq]," ","Length of Eqs2W is ",Length[Eqs2W]]];
];
If[Length[Eqs2W]==3,
Print["done"];
j1=21;j2=21;j3=21;
];
];
];
];
];
];
];

We are looking to find nicer degree three equations by randomly selecting some terms to be zero.
We will also make sure that the 24 coefficients by W_i^2 W_j for 1<=i<=3 are zero.
We start by looking for two odd equations. One may need to rerun the code, if not lucky enough, or use our precomputed
values below.

(* sorting monomials of degree three in W according to parity *)
AllMons3even={};
AllMons3odd={};
For[i=1,i≤8,i++,For[j=1,j≤i,j++,For[k=1,k≤j,k++,
sgn=Mod[Floor[(i-1)/4]+Floor[(j-1)/4]+Floor[(k-1)/4],2];
If[sgn==0,
AppendTo[AllMons3even,VarsW[[i]]VarsW[[j]]VarsW[[k]]];
];
If[sgn==1,
AppendTo[AllMons3odd,VarsW[[i]]VarsW[[j]]VarsW[[k]]];
];
]]];
AllMons3even
AllMons3odd
AllMons3=Join[AllMons3even,AllMons3odd];

The trouble is that a lot of simple equations are just products of a degree two equation and a variable. So we will just have to make some reasonable choices of getting rid of these uniformly.

(* looking for nice odd equations *)

UnivPolyW3odd  = {c1,c2,c3,c4,c5,c6,c7,c8,c9,c10,c11,c12,c13,c14}.
{AllEqs23W[[4]],AllEqs23W[[5]],
W4 AllEqs23W[[1]],W5 AllEqs23W[[1]],W6 AllEqs23W[[1]],W7 AllEqs23W[[1]],
W4 AllEqs23W[[2]],W5 AllEqs23W[[2]],W6 AllEqs23W[[2]],W7 AllEqs23W[[2]],
W4 AllEqs23W[[3]],W5 AllEqs23W[[3]],W6 AllEqs23W[[3]],W7 AllEqs23W[[3]]};

(* we start with the products of degree two equations and odd variables *)
GoodOddEqs={};
For[j=5,j≤8,j++,
AppendTo[GoodOddEqs,VarsW[[j]] Eqs2W[[1]]];
AppendTo[GoodOddEqs,VarsW[[j]] Eqs2W[[2]]];
AppendTo[GoodOddEqs,VarsW[[j]] Eqs2W[[3]]];
];

For[j1=1,j1≤60,j1++,
If[Mod[j1,1]==0,Print[j1]];
For[j2=1,j2≤60,j2++,
If[j1≠j2,

EqsToSolve= {1==Coefficient[UnivPolyW3odd,AllMons3odd[[j1]]],0==Coefficient[UnivPolyW3odd,AllMons3odd[[j2]]]};

(* a uniform way of getting rid of the simple equations *)
For[j=5,j≤8,j++,
AppendTo[EqsToSolve,0==Coefficient[UnivPolyW3odd,VarsW[[j]] W4^2]];
AppendTo[EqsToSolve,0==Coefficient[UnivPolyW3odd,VarsW[[j]] W5^2]];
AppendTo[EqsToSolve,0==Coefficient[UnivPolyW3odd,VarsW[[j]] W6^2]];
];

sols=Solve[EqsToSolve];
(* perhaps we didn't get linearly independent conditions *)
If[Length[sols]>0,sols=sols[[1]]];
If[Length[sols]==14 ,
count++;
Polysolve=UnivPolyW3odd/.sols;
goodeq=Numerator[Factor[Sum[RootApproximant[Chop[Coefficient[Polysolve, AllMons3[[k]]]],2] AllMons3[[k]],{k,1,Length[AllMons3]}]]];

(* we pick 16280 after experimenting with others; this seems to be the lowest number that gets the results *)
newGoodOddEqs=Append[GoodOddEqs,goodeq];
If[ByteCount[goodeq]≤16280,
If[MatrixRank[Table[Coefficient[GoodOddEqs[[i]],AllMons3odd[[j]]],{i,1,Length[GoodOddEqs]},{j,1,60}]]<
MatrixRank[Table[Coefficient[newGoodOddEqs[[i]],AllMons3odd[[j]]],{i,1,Length[newGoodOddEqs]},{j,1,60}]],
GoodOddEqs=newGoodOddEqs;
 Print[ByteCount[goodeq]," ","Length of GoodOddEqs is ",Length[GoodOddEqs]];
];
If[Length[GoodOddEqs]==14,j1=61;j2=62;];
];
];
]]];

Since the GoodEvenEqs is computed randomly one may want to use our precomputed equations, which is suggested.

(* we start with the products of degree two equations and even variables *)
GoodEvenEqs={};
For[j=1,j≤4,j++,
AppendTo[GoodEvenEqs,VarsW[[j]] Eqs2W[[1]]];
AppendTo[GoodEvenEqs,VarsW[[j]] Eqs2W[[2]]];
AppendTo[GoodEvenEqs,VarsW[[j]] Eqs2W[[3]]];
];

(* looking for further nice even equations, this had to be run multiple times *)
UnivPolyW3even  = {c1,c2,c3,c4,c5,c6,c7,c8,c9,c10,c11,c12,c13,c14,c15,c16}.
{AllEqs23W[[6]],AllEqs23W[[7]],AllEqs23W[[8]],AllEqs23W[[9]],
W0 AllEqs23W[[1]],W1 AllEqs23W[[1]],W2 AllEqs23W[[1]],W3 AllEqs23W[[1]],
W0 AllEqs23W[[2]],W1 AllEqs23W[[2]],W2 AllEqs23W[[2]],W3 AllEqs23W[[2]],
W0 AllEqs23W[[3]],W1 AllEqs23W[[3]],W2 AllEqs23W[[3]],W3 AllEqs23W[[3]]};

