This notebook computes a family of Dolgachev surfaces with  a double and a triple fiber.
The main idea is to compute the ring \oplus H^0(aF+bS) where F is the canonical class and S
is one of the 6-sections.

(* we start by computing the generating series *)
GenSeries=Factor[1/(1-t^2)/(1-t^3)+Sum[Sum[(((2 a -1 )b -3 b^2)/2+1)t^a s^b,{a,3b,Infinity}] ,{b,1,Infinity}]+ Sum[ (((6b -1 )b -3 b^2)/2+1)  t^(3b) s^b s/(1-s), {b,1,Infinity}]+ Sum[ (((6b+2 -1 )b -3 b^2)/2+1)  t^(3b+1)s^b s/(1-s), {b,1,Infinity}]+ Sum[ (((6b+4 -1 )b -3 b^2)/2+1)  t^(3b+2)s^b s/(1-s), {b,1,Infinity}]
+ (1+t^2)s/(1-s)
]

We are going to have variables u0,u1,v1,v2,v3,v4,v5,v6 of weights 
(2,0),(3,0),(0,1),(3,1),(4,1),(4,1),(5,1),(5,1) respectively.

(* we see that there are three equations of degrees (2,8),(2,9) and (2,10) *)
Series[SeriesCoefficient[1/((1-t^2)(1-t^3)(1-s)(1-s t^3) (1- s t^4)^2 (1 - s t^5)^2) - GenSeries,{s,0,2}],{t,0,21}]

We will set up the quadrics.

Coeffs=Table[ToExpression[StringJoin["c",ToString[k]]],{k,1,500}];
uvs={u0,u1,v1,v2,v3,v4,v5,v6};
degs={{2,0},{3,0},{0,1},{3,1},{4,1},{4,1},{5,1},{5,1}};
as={a1,a2,a3,a4,a5,a6,a7,a8};

dega=as.degs;
count=0;
deg28=Solve[Join[{dega[[1]]==8,dega[[2]]==2},{a1≥0,a2≥0,a3≥0,a4≥0,a5≥0,a6≥0,a7≥0,a8≥0}],Integers];
deg29=Solve[Join[{dega[[1]]==9,dega[[2]]==2},{a1≥0,a2≥0,a3≥0,a4≥0,a5≥0,a6≥0,a7≥0,a8≥0}],Integers];
deg210=Solve[Join[{dega[[1]]==10,dega[[2]]==2},{a1≥0,a2≥0,a3≥0,a4≥0,a5≥0,a6≥0,a7≥0,a8≥0}],Integers];

Quadrics81=0;Quadrics82=0;Quadrics83=0;
For[i=1,i≤Length[deg28],i++,count++;Quadrics81+=Coeffs[[count]] Product[uvs[[k]]^(as[[k]]/.deg28[[i]]),{k,1,8}]];
For[i=1,i≤Length[deg28],i++,count++;Quadrics82+=Coeffs[[count]] Product[uvs[[k]]^(as[[k]]/.deg28[[i]]),{k,1,8}]];
For[i=1,i≤Length[deg28],i++,count++;Quadrics83+=Coeffs[[count]] Product[uvs[[k]]^(as[[k]]/.deg28[[i]]),{k,1,8}]];

Quadrics91=0;Quadrics92=0;Quadrics93=0;
For[i=1,i≤Length[deg29],i++,count++;Quadrics91+=Coeffs[[count]] Product[uvs[[k]]^(as[[k]]/.deg29[[i]]),{k,1,8}]];
For[i=1,i≤Length[deg29],i++,count++;Quadrics92+=Coeffs[[count]] Product[uvs[[k]]^(as[[k]]/.deg29[[i]]),{k,1,8}]];
For[i=1,i≤Length[deg29],i++,count++;Quadrics93+=Coeffs[[count]] Product[uvs[[k]]^(as[[k]]/.deg29[[i]]),{k,1,8}]];

Quadrics101=0;Quadrics102=0;Quadrics103=0;
For[i=1,i≤Length[deg210],i++,count++;Quadrics101+=Coeffs[[count]] Product[uvs[[k]]^(as[[k]]/.deg210[[i]]),{k,1,8}]];
For[i=1,i≤Length[deg210],i++,count++;Quadrics102+=Coeffs[[count]] Product[uvs[[k]]^(as[[k]]/.deg210[[i]]),{k,1,8}]];
For[i=1,i≤Length[deg210],i++,count++;Quadrics103+=Coeffs[[count]] Product[uvs[[k]]^(as[[k]]/.deg210[[i]]),{k,1,8}]];

count

Now we simplify the quadrics using various symmetries of the problem.

(* fix coefficients of Quadrics81,82,83 at v3^2, v3 v4, v4^2 *)
sls=Solve[{Coefficient[Quadrics81,v3^2]==1,Coefficient[Quadrics81,v3 v4]==0, Coefficient[Quadrics81,v4^2]==0,
  Coefficient[Quadrics82,v3^2]==0,Coefficient[Quadrics82,v3 v4]==1, Coefficient[Quadrics82,v4^2]==0,
  Coefficient[Quadrics83,v3^2]==0,Coefficient[Quadrics83,v3 v4]==0, Coefficient[Quadrics83,v4^2]==1}][[1]];
Quadrics81=Quadrics81/.sls;
Quadrics82=Quadrics82/.sls;
Quadrics83=Quadrics83/.sls;