MaxTry=5000;
For[numtry=1,numtry≤MaxTry,numtry++,
If[Mod[numtry,MaxTry/10]==0,Print[numtry]];

{m1,m2,m3,m4}=RandomSample[AllMons3even,4];

EqsToSolve= {1==Coefficient[UnivPolyW3even,m1],0==Coefficient[UnivPolyW3even,m2],
0==Coefficient[UnivPolyW3even,m3],0==Coefficient[UnivPolyW3even,m4]};

(* a uniform way of getting rid of the simple equations *)
For[j=1,j≤4,j++,
AppendTo[EqsToSolve,0==Coefficient[UnivPolyW3even,VarsW[[j]] W4^2]];
AppendTo[EqsToSolve,0==Coefficient[UnivPolyW3even,VarsW[[j]] W5^2]];
AppendTo[EqsToSolve,0==Coefficient[UnivPolyW3even,VarsW[[j]] W6^2]];
];

sols=Solve[EqsToSolve];
(* perhaps we didn't get linearly independent conditions *)
If[Length[sols]>0,sols=sols[[1]]];
If[Length[sols]==16 ,
count++;
Polysolve=UnivPolyW3even/.sols;
goodeq=Numerator[Factor[Sum[RootApproximant[Chop[Coefficient[Polysolve, AllMons3[[k]]]],2] AllMons3[[k]],{k,1,Length[AllMons3]}]]];

(* we ran it multiple times at different cut-off values; the lowest observed ByteCount was 15944 *)
newGoodEvenEqs=Append[GoodEvenEqs,goodeq];
If[ByteCount[goodeq]≤17500,
If[MatrixRank[Table[Coefficient[GoodEvenEqs[[i]],AllMons3even[[j]]],{i,1,Length[GoodEvenEqs]},{j,1,60}]]<
MatrixRank[Table[Coefficient[newGoodEvenEqs[[i]],AllMons3even[[j]]],{i,1,Length[newGoodEvenEqs]},{j,1,60}]],
GoodEvenEqs=newGoodEvenEqs;
 Print[ByteCount[goodeq]," ",{m1,m2,m3,m3}," ","Length of GoodEvenEqs is ",Length[GoodEvenEqs]];
];
If[Length[GoodEvenEqs]==16,numtry=MaxTry+1;];
];
];
];

Length[GoodOddEqs]
Length[GoodEvenEqs]

Since the GoodEvenEqs is computed randomly one may want to use our precomputed equations, which is suggested.

Eqs23W = {Eqs2W[[1]],Eqs2W[[2]],Eqs2W[[3]],GoodOddEqs[[13]],GoodOddEqs[[14]],GoodEvenEqs[[13]],GoodEvenEqs[[14]],GoodEvenEqs[[15]],GoodEvenEqs[[16]]};