(* Set some other coefficients using v3 -> b1 v3 + b2 v4 +  b3 v1 u0^2, etc., *)
Eqs={};
VarChange={v3->b1 v3 + b2 v4 + b3 v1 u0^2, v4->b4 v3 + b5 v4 + b6 v1 u0^2};
{Quadrics81n,Quadrics82n,Quadrics83n}={Quadrics81,Quadrics82,Quadrics83}/.VarChange;
lincomb= {a1,a2,a3}.{Quadrics81n,Quadrics82n,Quadrics83n};
sls=Solve[{Coefficient[lincomb,v3^2]==1,Coefficient[lincomb,v3 v4]==0, Coefficient[lincomb,v4^2]==0},{a1,a2,a3}][[1]];
AppendTo[Eqs,Coefficient[lincomb/.sls,v3 v1 u0^2]==1];
AppendTo[Eqs,Coefficient[lincomb/.sls,v4 v1 u0^2]==0];
lincomb= {a1,a2,a3}.{Quadrics81n,Quadrics82n,Quadrics83n};
sls=Solve[{Coefficient[lincomb,v3^2]==0,Coefficient[lincomb,v3 v4]==1, Coefficient[lincomb,v4^2]==0},{a1,a2,a3}][[1]];
AppendTo[Eqs,Coefficient[lincomb/.sls,v3 v1 u0^2]==0];
AppendTo[Eqs,Coefficient[lincomb/.sls,v4 v1 u0^2]==0];
lincomb= {a1,a2,a3}.{Quadrics81n,Quadrics82n,Quadrics83n};
sls=Solve[{Coefficient[lincomb,v3^2]==0,Coefficient[lincomb,v3 v4]==0, Coefficient[lincomb,v4^2]==1},{a1,a2,a3}][[1]];
AppendTo[Eqs,Coefficient[lincomb/.sls,v3 v1 u0^2]==0];
AppendTo[Eqs,Coefficient[lincomb/.sls,v4 v1 u0^2]==1];
(*
sls=Solve[Eqs,{b1,b2,b3,b4,b5,b6}];
Table[Length[sls[[l]]],{l,1,Length[sls]}]
*)
(* this works but takes some time, which means we will not do it every time *)
sls=Solve[{Coefficient[Quadrics81,v3 v1 u0^2]==1,Coefficient[Quadrics81,v4 v1 u0^2]==0,Coefficient[Quadrics82,v3 v1 u0^2]==0,Coefficient[Quadrics82,v4 v1 u0^2]==0,Coefficient[Quadrics83,v3 v1 u0^2]==0,Coefficient[Quadrics83,v4 v1 u0^2]==1}][[1]];
Quadrics81=Quadrics81/.sls;
Quadrics82=Quadrics82/.sls;
Quadrics83=Quadrics83/.sls;

(* we will have the (v5,v6) a dual basis to (v3,v4) in the pairing in the Gorenstein ring. Thus the v3 v5, v3 v6, v4 v5, v4 v6 coefficients of the (2,9) quadrics can be set *)
sls=Solve[{Coefficient[Quadrics91,v3 v5]==0,Coefficient[Quadrics91,v3 v6]==1, Coefficient[Quadrics91,v4 v5]==0,Coefficient[Quadrics91,v4 v6]==0,
Coefficient[Quadrics92,v3 v5]==1,Coefficient[Quadrics92,v3 v6]==0, Coefficient[Quadrics92,v4 v5]==0,Coefficient[Quadrics92,v4 v6]==-1,
Coefficient[Quadrics93,v3 v5]==0,Coefficient[Quadrics93,v3 v6]==0, Coefficient[Quadrics93,v4 v5]==1,Coefficient[Quadrics93,v4 v6]==0 
}][[1]];
Quadrics91=Quadrics91/.sls;
Quadrics92=Quadrics92/.sls;
Quadrics93=Quadrics93/.sls;

(* we also have symmetries v5 -> v5 + b1 u0 v2 + b2 u0 u1 v1 and similar for v6 which allows us to get rid of u0 v3 v2, u0 u1 v1 v2 in Quad91 and similar in Quad93 *)
VarChange={v5->v5 + b1 u0 v2 + b2 u0 u1 v1 , v6->v6 + b3 u0 v2 + b4 u0 u1 v1};
Quadrics91n=Quadrics91/.VarChange;
Quadrics93n=Quadrics93/.VarChange;
sls=Solve[{Coefficient[Quadrics91n,v3 u0 v2]==0,Coefficient[Quadrics91n,v3 u0 u1 v1]==0, Coefficient[Quadrics93n,v4 u0 v2]==0,Coefficient[Quadrics93n,v4 u0 u1 v1]==0},{b1,b2,b3,b4}];
(* it works *)
sls=Solve[{Coefficient[Quadrics91,v3 u0 v2]==0,Coefficient[Quadrics91,v3 u0 u1 v1]==0, Coefficient[Quadrics93,v4 u0 v2]==0,Coefficient[Quadrics93,v4 u0 u1 v1]==0}][[1]];
Quadrics91=Quadrics91/.sls;
Quadrics93=Quadrics93/.sls;

(* now for the last (2,10) quadrics we can make sure that we don't have any combinations of u0 Quad8i, 
and the standard coefficients at v5^2, v5 v6, v6^2*)

sls=Solve[{Coefficient[Quadrics101,u0 v3^2]==0,Coefficient[Quadrics101,u0 v3 v4]==0, Coefficient[Quadrics101,u0 v4^2]==0,
Coefficient[Quadrics102,u0 v3^2]==0,Coefficient[Quadrics102,u0 v3 v4]==0, Coefficient[Quadrics102,u0 v4^2]==0,
Coefficient[Quadrics103,u0 v3^2]==0,Coefficient[Quadrics103,u0 v3 v4]==0, Coefficient[Quadrics103,u0 v4^2]==0
}][[1]];
Quadrics101=Quadrics101/.sls;
Quadrics102=Quadrics102/.sls;
Quadrics103=Quadrics103/.sls;
sls=Solve[{Coefficient[Quadrics101,v5^2]==1,Coefficient[Quadrics101,v5 v6]==0, Coefficient[Quadrics101,v6^2]==0,
Coefficient[Quadrics102,v5^2]==0,Coefficient[Quadrics102,v5 v6]==1, Coefficient[Quadrics102,v6^2]==0,
Coefficient[Quadrics103,v5^2]==0,Coefficient[Quadrics103,v5 v6]==0, Coefficient[Quadrics103,v6^2]==1
}][[1]];
Quadrics101=Quadrics101/.sls;
Quadrics102=Quadrics102/.sls;
Quadrics103=Quadrics103/.sls;

(* Finally, there are symmetries v2-> a1 v2 + a1 v1 u1, v1 -> a3 v1. We will use them to set the coeefficients of  *)
VarChange={v2->a1 v2+ a2 v1 u1, v1-> a3 v1};
sls=Solve[{Coefficient[Quadrics91/.VarChange, u1 v2^2]==1,Coefficient[Quadrics91/.VarChange, u1^2 v1 v2]==1,Coefficient[Quadrics91/.VarChange, u1^3 v1^2]==0},{a1,a2,a3}];
(* it works *)
sls=Solve[{Coefficient[Quadrics91, u1 v2^2]==1,Coefficient[Quadrics91, u1^2 v1 v2]==1,Coefficient[Quadrics91, u1^3 v1^2]==0}][[1]];
Quadrics91=Quadrics91/.sls;

We will  use the quadrics to construct the multiplication table of the free module with respect to u0, u1, v1, v2. The basis will be denoted by 1, v3, v4, v5, v6, w.

MultTable=Table[0,{i,1,6},{j,1,6}];
MultTable[[1]]={1,v3, v4, v5, v6, w};
For[k=1,k≤6,k++,MultTable[[k]][[1]]=MultTable[[1]][[k]]];
(* we will use v3 v5 = w *)
MultTable[[2]][[4]]=w;MultTable[[4]][[2]]=w;
MultTable//MatrixForm

(* we build products of v3, v4, v5, v6 using the quadric relations *)
MultTable[[2]][[2]] = v3^2 - Quadrics81;
MultTable[[2]][[3]]= v3 v4 - Quadrics82;MultTable[[3]][[2]]=MultTable[[2]][[3]];
MultTable[[3]][[3]] = v4^2 - Quadrics83;
MultTable[[2]][[5]] = v3 v6  - Quadrics91;MultTable[[5]][[2]] =MultTable[[2]][[5]];
MultTable[[3]][[4]] = v4 v5 - Quadrics93;MultTable[[4]][[3]] =MultTable[[3]][[4]];
MultTable[[3]][[5]] =v4 v6 + Quadrics92 + w - v3 v5;MultTable[[5]][[3]]=MultTable[[3]][[5]];
MultTable[[4]][[4]] = v5^2 - Quadrics101;
MultTable[[4]][[5]] = v5 v6- Quadrics102; MultTable[[5]][[4]] =MultTable[[4]][[5]] ;
MultTable[[5]][[5]] = v6^2 - Quadrics103;

(* nNw we construct multiplication with w. The idea is that v3 w = v3 v3 v5 = MT[[2]][[2]] v5, v4 w = v3 v4 v5, v5 w = v5 v5 v3, v6 w = v3 v6 v5 *)
v3w=MultTable[[2]][[2]] v5;
MultTable[[2]][[6]] =Expand[ v3w+
Coefficient[v3w, v5 v3] (MultTable[[2]][[4]]-v5 v3) + Coefficient[v3w, v5 v4 ]  (MultTable[[3]][[4]]-v5 v4)   + Coefficient[v3w, v5^2 ] (MultTable[[4]][[4]] -v5^2)+
Coefficient[v3w, v5 v6] (MultTable[[4]][[5]]-v5 v6) ];MultTable[[6]][[2]] =MultTable[[2]][[6]] ;
v4w=MultTable[[2]][[3]] v5;
MultTable[[3]][[6]] =Expand[ v4w+
Coefficient[v4w, v5 v3] (MultTable[[2]][[4]]-v5 v3) + Coefficient[v4w, v5 v4 ]  (MultTable[[3]][[4]]-v5 v4)   + Coefficient[v4w, v5^2 ] (MultTable[[4]][[4]] -v5^2)+
Coefficient[v4w, v5 v6] (MultTable[[4]][[5]]-v5 v6) ];MultTable[[6]][[3]] =MultTable[[3]][[6]];
v6w = MultTable[[2]][[5]] v5;
MultTable[[5]][[6]] =Expand[ v6w+
Coefficient[v6w, v5 v3] (MultTable[[2]][[4]]-v5 v3) + Coefficient[v6w, v5 v4 ]  (MultTable[[3]][[4]]-v5 v4)   + Coefficient[v6w, v5^2 ] (MultTable[[4]][[4]] -v5^2)+
Coefficient[v6w, v5 v6] (MultTable[[4]][[5]]-v5 v6) ];MultTable[[6]][[5]] =MultTable[[5]][[6]];
v5w = MultTable[[4]][[4]]v3;
MultTable[[4]][[6]] =Expand[v5w +
Coefficient[v5w, v3^2](MultTable[[2]][[2]]-v3^2) + Coefficient[v5w, v3 v4](MultTable[[2]][[3]]-v3 v4) + Coefficient[v5w, v3 v5](MultTable[[2]][[4]]-v3 v5) +Coefficient[v5w, v3 v6](MultTable[[2]][[5]]-v3 v6)  
];MultTable[[6]][[4]] =MultTable[[4]][[6]];


(* now we need to compute w w = v3 v3 v5 v5 = (v3 w) v5 = MT[[2]][[6] v5*)
ww = MultTable[[2]][[6]] v5;
MultTable[[6]][[6]] = Expand[ ww+Coefficient[ww, v3 v3] (MultTable[[2]][[2]]-v3 v3) + Coefficient[ww, v3 v4] (MultTable[[2]][[3]]-v3 v4)
+ Coefficient[ww, v3 v5] (MultTable[[2]][[4]]-v3 v5) + Coefficient[ww, v3 v6] (MultTable[[2]][[5]]-v3 v6)
+ Coefficient[ww, v4 v4] (MultTable[[3]][[3]]-v4 v4)+ Coefficient[ww, v4 v5] (MultTable[[3]][[4]]-v4 v5)+ Coefficient[ww, v4 v6] (MultTable[[3]][[5]]-v4 v6)
+ Coefficient[ww, v5 v5] (MultTable[[4]][[4]]-v5 v5)+ Coefficient[ww, v5 v6] (MultTable[[4]][[5]]-v5 v6) + Coefficient[ww, v6 v6] (MultTable[[5]][[5]]-v6 v6)
+Coefficient[ww,v5 w](MultTable[[4]][[6]]-v5 w)
];

Table[ByteCount[MultTable[[k]][[l]]],{k,1,6},{l,1,6}]//MatrixForm

Let us now compute the associativity relations.

(* some basic sanity check: we want to make sure that the terms of the multiplication table are linear *)
For[i=1,i≤6,i++,For[j=1,j≤6,j++,
entry = MultTable[[i]][[j]];
If[0≠ tocheck - Coefficient[entry,v3] v3 -  Coefficient[entry,v4] v4 -  Coefficient[entry,v5] v5 -  Coefficient[entry,v6] v6 - Coefficient[entry,w] w - entry/.{v3->0,v4->0,v5->0,v6->0,w->0},Print["Waah ",{i,j}]]
]];

(* now convert the table into a more useful format *)
MultTableVector=Table[0,{i,1,6},{j,1,6}];
For[i=1,i≤6,i++,For[j=1,j≤6,j++,
entry=MultTable[[i]][[j]];
MultTableVector[[i]][[j]]={entry/.{v3->0,v4->0,v5->0,v6->0,w->0},Coefficient[entry,v3] ,  Coefficient[entry,v4],Coefficient[entry,v5] , Coefficient[entry,v6] , Coefficient[entry,w] }; 
]]

(* set up associativity relations v_i (v_j v_k ) = v_j (v_i v_k); we may assume that i<j and k > 1*)
Eqs={};
For[k=2,k≤6,k++,
For[i=1,i≤6,i++,For[j=1,j<i,j++,
tbz =Expand[Sum[(MultTableVector[[j]][[k]] )[[m]] MultTableVector[[m]][[i]],{m,1,6}]-Sum[(MultTableVector[[i]][[k]] )[[m]] MultTableVector[[m]][[j]],{m,1,6}]];
(* we want polynomials in u0, u1, v1, v2 to be zero *)
For[n=1,n≤6,n++,
Eqs=Join[Eqs,Flatten[CoefficientList[tbz[[n]],{u0,u1,v1,v2}]]];
];
]];
];
Eqs=Complement[DeleteDuplicates[Eqs],{0}];

We will now work to solve these equations. We will keep track of the quadrics as we do so. This calculation has been attempted many times and there are some idiosyncratic choices made in the process that allowed it to work. There is no claim that we get the only nine-dimensional component of the solution locus, but it may well be the case. 