Eqs23W={1706103025 W0 W1-826261465 I (Sqrt[15]) W0 W1+12001941280 (W1^2)+905221605 W0 W2+967261715 I (Sqrt[15]) W0 W2+14726066110 W1 W2-9988457710 I (Sqrt[15]) W1 W2+3014585345 W0 W3-1051861865 I (Sqrt[15]) W0 W3-52547973170 W1 W3-3355805950 I (Sqrt[15]) W1 W3+14568145830 W2 W3+6593171690 I (Sqrt[15]) W2 W3-12001941280 (W4^2)-19345234300 W4 W5-4279639588 I (Sqrt[15]) W4 W5-20766516820 W4 W6+931729652 I (Sqrt[15]) W4 W6+2605684620 W5 W6-4753400428 I (Sqrt[15]) W5 W6+55004411560 W4 W7+586891480 I (Sqrt[15]) W4 W7+1647964360 W5 W7+4419875320 I (Sqrt[15]) W5 W7+6621506920 W6 W7-967498280 I (Sqrt[15]) W6 W7-25469063760 (W7^2)-1367727312 I (Sqrt[15]) (W7^2),53321 W0 W1-119409 I (Sqrt[15]) W0 W1+799064 (W1^2)-140437 W0 W2+134429 I (Sqrt[15]) W0 W2+10514 W1 W2-1103970 I (Sqrt[15]) W1 W2+87116 W0 W3-15020 I (Sqrt[15]) W0 W3-4478964 W1 W3+441588 I (Sqrt[15]) W1 W3+4468450 W2 W3+662382 I (Sqrt[15]) W2 W3-799064 (W3^2)-799064 (W4^2)-1461446 W4 W5+31542 I (Sqrt[15]) W4 W5-94626 W4 W6+346962 I (Sqrt[15]) W4 W6+1556072 W5 W6-378504 I (Sqrt[15]) W5 W6+799064 (W6^2)+3552360 W4 W7-255192 I (Sqrt[15]) W4 W7-331128 W5 W7+358680 I (Sqrt[15]) W5 W7-3221232 W6 W7-103488 I (Sqrt[15]) W6 W7,-87116 W0 W1+15020 I (Sqrt[15]) W0 W1+799064 (W1^2)-53321 W0 W2+119409 I (Sqrt[15]) W0 W2+4478964 W1 W2-441588 I (Sqrt[15]) W1 W2-799064 (W2^2)+140437 W0 W3-134429 I (Sqrt[15]) W0 W3-4468450 W1 W3-662382 I (Sqrt[15]) W1 W3-10514 W2 W3+1103970 I (Sqrt[15]) W2 W3-799064 (W4^2)+94626 W4 W5-346962 I (Sqrt[15]) W4 W5+799064 (W5^2)-1556072 W4 W6+378504 I (Sqrt[15]) W4 W6+1461446 W5 W6-31542 I (Sqrt[15]) W5 W6+3221232 W4 W7+103488 I (Sqrt[15]) W4 W7-3552360 W5 W7+255192 I (Sqrt[15]) W5 W7+331128 W6 W7-358680 I (Sqrt[15]) W6 W7,1753408974347648 (W0^2) W4+4624568052886208 W0 W1 W4+1836658084768192 I (Sqrt[15]) W0 W1 W4-25723910738105944 W0 W2 W4-4926352855196696 I (Sqrt[15]) W0 W2 W4+307036825631433576 W1 W2 W4+57793461876095400 I (Sqrt[15]) W1 W2 W4-42812288376882136 W0 W3 W4-3212847068911256 I (Sqrt[15]) W0 W3 W4+146468880652808448 W1 W3 W4-67854829136903424 I (Sqrt[15]) W1 W3 W4-263825659127933973 W2 W3 W4+97123836640946787 I (Sqrt[15]) W2 W3 W4+244274695087257576 (W3^2) W4+52031300557563432 I (Sqrt[15]) (W3^2) W4+1753408974347648 (W0^2) W5-42812288376882136 W0 W1 W5-3212847068911256 I (Sqrt[15]) W0 W1 W5+244274695087257576 (W1^2) W5+52031300557563432 I (Sqrt[15]) (W1^2) W5+4624568052886208 W0 W2 W5+1836658084768192 I (Sqrt[15]) W0 W2 W5+146468880652808448 W1 W2 W5-67854829136903424 I (Sqrt[15]) W1 W2 W5-25723910738105944 W0 W3 W5-4926352855196696 I (Sqrt[15]) W0 W3 W5-263825659127933973 W1 W3 W5+97123836640946787 I (Sqrt[15]) W1 W3 W5+307036825631433576 W2 W3 W5+57793461876095400 I (Sqrt[15]) W2 W3 W5+1753408974347648 (W0^2) W6-25723910738105944 W0 W1 W6-4926352855196696 I (Sqrt[15]) W0 W1 W6-42812288376882136 W0 W2 W6-3212847068911256 I (Sqrt[15]) W0 W2 W6-263825659127933973 W1 W2 W6+97123836640946787 I (Sqrt[15]) W1 W2 W6+244274695087257576 (W2^2) W6+52031300557563432 I (Sqrt[15]) (W2^2) W6+4624568052886208 W0 W3 W6+1836658084768192 I (Sqrt[15]) W0 W3 W6+307036825631433576 W1 W3 W6+57793461876095400 I (Sqrt[15]) W1 W3 W6+146468880652808448 W2 W3 W6-67854829136903424 I (Sqrt[15]) W2 W3 W6+413162544996145035 W4 W5 W6+86812915194403587 I (Sqrt[15]) W4 W5 W6-2219261238917376 (W0^2) W7+41371720329984 I (Sqrt[15]) (W0^2) W7+112661548728436848 W0 W1 W7-15940846449910032 I (Sqrt[15]) W0 W1 W7-123783199950329640 (W1^2) W7+32171290678282392 I (Sqrt[15]) (W1^2) W7+112661548728436848 W0 W2 W7-15940846449910032 I (Sqrt[15]) W0 W2 W7-668268262244236563 W1 W2 W7-103644995514656715 I (Sqrt[15]) W1 W2 W7-123783199950329640 (W2^2) W7+32171290678282392 I (Sqrt[15]) (W2^2) W7+112661548728436848 W0 W3 W7-15940846449910032 I (Sqrt[15]) W0 W3 W7-668268262244236563 W1 W3 W7-103644995514656715 I (Sqrt[15]) W1 W3 W7-668268262244236563 W2 W3 W7-103644995514656715 I (Sqrt[15]) W2 W3 W7-123783199950329640 (W3^2) W7+32171290678282392 I (Sqrt[15]) (W3^2) W7-574830176340661227 W4 W5 W7-133758869453715555 I (Sqrt[15]) W4 W5 W7-574830176340661227 W4 W6 W7-133758869453715555 I (Sqrt[15]) W4 W6 W7-574830176340661227 W5 W6 W7-133758869453715555 I (Sqrt[15]) W5 W6 W7+642303707483671947 W4 (W7^2)+134647945440329475 I (Sqrt[15]) W4 (W7^2)+642303707483671947 W5 (W7^2)+134647945440329475 I (Sqrt[15]) W5 (W7^2)+642303707483671947 W6 (W7^2)+134647945440329475 I (Sqrt[15]) W6 (W7^2)-1138685084396455995 (W7^3)-184628475341453619 I (Sqrt[15]) (W7^3),-98948224478443260 W0 W1 W4-3058298456825380 I (Sqrt[15]) W0 W1 W4+81462632898956280 W0 W2 W4-23953315200421720 I (Sqrt[15]) W0 W2 W4-1094836267717214400 W1 W2 W4+398810644281337680 I (Sqrt[15]) W1 W2 W4+138080956729877280 (W2^2) W4-1543563136889220 W0 W3 W4-50501498192304700 I (Sqrt[15]) W0 W3 W4+1373637819762610920 W1 W3 W4+199151284945609080 I (Sqrt[15]) W1 W3 W4-1695126293066323845 W2 W3 W4-246880524310909965 I (Sqrt[15]) W2 W3 W4-551526491689636680 (W3^2) W4+326575488081046680 I (Sqrt[15]) (W3^2) W4-1543563136889220 W0 W1 W5-50501498192304700 I (Sqrt[15]) W0 W1 W5-551526491689636680 (W1^2) W5+326575488081046680 I (Sqrt[15]) (W1^2) W5-98948224478443260 W0 W2 W5-3058298456825380 I (Sqrt[15]) W0 W2 W5+1373637819762610920 W1 W2 W5+199151284945609080 I (Sqrt[15]) W1 W2 W5+81462632898956280 W0 W3 W5-23953315200421720 I (Sqrt[15]) W0 W3 W5-1695126293066323845 W1 W3 W5-246880524310909965 I (Sqrt[15]) W1 W3 W5-1094836267717214400 W2 W3 W5+398810644281337680 I (Sqrt[15]) W2 W3 W5+138080956729877280 (W3^2) W5+81462632898956280 W0 W1 W6-23953315200421720 I (Sqrt[15]) W0 W1 W6+138080956729877280 (W1^2) W6-1543563136889220 W0 W2 W6-50501498192304700 I (Sqrt[15]) W0 W2 W6-1695126293066323845 W1 W2 W6-246880524310909965 I (Sqrt[15]) W1 W2 W6-551526491689636680 (W2^2) W6+326575488081046680 I (Sqrt[15]) (W2^2) W6-98948224478443260 W0 W3 W6-3058298456825380 I (Sqrt[15]) W0 W3 W6-1094836267717214400 W1 W3 W6+398810644281337680 I (Sqrt[15]) W1 W3 W6+1373637819762610920 W2 W3 W6+199151284945609080 I (Sqrt[15]) W2 W3 W6-1690952291170184385 W4 W5 W6+369304304128884495 I (Sqrt[15]) W4 W5 W6-5328591207840000 (W0^2) W7+2144900495488320 I (Sqrt[15]) (W0^2) W7+284310840392140950 W0 W1 W7+106996549176297210 I (Sqrt[15]) W0 W1 W7-688957430544122040 (W1^2) W7-158946397497914040 I (Sqrt[15]) (W1^2) W7+284310840392140950 W0 W2 W7+106996549176297210 I (Sqrt[15]) W0 W2 W7+1200158218487790765 W1 W2 W7-641150643161964915 I (Sqrt[15]) W1 W2 W7-688957430544122040 (W2^2) W7-158946397497914040 I (Sqrt[15]) (W2^2) W7+284310840392140950 W0 W3 W7+106996549176297210 I (Sqrt[15]) W0 W3 W7+1200158218487790765 W1 W3 W7-641150643161964915 I (Sqrt[15]) W1 W3 W7+1200158218487790765 W2 W3 W7-641150643161964915 I (Sqrt[15]) W2 W3 W7-688957430544122040 (W3^2) W7-158946397497914040 I (Sqrt[15]) (W3^2) W7+1808067319615295865 W4 W5 W7-379632318487526223 I (Sqrt[15]) W4 W5 W7+1808067319615295865 W4 W6 W7-379632318487526223 I (Sqrt[15]) W4 W6 W7+1808067319615295865 W5 W6 W7-379632318487526223 I (Sqrt[15]) W5 W6 W7-1999928222013559005 W4 (W7^2)+443220177629557755 I (Sqrt[15]) W4 (W7^2)-1999928222013559005 W5 (W7^2)+443220177629557755 I (Sqrt[15]) W5 (W7^2)-1999928222013559005 W6 (W7^2)+443220177629557755 I (Sqrt[15]) W6 (W7^2)+2471992417171938465 (W7^3)-903652627464942327 I (Sqrt[15]) (W7^3),-1666764770896080 (W0^3)+24780099758160 I (Sqrt[15]) (W0^3)-26655160103850225 (W0^2) W1-141099466791655 I (Sqrt[15]) (W0^2) W1+68978091270130770 W0 (W1^2)-43638746552374050 I (Sqrt[15]) W0 (W1^2)-14960300456911665 (W0^2) W2+243600535544345 I (Sqrt[15]) (W0^2) W2-136997854837433730 W0 W1 W2+100084814202953370 I (Sqrt[15]) W0 W1 W2-2166930718788380580 (W1^2) W2-114990006031685820 I (Sqrt[15]) (W1^2) W2+188045734378652430 W0 (W2^2)-11700924076935870 I (Sqrt[15]) W0 (W2^2)-334721551455006660 W1 (W2^2)+371044998805248180 I (Sqrt[15]) W1 (W2^2)-3471146893213005 (W0^2) W3+3488453404965845 I (Sqrt[15]) (W0^2) W3+370759844061735240 W0 W1 W3+142387946334768960 I (Sqrt[15]) W0 W1 W3+1178593751048740740 (W1^2) W3+258386774372850060 I (Sqrt[15]) (W1^2) W3+337733550357073080 W0 W2 W3-23905566679900920 I (Sqrt[15]) W0 W2 W3+3735489413393712765 W1 W2 W3-810504711260264475 I (Sqrt[15]) W1 W2 W3-1479640375942060500 (W2^2) W3-541366284526196220 I (Sqrt[15]) (W2^2) W3+88187928349336200 W0 W4 W5+45648803952015264 I (Sqrt[15]) W0 W4 W5-25167050881043400 W1 W4 W5+54618239319319368 I (Sqrt[15]) W1 W4 W5+691467466176945360 W2 W4 W5-135405264642391296 I (Sqrt[15]) W2 W4 W5+731586446190101565 W3 W4 W5-157420661770451835 I (Sqrt[15]) W3 W4 W5-22789739259859320 W0 W4 W6+14076296148587904 I (Sqrt[15]) W0 W4 W6+471919011020770320 W1 W4 W6-286625927180712672 I (Sqrt[15]) W1 W4 W6-52065650645825355 W2 W4 W6-336326621647458771 I (Sqrt[15]) W2 W4 W6-71375582659697820 W0 W5 W6+34183213308069084 I (Sqrt[15]) W0 W5 W6+544930562815031925 W1 W5 W6-262568409537346227 I (Sqrt[15]) W1 W5 W6-773851724743350960 W2 W5 W6-167052236438578176 I (Sqrt[15]) W2 W5 W6-83927877818450400 W0 W4 W7-17377009952576400 I (Sqrt[15]) W0 W4 W7-36849898243929960 W1 W4 W7+28649673281993400 I (Sqrt[15]) W1 W4 W7-549479940520862295 W2 W4 W7+243736669319493105 I (Sqrt[15]) W2 W4 W7-796822141111243695 W3 W4 W7+45136039971948345 I (Sqrt[15]) W3 W4 W7+53307405771262020 W0 W5 W7-21625406694472140 I (Sqrt[15]) W0 W5 W7+832604580620480745 W1 W5 W7+48931912928647665 I (Sqrt[15]) W1 W5 W7+356264465750010720 W2 W5 W7-313200156644634105 W3 W5 W7+130270143553698015 I (Sqrt[15]) W3 W5 W7+42156434307162420 W0 W6 W7+3936635595601380 I (Sqrt[15]) W0 W6 W7-1166733390875518935 W1 W6 W7+266638938070257585 I (Sqrt[15]) W1 W6 W7+988209940857367095 W2 W6 W7+425481801320871855 I (Sqrt[15]) W2 W6 W7-142547635521932760 W0 (W7^2)-5281782721276824 I (Sqrt[15]) W0 (W7^2)+232999883627690025 W1 (W7^2)-74107682752546863 I (Sqrt[15]) W1 (W7^2)-876915653107092165 W2 (W7^2)-319487180407733709 I (Sqrt[15]) W2 (W7^2)+314943079584183435 W3 (W7^2)-119573778905614845 I (Sqrt[15]) W3 (W7^2),-5013788093839440 (W0^3)-2616279996930096 I (Sqrt[15]) (W0^3)-45513288110636265 (W0^2) W1-23426699358079343 I (Sqrt[15]) (W0^2) W1+780595279904010270 W0 (W1^2)+271578555264551346 I (Sqrt[15]) W0 (W1^2)-93393081166855845 (W0^2) W2+3656535225002989 I (Sqrt[15]) (W0^2) W2+1471728092428425720 W0 W1 W2+480102333223826376 I (Sqrt[15]) W0 W1 W2+9146428748595735660 (W1^2) W2-3815055396059983740 I (Sqrt[15]) (W1^2) W2-67210942787666985 (W0^2) W3-43263632725852655 I (Sqrt[15]) (W0^2) W3-2778584336999918370 W0 W1 W3+44712791412074394 I (Sqrt[15]) W0 W1 W3-9842285067436547460 (W1^2) W3+444407576155890228 I (Sqrt[15]) (W1^2) W3-2454426866328969000 W0 W2 W3+973798846404589968 I (Sqrt[15]) W0 W2 W3+29458584503171418105 W1 W2 W3+3863397004095234753 I (Sqrt[15]) W1 W2 W3+1236003231613165890 W0 (W3^2)-4620414488953746 I (Sqrt[15]) W0 (W3^2)-2965375983927514500 W1 (W3^2)-3791366028666657756 I (Sqrt[15]) W1 (W3^2)-3114003711146586300 W2 (W3^2)+2580965557251104268 I (Sqrt[15]) W2 (W3^2)-1014136630921693692 W0 W4 W5-24287487279628356 I (Sqrt[15]) W0 W4 W5+1981826568032775408 W1 W4 W5-1687741120523745504 I (Sqrt[15]) W1 W4 W5+7780999414807476081 W3 W4 W5+181214837656337961 I (Sqrt[15]) W3 W4 W5-868143844238893992 W0 W4 W6+262835559791381808 I (Sqrt[15]) W0 W4 W6+5100012058333450608 W1 W4 W6+753890426588063520 I (Sqrt[15]) W1 W4 W6+5737660067056849785 W2 W4 W6+760119818012636193 I (Sqrt[15]) W2 W4 W6-1384936673551780104 W3 W4 W6+104202680730502536 I (Sqrt[15]) W3 W4 W6-400120638390226152 W0 W5 W6+621682050047184 I (Sqrt[15]) W0 W5 W6+7979623637881267953 W1 W5 W6-974794338205463223 I (Sqrt[15]) W1 W5 W6+8168223467826028656 W3 W5 W6+663174822276528660 W0 W4 W7+28483582014993924 I (Sqrt[15]) W0 W4 W7+943838022041540640 W1 W4 W7+573251854361425344 I (Sqrt[15]) W1 W4 W7-3973941756361724805 W2 W4 W7-158838900312172941 I (Sqrt[15]) W2 W4 W7+1024770214986179445 W3 W4 W7+1469346426682676829 I (Sqrt[15]) W3 W4 W7+16669051552584900 W0 W5 W7+78261525283472244 I (Sqrt[15]) W0 W5 W7-7651375930474067925 W1 W5 W7+2717304631631549379 I (Sqrt[15]) W1 W5 W7-9526564245341702475 W3 W5 W7-1170950661832272867 I (Sqrt[15]) W3 W5 W7+197063508246494400 W0 W6 W7-181081473165920880 I (Sqrt[15]) W0 W6 W7-7338530155281953355 W1 W6 W7-238425291137764899 I (Sqrt[15]) W1 W6 W7-3200391625843951875 W2 W6 W7-1162559556498058299 I (Sqrt[15]) W2 W6 W7-789111845649315720 W3 W6 W7+16606703811437208 I (Sqrt[15]) W3 W6 W7-250165286848077048 W0 (W7^2)-243360907666624440 I (Sqrt[15]) W0 (W7^2)+5387943868144889247 W1 (W7^2)-2257417592834519241 I (Sqrt[15]) W1 (W7^2)+3720392399673582255 W2 (W7^2)+189980848292392647 I (Sqrt[15]) W2 (W7^2)+2405936404293017589 W3 (W7^2)+178580725176692349 I (Sqrt[15]) W3 (W7^2),9835587399073830 (W0^2) W1+1578420676125258 I (Sqrt[15]) (W0^2) W1+35745315683806925 W0 (W1^2)+1688198228385963 I (Sqrt[15]) W0 (W1^2)+7582977901132440 (W0^2) W2-2163669722335032 I (Sqrt[15]) (W0^2) W2-17299113918193415 W0 W1 W2-60268666487376277 I (Sqrt[15]) W0 W1 W2-133603664158909350 (W1^2) W2-19184371325606466 I (Sqrt[15]) (W1^2) W2-37183010128126030 W0 (W2^2)+12853731757328126 I (Sqrt[15]) W0 (W2^2)-84961513178793720 W1 (W2^2)-92450186250447312 I (Sqrt[15]) W1 (W2^2)-17418565300206270 (W0^2) W3+585249046209774 I (Sqrt[15]) (W0^2) W3-70086431163261590 W0 W1 W3+2414743691108854 I (Sqrt[15]) W0 W1 W3-342479032236890010 (W1^2) W3+89881732813011930 I (Sqrt[15]) (W1^2) W3+87385545081455005 W0 W2 W3+57853922796267423 I (Sqrt[15]) W0 W2 W3+1042730687524077720 (W2^2) W3+30748986824663520 I (Sqrt[15]) (W2^2) W3+1437694444319105 W0 (W3^2)-14541929985714089 I (Sqrt[15]) W0 (W3^2)-909127023365168370 W1 (W3^2)-11564615499057054 I (Sqrt[15]) W1 (W3^2)+427440545415683730 W2 (W3^2)+2568453437435382 I (Sqrt[15]) W2 (W3^2)-86212099312821090 W0 W4 W5+1570156818417054 I (Sqrt[15]) W0 W4 W5-732268420932010032 W1 W4 W5+41562839039886360 I (Sqrt[15]) W1 W4 W5+241198295104065348 W2 W4 W5+79920007050512916 I (Sqrt[15]) W2 W4 W5-438378169379583636 W3 W4 W5-142108461635806764 I (Sqrt[15]) W3 W4 W5+70867691964193320 W0 W4 W6-5264228916218664 I (Sqrt[15]) W0 W4 W6-100267650150454332 W1 W4 W6+89820718220390220 I (Sqrt[15]) W1 W4 W6+870472160553251088 W2 W4 W6+19650634639745004 W3 W4 W6-115234116091814412 I (Sqrt[15]) W3 W4 W6+15344407348627770 W0 W5 W6+3694072097801610 I (Sqrt[15]) W0 W5 W6-432093991173667452 W1 W5 W6+142108461635806764 I (Sqrt[15]) W1 W5 W6+712617786292265028 W2 W5 W6+73671277051928052 I (Sqrt[15]) W2 W5 W6-140930644953611016 W3 W5 W6-169740725270903136 I (Sqrt[15]) W3 W5 W6+57555874348012530 W0 W4 W7-5581566298411614 I (Sqrt[15]) W0 W4 W7+98107489272578940 W1 W4 W7-86621136621204276 I (Sqrt[15]) W1 W4 W7-515830664168907885 W2 W4 W7-118921503273724053 I (Sqrt[15]) W2 W4 W7+786000668209326360 W3 W4 W7+28867861737924552 I (Sqrt[15]) W3 W4 W7+349407354877504875 W1 W5 W7-50861884373130285 I (Sqrt[15]) W1 W5 W7-313056742736377080 W2 W5 W7+41140380485439144 I (Sqrt[15]) W2 W5 W7-149922813705251895 W3 W5 W7+219439466686437921 I (Sqrt[15]) W3 W5 W7-57555874348012530 W0 W6 W7+5581566298411614 I (Sqrt[15]) W0 W6 W7+665753477874159780 W1 W6 W7-100517963412713868 I (Sqrt[15]) W1 W6 W7-1135408023086831235 W2 W6 W7+21994022635205733 I (Sqrt[15]) W2 W6 W7+214949253463798140 W3 W6 W7+45480756135765132 I (Sqrt[15]) W3 W6 W7-893396235162453183 W1 (W7^2)+106129289751482025 I (Sqrt[15]) W1 (W7^2)+1050816763479352212 W2 (W7^2)+51374298411415476 I (Sqrt[15]) W2 (W7^2)-157420528316899029 W3 (W7^2)-157503588162897501 I (Sqrt[15]) W3 (W7^2),324554939451 (W0^3)-70795097523 I (Sqrt[15]) (W0^3)+1529614704078 (W0^2) W1-1205210352894 I (Sqrt[15]) (W0^2) W1-5415939170712 W0 (W1^2)+1474501654360 I (Sqrt[15]) W0 (W1^2)+1529614704078 (W0^2) W2-1205210352894 I (Sqrt[15]) (W0^2) W2-62405246503404 W0 W1 W2-10367134749412 I (Sqrt[15]) W0 W1 W2+180800780856192 (W1^2) W2-5415939170712 W0 (W2^2)+1474501654360 I (Sqrt[15]) W0 (W2^2)-146202216215328 W1 (W2^2)-15520405118400 I (Sqrt[15]) W1 (W2^2)+1529614704078 (W0^2) W3-1205210352894 I (Sqrt[15]) (W0^2) W3-62405246503404 W0 W1 W3-10367134749412 I (Sqrt[15]) W0 W1 W3-146202216215328 (W1^2) W3-15520405118400 I (Sqrt[15]) (W1^2) W3-62405246503404 W0 W2 W3-10367134749412 I (Sqrt[15]) W0 W2 W3-341956310464440 W1 W2 W3+182430423393624 I (Sqrt[15]) W1 W2 W3+180800780856192 (W2^2) W3-5415939170712 W0 (W3^2)+1474501654360 I (Sqrt[15]) W0 (W3^2)+180800780856192 W1 (W3^2)-146202216215328 W2 (W3^2)-15520405118400 I (Sqrt[15]) W2 (W3^2)-24797684639448 W0 W4 W5-4795122965976 I (Sqrt[15]) W0 W4 W5-36568462213584 W2 W4 W5+31673565214704 I (Sqrt[15]) W2 W4 W5-58201519194000 W3 W4 W5+40317236988336 I (Sqrt[15]) W3 W4 W5-24797684639448 W0 W4 W6-4795122965976 I (Sqrt[15]) W0 W4 W6-36568462213584 W1 W4 W6+31673565214704 I (Sqrt[15]) W1 W4 W6-58201519194000 W2 W4 W6+40317236988336 I (Sqrt[15]) W2 W4 W6-24797684639448 W0 W5 W6-4795122965976 I (Sqrt[15]) W0 W5 W6-58201519194000 W1 W5 W6+40317236988336 I (Sqrt[15]) W1 W5 W6-36568462213584 W3 W5 W6+31673565214704 I (Sqrt[15]) W3 W5 W6+19496920450896 W0 W4 W7+2134605859344 I (Sqrt[15]) W0 W4 W7+21063286582128 W1 W4 W7-6992925745680 I (Sqrt[15]) W1 W4 W7+99385147864044 W2 W4 W7-46684295204148 I (Sqrt[15]) W2 W4 W7+1823973493356 W3 W4 W7-15659763667380 I (Sqrt[15]) W3 W4 W7+19496920450896 W0 W5 W7+2134605859344 I (Sqrt[15]) W0 W5 W7+1823973493356 W1 W5 W7-15659763667380 I (Sqrt[15]) W1 W5 W7+21063286582128 W2 W5 W7-6992925745680 I (Sqrt[15]) W2 W5 W7+99385147864044 W3 W5 W7-46684295204148 I (Sqrt[15]) W3 W5 W7+19496920450896 W0 W6 W7+2134605859344 I (Sqrt[15]) W0 W6 W7+99385147864044 W1 W6 W7-46684295204148 I (Sqrt[15]) W1 W6 W7+1823973493356 W2 W6 W7-15659763667380 I (Sqrt[15]) W2 W6 W7+21063286582128 W3 W6 W7-6992925745680 I (Sqrt[15]) W3 W6 W7-95094214488 W0 (W7^2)-2876821724952 I (Sqrt[15]) W0 (W7^2)-38259379316088 W1 (W7^2)+23825658096072 I (Sqrt[15]) W1 (W7^2)-38259379316088 W2 (W7^2)+23825658096072 I (Sqrt[15]) W2 (W7^2)-38259379316088 W3 (W7^2)+23825658096072 I (Sqrt[15]) W3 (W7^2)};