The calculation runs under a minute. In the process it keeps printing out the number of equations left, their byte count and the number of variables left.

Quadrics={Quadrics81,Quadrics82,Quadrics83,Quadrics91,Quadrics92,Quadrics93,Quadrics101,Quadrics102,Quadrics103};
{Length[Eqs],ByteCount[Eqs],Length[Variables[Eqs]]}

GoodEqsList={};
For[i=1,i≤Length[Eqs],i++,
If[ByteCount[Eqs[[i]]]≤160,AppendTo[GoodEqsList,i];]
];
GoodEqsList;
sols=Solve[Table[0== Eqs[[GoodEqsList[[k]]]],{k,1,Length[GoodEqsList]}],{c102,c115,c116,c117,c118,c119,c124,c129,c13,c131,c136,c139,c143,c18,c20,c26,c31,c33,c39,c49,c51,c52,c62,c64,c66,c67,c81,c82,c88,c90,c95,c96,c97,c98,c99}][[1]]
Quadrics1=Quadrics/.sols;
Eqs1=Complement[DeleteDuplicates[Eqs/.sols],{0}];
{Length[Eqs1],ByteCount[Eqs1],Length[Variables[Eqs1]]}

GoodEqsList={};
For[i=1,i≤Length[Eqs1],i++,
If[ByteCount[Eqs1[[i]]]≤160,AppendTo[GoodEqsList,i];]
];
GoodEqsList;
sols1=Solve[Table[0== Eqs1[[GoodEqsList[[k]]]],{k,1,Length[GoodEqsList]}],{c104,c109,c111,c122,c144,c17,c6}][[1]]
Quadrics2=Quadrics1/.sols1;
Eqs2=Complement[DeleteDuplicates[Eqs1/.sols1],{0}];
{Length[Eqs2],ByteCount[Eqs2],Length[Variables[Eqs2]]}

ts=Sort[Table[{ByteCount[Eqs2[[k]]],Eqs2[[k]],k},{k,1,Length[Eqs2]}]];
Table[ts[[k]],{k,1,50}];
GoodEqsList={4,15,442,78,678}
sols2=Solve[Join[{c110≠0},Table[0== Eqs2[[GoodEqsList[[k]]]],{k,1,Length[GoodEqsList]}]]][[1]]
Quadrics3=Quadrics2/.sols2;
Eqs3=Complement[DeleteDuplicates[Eqs2/.sols2],{0}];
{Length[Eqs3],ByteCount[Eqs3],Length[Variables[Eqs3]]}

ts=Sort[Table[{Length[Eqs3[[k]]],Eqs3[[k]],k},{k,1,Length[Eqs3]}]];
Table[ts[[k]],{k,1,50}];
GoodEqsList={62,38,3,8};
sols3=Solve[Join[{c17≠0,c19≠0,c123≠0},Table[0== Eqs3[[GoodEqsList[[k]]]],{k,1,Length[GoodEqsList]}]]][[1]]
Quadrics4=Quadrics3/.sols3;
Eqs4=Complement[DeleteDuplicates[Factor[Eqs3/.sols3]],{0}];
{Length[Eqs4],ByteCount[Eqs4],Length[Variables[Eqs4]]}

ts=Sort[Table[{ByteCount[Eqs4[[k]]],Eqs4[[k]],k},{k,1,Length[Eqs4]}]];
Table[ts[[k]],{k,1,50}];
GoodEqsList={40,113,193,54,87,341,58};
sols4=Solve[Join[{c17≠0,c19≠0,c123≠0},Table[0== Eqs4[[GoodEqsList[[k]]]],{k,1,Length[GoodEqsList]}]]][[1]]
Quadrics5=Quadrics4/.sols4;
Eqs5=Complement[DeleteDuplicates[Factor[Eqs4/.sols4]],{0}];
{Length[Eqs5],ByteCount[Eqs5],Length[Variables[Eqs5]]}

ts=Sort[Table[{ByteCount[Eqs5[[k]]],Eqs5[[k]],k},{k,1,Length[Eqs5]}]];
Table[ts[[k]],{k,1,50}];
GoodEqsList={2,46,503,158};
sols5=Solve[Join[{c17≠0,c19≠0,c123≠0,c110≠0,c108≠0},Table[0== Eqs5[[GoodEqsList[[k]]]],{k,1,Length[GoodEqsList]}]],{c108,c128,c130,c5}][[1]]
Quadrics6=Quadrics5/.sols5;
Eqs6=Complement[DeleteDuplicates[Factor[Eqs5/.sols5]],{0}];
{Length[Eqs6],ByteCount[Eqs6],Length[Variables[Eqs6]]}

ts=Sort[Table[{ByteCount[Eqs6[[k]]],Eqs6[[k]],k},{k,1,Length[Eqs6]}]];
Table[ts[[k]],{k,1,50}];
GoodEqsList={28,436,117};
sols6=Solve[Join[{c17≠0,c19≠0,c123≠0,c110≠0,c108≠0},Table[0== Eqs6[[GoodEqsList[[k]]]],{k,1,Length[GoodEqsList]}]],{c19,c78,c91}][[1]]
Quadrics7=Quadrics6/.sols6;
Eqs7=Complement[DeleteDuplicates[Factor[Eqs6/.sols6]],{0}];
{Length[Eqs7],ByteCount[Eqs7],Length[Variables[Eqs7]]}

ts=Sort[Table[{ByteCount[Eqs7[[k]]],Eqs7[[k]],k},{k,1,Length[Eqs7]}]];
Table[ts[[k]],{k,1,50}];
GoodEqsList={304,190,312};
sols7=Solve[Join[{c17≠0,c19≠0,c123≠0,c110≠0,c108≠0},Table[0== Eqs7[[GoodEqsList[[k]]]],{k,1,Length[GoodEqsList]}]],{c89,c68,c75}][[1]]
Quadrics8=Quadrics7/.sols7;
Eqs8=Complement[DeleteDuplicates[Factor[Eqs7/.sols7]],{0}];
{Length[Eqs8],ByteCount[Eqs8],Length[Variables[Eqs8]]}