(* writing to a file *)
fname="EquationsOfXintermsofW";
OpenWrite[fname];
Write[fname,Eqs23W];
Close[fname];

(* writing to a file a Magma-friendly version *)
Rels23Withrr={};
For[i=1,i≤3,i++,
newrel=0;
For[j=1,j≤Length[Mons2W],j++,
cf=Coefficient[Eqs23W[[i]] ,Mons2W[[j]]];
newrel+=(Re[cf]+Im[cf] (rr/Sqrt[15])) Mons2W[[j]] ;
];
AppendTo[Rels23Withrr,newrel];
];
For[i=4,i≤5,i++,
newrel=0;
For[j=1,j≤60,j++,
cf=Coefficient[Eqs23W[[i]] ,AllMons3odd[[j]]];
newrel+=(Re[cf]+Im[cf] (rr/Sqrt[15])) AllMons3odd[[j]] ;
];
AppendTo[Rels23Withrr,newrel];
];
For[i=6,i≤9,i++,
newrel=0;
For[j=1,j≤60,j++,
cf=Coefficient[Eqs23W[[i]] ,AllMons3even[[j]]];
newrel+=(Re[cf]+Im[cf] (rr/Sqrt[15])) AllMons3even[[j]] ;
];
AppendTo[Rels23Withrr,newrel];
];

fname="EquationsOfXmagma";
OpenWrite[fname];
Write[fname,
Rels23Withrr];
Close[fname];

The below code can be ignored. This was an attempt of getting smaller coefficients in the equations, but the price paid was more terms. It does not seem worth it.

(* an attempt at finding smaller equations for deg 2, but perhaps with more terms, moderately sucessful *)

BigMatrix=Table[Coefficient[Eqs23W[[k]],Mons2W[[j]]],{k,1,3},{j,1,Length[Mons2W]}];
(* writing each equation in terms of real and imaginary parts, same for Sqrt[-15] times the equation; get two rows at a time *)
BigMatrixReIm={};
For[i=1,i≤3,i++,
NewRow1={};
NewRow2={};
For[j=1,j≤Length[Mons2W],j++,
AppendTo[NewRow1,Re[BigMatrix[[i]][[j]]]];
AppendTo[NewRow1,Im[BigMatrix[[i]][[j]]]/Sqrt[15]];
AppendTo[NewRow2,Im[BigMatrix[[i]][[j]]](-Sqrt[15])];
AppendTo[NewRow2,Re[BigMatrix[[i]][[j]]]];
];
AppendTo[BigMatrixReIm,NewRow1];
AppendTo[BigMatrixReIm,NewRow2];
];
(* finding a better basis *)
BestRels=LatticeReduce[BigMatrixReIm];

(* doing Smith decomposition to find a basis of the refinement *)
{u,r, v} = SmithDecomposition[BestRels];
vt=Transpose[v];
SmallBestRels={};
For[i=1,i≤6,i++,
basisvec=Table[0,{k,1,40}];
basisvec[[i]]=1;
AppendTo[SmallBestRels,LinearSolve[vt,basisvec]];
];
SmallerBestRels=LatticeReduce[SmallBestRels];

(* back to complex matrix *)
BackToCxMatrix={};
For[i=1,i≤6,i++,
newrow=Table[SmallerBestRels[[i]][[2k-1]]+ I Sqrt[15] SmallerBestRels[[i]][[2k]],{k,1,20}];
AppendTo[BackToCxMatrix,newrow];
];
MatrixRank[N[BackToCxMatrix,1000]]
BackToCxReduced={};
rk=0;
For[i=1,i≤6,i++,
newred=Append[BackToCxReduced,BackToCxMatrix[[i]]];
If[MatrixRank[N[newred,1000]]>rk, rk++;Print[i," ",rk];BackToCxReduced=newred;];
If[rk==3,i=120];
];

Eqs2Wimproved = Expand[Table[Sum[BackToCxReduced[[i]][[k]] Mons2W[[k]],{k,1,20}],{i,1,3}]]
Table[ByteCount[Eqs2Wimproved[[i]]],{i,1,3}]
Table[Length[MonomialList[Eqs2Wimproved[[i]]]],{i,1,3}]
Eqs2W
Table[ByteCount[Eqs2W[[i]]],{i,1,3}]
Table[Length[MonomialList[Eqs2W[[i]]]],{i,1,3}]

(* an attempt at finding smaller odd equations for deg 3, but perhaps with more terms, not particularly successful, since the byte count grew (only the last three equations are new) *)

BigMatrix=Table[Coefficient[GoodOddEqs[[k]], AllMons3odd[[j]]],{k,1,14},{j,1,60}];
(* writing each equation in terms of real and imaginary parts, same for Sqrt[-15] times the equation; get two rows at a time *)
BigMatrixReIm={};
For[i=1,i≤14,i++,
NewRow1={};
NewRow2={};
For[j=1,j≤60,j++,
AppendTo[NewRow1,Re[BigMatrix[[i]][[j]]]];
AppendTo[NewRow1,Im[BigMatrix[[i]][[j]]]/Sqrt[15]];
AppendTo[NewRow2,Im[BigMatrix[[i]][[j]]](-Sqrt[15])];
AppendTo[NewRow2,Re[BigMatrix[[i]][[j]]]];
];
AppendTo[BigMatrixReIm,NewRow1];
AppendTo[BigMatrixReIm,NewRow2];
];
(* finding a better basis *)
BestRels=LatticeReduce[BigMatrixReIm];

(* doing Smith decomposition to find a basis of the refinement *)
{u,r, v} = SmithDecomposition[BestRels];
vt=Transpose[v];
SmallBestRels={};
For[i=1,i≤28,i++,
basisvec=Table[0,{k,1,120}];
basisvec[[i]]=1;
AppendTo[SmallBestRels,LinearSolve[vt,basisvec]];
];
SmallerBestRels=LatticeReduce[SmallBestRels];

(* back to complex matrix *)
BackToCxMatrix={};
For[i=1,i≤28,i++,
newrow=Table[SmallerBestRels[[i]][[2k-1]]+ I Sqrt[15] SmallerBestRels[[i]][[2k]],{k,1,60}];
AppendTo[BackToCxMatrix,newrow];
];
MatrixRank[N[BackToCxMatrix,1000]]
BackToCxReduced={};
rk=0;
For[i=1,i≤28,i++,
newred=Append[BackToCxReduced,BackToCxMatrix[[i]]];
If[MatrixRank[N[newred,1000]]>rk, rk++;Print[i," ",rk];BackToCxReduced=newred;];
If[rk==14,i=120];
];

Eqs3Woddimproved = Expand[Table[Sum[BackToCxReduced[[i]][[k]] AllMons3odd[[k]],{k,1,60}],{i,1,14}]];
Table[ByteCount[Eqs3Woddimproved[[i]]],{i,1,14}]
Table[Length[MonomialList[Eqs3Woddimproved[[i]]]],{i,1,14}]
GoodOddEqs;
Table[ByteCount[GoodOddEqs[[i]]],{i,1,14}]
Table[Length[MonomialList[GoodOddEqs[[i]]]],{i,1,14}]