ts=Sort[Table[{ByteCount[Eqs8[[k]]],Eqs8[[k]],k},{k,1,Length[Eqs8]}]];
Table[ts[[k]],{k,1,50}];
GoodEqsList={246,118};
sols8=Solve[Join[{c17≠0,c19≠0,c123≠0,c110≠0,c108≠0},Table[0== Eqs8[[GoodEqsList[[k]]]],{k,1,Length[GoodEqsList]}]],{c69,c65}][[1]]
Quadrics9=Quadrics8/.sols8;
Eqs9=Complement[DeleteDuplicates[Factor[Eqs8/.sols8]],{0}];
{Length[Eqs9],ByteCount[Eqs9],Length[Variables[Eqs9]]}

ts=Sort[Table[{ByteCount[Eqs9[[k]]],Eqs9[[k]],k},{k,1,Length[Eqs9]}]];
Table[ts[[k]],{k,1,50}];
GoodEqsList={4,21,1};
sols9=Solve[Join[{c17≠0,c19≠0,c123≠0,c110≠0,c108≠0},Table[0== Eqs9[[GoodEqsList[[k]]]],{k,1,Length[GoodEqsList]}]],{c101,c36,c121}][[1]]
Quadrics10=Quadrics9/.sols9;
Eqs10=Complement[DeleteDuplicates[Factor[Eqs9/.sols9]],{0}];
{Length[Eqs10],ByteCount[Eqs10],Length[Variables[Eqs10]]}

ts=Sort[Table[{ByteCount[Eqs10[[k]]],Eqs10[[k]],k},{k,1,Length[Eqs10]}]];
Table[ts[[k]],{k,1,50}];
GoodEqsList={5,66};
sols10=Solve[Join[{c17≠0,c19≠0,c123≠0,c110≠0,c108≠0},Table[0== Eqs10[[GoodEqsList[[k]]]],{k,1,Length[GoodEqsList]}]],{c100,c60}][[1]]
Quadrics11=Quadrics10/.sols10;
Eqs11=Complement[DeleteDuplicates[Factor[Eqs10/.sols10]],{0}];
{Length[Eqs11],ByteCount[Eqs11],Length[Variables[Eqs11]]}

ts=Sort[Table[{ByteCount[Eqs11[[k]]],Eqs11[[k]],k},{k,1,Length[Eqs11]}]];
Table[ts[[k]],{k,1,50}];
GoodEqsList={60,3};
sols11=Solve[Join[{c17≠0,c19≠0,c123≠0,c110≠0,c108≠0},Table[0== Eqs11[[GoodEqsList[[k]]]],{k,1,Length[GoodEqsList]}]],{c63,c22}][[1]](*it's an Occam's razor choice *)
Quadrics12=Quadrics11/.sols11;
Eqs12=Complement[DeleteDuplicates[Factor[Eqs11/.sols11]],{0}];
{Length[Eqs12],ByteCount[Eqs12],Length[Variables[Eqs12]]}

ts=Sort[Table[{ByteCount[Eqs12[[k]]],Eqs12[[k]],k},{k,1,Length[Eqs12]}]];
Table[ts[[k]],{k,1,50}];
GoodEqsList={123,195};
sols12=Solve[Join[{c17≠0,c19≠0,c123≠0,c110≠0,c108≠0},Table[0== Eqs12[[GoodEqsList[[k]]]],{k,1,Length[GoodEqsList]}]],{c8,c9}][[1]]
Quadrics13=Quadrics12/.sols12;
Eqs13=Complement[DeleteDuplicates[Factor[Eqs12/.sols12]],{0}];
{Length[Eqs13],ByteCount[Eqs13],Length[Variables[Eqs13]]}

ts=Sort[Table[{ByteCount[Eqs13[[k]]],k},{k,1,Length[Eqs13]}]];
Table[ts[[k]],{k,1,50}];
GoodEqsList={4};
sols13=Solve[Join[{c17≠0,c19≠0,c123≠0,c110≠0,c108≠0},Table[0== Eqs13[[GoodEqsList[[k]]]],{k,1,Length[GoodEqsList]}]]][[1]]
Quadrics14=Quadrics13/.sols13;
Eqs14=Complement[DeleteDuplicates[Factor[Eqs13/.sols13]],{0}];
{Length[Eqs14],ByteCount[Eqs14],Length[Variables[Eqs14]]}

sols14=Solve[Table[0==Eqs14[[k]],{k,1,Length[Eqs14]}],{c120,c140,c141}][[1]]
Complement[DeleteDuplicates[Factor[Eqs14/.sols14]],{0}] (* we are done *)
Quadrics15=Simplify[Quadrics14/.sols14];

We can verify that a generic member of this nine-parameter family is a (2,3) Dolgachev surface. The next step is to impose a condition that the fiber at u0=u1=1 contains a line. This leads to a seven-parameter family.

(* we define a line in P^5 *)
vs={v1,v2,v3,v4,v5,v6};
LineInP5={v1->1,v2->t,v3->b1+b5 t,v4->b2+b6 t,v5->b3+b7 t,v6->b4+b8 t};
Eqs=Flatten[Table[0==SeriesCoefficient[Quadrics15[[k]]/.Join[LineInP5,{u0->1,u1->1}],{t,0,l}],{k,1,9},{l,0,2}]];
(* The choice of variables to solve for is a consequence of some trial and error approach, similar to solving for the nine-parameter family. *)
containsline=Solve[Eqs,{b3,b7,b8,c30,c32,c103,c123,c135,c137,c142}][[1]];
QuadricsLine=Quadrics15/.containsline;
Complement[Variables[QuadricsLine],uvs]

We will now consider a condition of having two lines in this fiber. This is tricky and there are likely several cases. We have found at least two different 5-parameter families. In one family the degrees of the components of the u0=u1=1 fiber are 1,1,1,3. The one we present below has degrees 1,2,1,2. It appears to be more accessible, although still very complicated. It has been found by creatively looking for solutions of the equations below, but details are lost to time.

Line2InP5={v1->1,v2->t,v3->r1+r5 t,v4->r2+r6 t,v5->r3+r7 t,v6->r4+r8 t};
containsline2=containsline/.{b1->r1,b2->r2,b3->r3,b4->r4,b5->r5,b6->r6,b7->r7,b8->r8};
Eqs=Factor[Flatten[Table[SeriesCoefficient[Quadrics15[[k]]/.Join[Line2InP5,{u0->1,u1->1}],{t,0,l}],{k,1,9},{l,0,2}]]];
Table[containsline[[k]][[1]],{k,1,Length[containsline]}]
For[i=4,i≤10,i++,
AppendTo[Eqs,Factor[containsline[[i]][[2]]-containsline2[[i]][[2]]]];
];

Solution1212={r1->r1,r2->-((-b1-(b1^2)-b1 b2+r1+(r1^2))/r1),r3->-(((b1+r1) (b1 r1+(b1^2) r1+b2 b5 r1-(r1^2)-(r1^3)-b1 r5-(b1^2) r5-b1 b2 r5+r1 r5+(r1^2) r5) ((b1^2) (r1^2)+(b1^3) (r1^2)+2 b1 b2 b5 (r1^2)-b2 (b5^2) (r1^2)-2 b1 (r1^3)-2 (b1^2) (r1^3)-2 b2 b5 (r1^3)+(r1^4)+b1 (r1^4)-2 (b1^2) r1 r5-2 (b1^3) r1 r5-2 (b1^2) b2 r1 r5+4 b1 (r1^2) r5+4 (b1^2) (r1^2) r5+2 b1 b2 (r1^2) r5+2 b2 b5 (r1^2) r5-2 (r1^3) r5-2 b1 (r1^3) r5+(b1^2) (r5^2)+(b1^3) (r5^2)+(b1^2) b2 (r5^2)-2 b1 r1 (r5^2)-2 (b1^2) r1 (r5^2)-2 b1 b2 r1 (r5^2)+(r1^2) (r5^2)+b1 (r1^2) (r5^2)))/((-b5 r1+b1 r5) (2 b1 r1+(b1^2) r1+b1 b5 r1+2 b2 b5 r1-2 (r1^2)-b5 (r1^2)-(r1^3)-2 b1 r5-2 (b1^2) r5-2 b1 b2 r5+2 r1 r5+b1 r1 r5+(r1^2) r5) (2 (b1^2) (r1^2)+(b1^3) (r1^2)-b1 b5 (r1^2)+2 b1 b2 b5 (r1^2)-b2 (b5^2) (r1^2)-2 b1 (r1^3)+b5 (r1^3)-b1 (r1^4)-3 (b1^2) r1 r5-2 (b1^3) r1 r5-2 (b1^2) b2 r1 r5+b1 b5 r1 r5+3 b1 (r1^2) r5-b5 (r1^2) r5+2 b1 (r1^3) r5+(b1^2) (r5^2)+(b1^3) (r5^2)+(b1^2) b2 (r5^2)-b1 r1 (r5^2)-b1 (r1^2) (r5^2)))),r4->-((b1 (b1+(b1^2)+b1 b2-r1-(r1^2)) (b1-b5-r1+r5))/(2 (b1^2) (r1^2)+(b1^3) (r1^2)-b1 b5 (r1^2)+2 b1 b2 b5 (r1^2)-b2 (b5^2) (r1^2)-2 b1 (r1^3)+b5 (r1^3)-b1 (r1^4)-3 (b1^2) r1 r5-2 (b1^3) r1 r5-2 (b1^2) b2 r1 r5+b1 b5 r1 r5+3 b1 (r1^2) r5-b5 (r1^2) r5+2 b1 (r1^3) r5+(b1^2) (r5^2)+(b1^3) (r5^2)+(b1^2) b2 (r5^2)-b1 r1 (r5^2)-b1 (r1^2) (r5^2))),r5->r5,r6->-(1/(r1 (b5 r1-b1 r5)))(-2 (b1^2) r1-(b1^3) r1+b1 b5 r1-2 b1 b2 b5 r1+b2 (b5^2) r1+2 b1 (r1^2)-b5 (r1^2)+b1 (r1^3)+2 (b1^2) r5+2 (b1^3) r5+2 (b1^2) b2 r5-b1 b5 r5-(b1^2) b5 r5-b1 b2 b5 r5-2 b1 r1 r5+b5 r1 r5-2 b1 (r1^2) r5+b5 (r1^2) r5),r7->-(((b1+r1) (b1 r1+(b1^2) r1+b2 b5 r1-(r1^2)-(r1^3)-b1 r5-(b1^2) r5-b1 b2 r5+r1 r5+(r1^2) r5) (2 (b1^2) (r1^2)+(b1^3) (r1^2)-b1 b5 (r1^2)+2 b1 b2 b5 (r1^2)-b2 (b5^2) (r1^2)-4 b1 (r1^3)-(b1^2) (r1^3)+b5 (r1^3)-b1 b5 (r1^3)-2 b2 b5 (r1^3)+2 (r1^4)-b1 (r1^4)+b5 (r1^4)+(r1^5)-3 (b1^2) r1 r5-2 (b1^3) r1 r5-2 (b1^2) b2 r1 r5+b1 b5 r1 r5+7 b1 (r1^2) r5+2 (b1^2) (r1^2) r5+2 b1 b2 (r1^2) r5-b5 (r1^2) r5+2 b1 b5 (r1^2) r5+2 b2 b5 (r1^2) r5-4 (r1^3) r5+3 b1 (r1^3) r5-2 b5 (r1^3) r5-3 (r1^4) r5+(b1^2) (r5^2)+(b1^3) (r5^2)+(b1^2) b2 (r5^2)-3 b1 r1 (r5^2)-(b1^2) r1 (r5^2)-2 b1 b2 r1 (r5^2)-b1 b5 r1 (r5^2)+2 (r1^2) (r5^2)-3 b1 (r1^2) (r5^2)+b5 (r1^2) (r5^2)+3 (r1^3) (r5^2)+b1 r1 (r5^3)-(r1^2) (r5^3)))/((-b5 r1+b1 r5) (2 b1 r1+(b1^2) r1+b1 b5 r1+2 b2 b5 r1-2 (r1^2)-b5 (r1^2)-(r1^3)-2 b1 r5-2 (b1^2) r5-2 b1 b2 r5+2 r1 r5+b1 r1 r5+(r1^2) r5) (2 (b1^2) (r1^2)+(b1^3) (r1^2)-b1 b5 (r1^2)+2 b1 b2 b5 (r1^2)-b2 (b5^2) (r1^2)-2 b1 (r1^3)+b5 (r1^3)-b1 (r1^4)-3 (b1^2) r1 r5-2 (b1^3) r1 r5-2 (b1^2) b2 r1 r5+b1 b5 r1 r5+3 b1 (r1^2) r5-b5 (r1^2) r5+2 b1 (r1^3) r5+(b1^2) (r5^2)+(b1^3) (r5^2)+(b1^2) b2 (r5^2)-b1 r1 (r5^2)-b1 (r1^2) (r5^2)))),r8->-((-2 (b1^4) r1-(b1^5) r1+3 (b1^3) b5 r1+(b1^4) b5 r1-2 (b1^3) b2 b5 r1-(b1^2) (b5^2) r1+3 (b1^2) b2 (b5^2) r1-b1 b2 (b5^3) r1+4 (b1^3) (r1^2)+(b1^4) (r1^2)-5 (b1^2) b5 (r1^2)-(b1^3) b5 (r1^2)+2 (b1^2) b2 b5 (r1^2)+2 b1 (b5^2) (r1^2)+(b1^2) (b5^2) (r1^2)-2 b1 b2 (b5^2) (r1^2)+b2 (b5^3) (r1^2)-2 (b1^2) (r1^3)+(b1^3) (r1^3)+2 b1 b5 (r1^3)-(b1^2) b5 (r1^3)-(b5^2) (r1^3)-(b1^2) (r1^4)+b1 b5 (r1^4)-(b5^2) (r1^4)+2 (b1^4) r5+2 (b1^5) r5+2 (b1^4) b2 r5-3 (b1^3) b5 r5-3 (b1^4) b5 r5-3 (b1^3) b2 b5 r5+(b1^2) (b5^2) r5+(b1^3) (b5^2) r5+(b1^2) b2 (b5^2) r5-5 (b1^3) r1 r5-2 (b1^4) r1 r5-2 (b1^3) b2 r1 r5+5 (b1^2) b5 r1 r5+(b1^3) b5 r1 r5+(b1^2) b2 b5 r1 r5-2 b1 (b5^2) r1 r5-(b1^2) (b5^2) r1 r5-b1 b2 (b5^2) r1 r5+3 (b1^2) (r1^2) r5-2 (b1^3) (r1^2) r5-2 b1 b5 (r1^2) r5+3 (b1^2) b5 (r1^2) r5+(b5^2) (r1^2) r5-b1 (b5^2) (r1^2) r5+2 (b1^2) (r1^3) r5-b1 b5 (r1^3) r5+(b5^2) (r1^3) r5+(b1^3) (r5^2)+(b1^4) (r5^2)+(b1^3) b2 (r5^2)-(b1^2) r1 (r5^2)-(b1^2) (r1^2) (r5^2))/((b5 r1-b1 r5) (-2 (b1^2) (r1^2)-(b1^3) (r1^2)+b1 b5 (r1^2)-2 b1 b2 b5 (r1^2)+b2 (b5^2) (r1^2)+2 b1 (r1^3)-b5 (r1^3)+b1 (r1^4)+3 (b1^2) r1 r5+2 (b1^3) r1 r5+2 (b1^2) b2 r1 r5-b1 b5 r1 r5-3 b1 (r1^2) r5+b5 (r1^2) r5-2 b1 (r1^3) r5-(b1^2) (r5^2)-(b1^3) (r5^2)-(b1^2) b2 (r5^2)+b1 r1 (r5^2)+b1 (r1^2) (r5^2)))),b1->b1,b2->b2,b3->((b1+r1) (b1 r1+(b1^2) r1+b2 b5 r1-(r1^2)-(r1^3)-b1 r5-(b1^2) r5-b1 b2 r5+r1 r5+(r1^2) r5) (-(b1^3) r1-2 (b1^3) b5 r1-2 (b1^2) b2 b5 r1+b1 (b5^2) r1+2 (b1^2) (b5^2) r1+2 b1 b2 (b5^2) r1+2 (b1^2) (r1^2)-(b1^3) (r1^2)+4 (b1^2) b5 (r1^2)+2 b1 b2 b5 (r1^2)-(b5^2) (r1^2)-2 b1 (b5^2) (r1^2)-b2 (b5^2) (r1^2)-b1 (r1^3)+2 (b1^2) (r1^3)-2 b1 b5 (r1^3)-b1 (r1^4)+2 (b1^3) r5+2 (b1^4) r5+2 (b1^3) b2 r5-2 (b1^2) b5 r5-2 (b1^3) b5 r5-2 (b1^2) b2 b5 r5-4 (b1^2) r1 r5-2 (b1^3) r1 r5-2 (b1^2) b2 r1 r5+2 b1 b5 r1 r5+2 b1 (r1^2) r5-2 (b1^2) (r1^2) r5+2 b1 b5 (r1^2) r5+2 b1 (r1^3) r5+(b1^2) (r5^2)+(b1^3) (r5^2)+(b1^2) b2 (r5^2)-b1 r1 (r5^2)-b1 (r1^2) (r5^2)))/((-b5 r1+b1 r5) (2 b1 r1+(b1^2) r1+b1 b5 r1+2 b2 b5 r1-2 (r1^2)-b5 (r1^2)-(r1^3)-2 b1 r5-2 (b1^2) r5-2 b1 b2 r5+2 r1 r5+b1 r1 r5+(r1^2) r5) (2 (b1^2) (r1^2)+(b1^3) (r1^2)-b1 b5 (r1^2)+2 b1 b2 b5 (r1^2)-b2 (b5^2) (r1^2)-2 b1 (r1^3)+b5 (r1^3)-b1 (r1^4)-3 (b1^2) r1 r5-2 (b1^3) r1 r5-2 (b1^2) b2 r1 r5+b1 b5 r1 r5+3 b1 (r1^2) r5-b5 (r1^2) r5+2 b1 (r1^3) r5+(b1^2) (r5^2)+(b1^3) (r5^2)+(b1^2) b2 (r5^2)-b1 r1 (r5^2)-b1 (r1^2) (r5^2))),b4->(b2 (r1^2) (-b1+b5+r1-r5))/(2 (b1^2) (r1^2)+(b1^3) (r1^2)-b1 b5 (r1^2)+2 b1 b2 b5 (r1^2)-b2 (b5^2) (r1^2)-2 b1 (r1^3)+b5 (r1^3)-b1 (r1^4)-3 (b1^2) r1 r5-2 (b1^3) r1 r5-2 (b1^2) b2 r1 r5+b1 b5 r1 r5+3 b1 (r1^2) r5-b5 (r1^2) r5+2 b1 (r1^3) r5+(b1^2) (r5^2)+(b1^3) (r5^2)+(b1^2) b2 (r5^2)-b1 r1 (r5^2)-b1 (r1^2) (r5^2)),b5->b5,b6->-((-2 b1 (r1^2)-(b1^2) (r1^2)-2 b2 b5 (r1^2)+2 (r1^3)+(r1^4)+3 b1 r1 r5+2 (b1^2) r1 r5+2 b1 b2 r1 r5+b2 b5 r1 r5-3 (r1^2) r5-2 (r1^3) r5-b1 (r5^2)-(b1^2) (r5^2)-b1 b2 (r5^2)+r1 (r5^2)+(r1^2) (r5^2))/(r1 (b5 r1-b1 r5))),b7->-(((b1+r1) (b1 r1+(b1^2) r1+b2 b5 r1-(r1^2)-(r1^3)-b1 r5-(b1^2) r5-b1 b2 r5+r1 r5+(r1^2) r5) (2 (b1^3) r1+(b1^4) r1-2 (b1^2) b5 r1-(b1^3) b5 r1+2 (b1^2) b2 b5 r1+(b1^2) (b5^2) r1-2 b1 b2 (b5^2) r1-b1 (b5^3) r1-4 (b1^2) (r1^2)-(b1^3) (r1^2)+3 b1 b5 (r1^2)+(b1^2) b5 (r1^2)-2 b1 b2 b5 (r1^2)-2 b1 (b5^2) (r1^2)+b2 (b5^2) (r1^2)+(b5^3) (r1^2)+2 b1 (r1^3)-(b1^2) (r1^3)-b5 (r1^3)+(b5^2) (r1^3)+b1 (r1^4)-2 (b1^3) r5-2 (b1^4) r5-2 (b1^3) b2 r5+2 (b1^2) b5 r5+2 (b1^3) b5 r5+2 (b1^2) b2 b5 r5+5 (b1^2) r1 r5+3 (b1^3) r1 r5+2 (b1^2) b2 r1 r5-3 b1 b5 r1 r5-2 (b1^2) b5 r1 r5+b1 (b5^2) r1 r5-3 b1 (r1^2) r5+(b1^2) (r1^2) r5+b5 (r1^2) r5-(b5^2) (r1^2) r5-2 b1 (r1^3) r5-(b1^2) (r5^2)-(b1^3) (r5^2)-(b1^2) b2 (r5^2)+b1 r1 (r5^2)+b1 (r1^2) (r5^2)))/((-b5 r1+b1 r5) (2 b1 r1+(b1^2) r1+b1 b5 r1+2 b2 b5 r1-2 (r1^2)-b5 (r1^2)-(r1^3)-2 b1 r5-2 (b1^2) r5-2 b1 b2 r5+2 r1 r5+b1 r1 r5+(r1^2) r5) (2 (b1^2) (r1^2)+(b1^3) (r1^2)-b1 b5 (r1^2)+2 b1 b2 b5 (r1^2)-b2 (b5^2) (r1^2)-2 b1 (r1^3)+b5 (r1^3)-b1 (r1^4)-3 (b1^2) r1 r5-2 (b1^3) r1 r5-2 (b1^2) b2 r1 r5+b1 b5 r1 r5+3 b1 (r1^2) r5-b5 (r1^2) r5+2 b1 (r1^3) r5+(b1^2) (r5^2)+(b1^3) (r5^2)+(b1^2) b2 (r5^2)-b1 r1 (r5^2)-b1 (r1^2) (r5^2)))),b8->-((-2 (b1^2) (r1^3)-(b1^3) (r1^3)+b1 b5 (r1^3)-2 b1 b2 b5 (r1^3)+b2 (b5^2) (r1^3)+4 b1 (r1^4)+(b1^2) (r1^4)-b5 (r1^4)+2 b2 b5 (r1^4)-2 (r1^5)+b1 (r1^5)-(r1^6)+4 (b1^2) (r1^2) r5+3 (b1^3) (r1^2) r5+2 (b1^2) b2 (r1^2) r5-b1 b5 (r1^2) r5+b1 b2 b5 (r1^2) r5-9 b1 (r1^3) r5-3 (b1^2) (r1^3) r5-2 b1 b2 (r1^3) r5+b5 (r1^3) r5-3 b2 b5 (r1^3) r5+5 (r1^4) r5-3 b1 (r1^4) r5+3 (r1^5) r5-3 (b1^2) r1 (r5^2)-3 (b1^3) r1 (r5^2)-2 (b1^2) b2 r1 (r5^2)-b1 b2 b5 r1 (r5^2)+7 b1 (r1^2) (r5^2)+3 (b1^2) (r1^2) (r5^2)+3 b1 b2 (r1^2) (r5^2)+b2 b5 (r1^2) (r5^2)-4 (r1^3) (r5^2)+3 b1 (r1^3) (r5^2)-3 (r1^4) (r5^2)+(b1^2) (r5^3)+(b1^3) (r5^3)+(b1^2) b2 (r5^3)-2 b1 r1 (r5^3)-(b1^2) r1 (r5^3)-b1 b2 r1 (r5^3)+(r1^2) (r5^3)-b1 (r1^2) (r5^3)+(r1^3) (r5^3))/((b5 r1-b1 r5) (-2 (b1^2) (r1^2)-(b1^3) (r1^2)+b1 b5 (r1^2)-2 b1 b2 b5 (r1^2)+b2 (b5^2) (r1^2)+2 b1 (r1^3)-b5 (r1^3)+b1 (r1^4)+3 (b1^2) r1 r5+2 (b1^3) r1 r5+2 (b1^2) b2 r1 r5-b1 b5 r1 r5-3 b1 (r1^2) r5+b5 (r1^2) r5-2 b1 (r1^3) r5-(b1^2) (r5^2)-(b1^3) (r5^2)-(b1^2) b2 (r5^2)+b1 r1 (r5^2)+b1 (r1^2) (r5^2)))),c4->((-b5 r1+b1 r5) (2 b1 r1+(b1^2) r1+b1 b5 r1+2 b2 b5 r1-2 (r1^2)-b5 (r1^2)-(r1^3)-2 b1 r5-2 (b1^2) r5-2 b1 b2 r5+2 r1 r5+b1 r1 r5+(r1^2) r5) (2 (b1^2) (r1^2)+(b1^3) (r1^2)-b1 b5 (r1^2)+2 b1 b2 b5 (r1^2)-b2 (b5^2) (r1^2)-2 b1 (r1^3)+b5 (r1^3)-b1 (r1^4)-3 (b1^2) r1 r5-2 (b1^3) r1 r5-2 (b1^2) b2 r1 r5+b1 b5 r1 r5+3 b1 (r1^2) r5-b5 (r1^2) r5+2 b1 (r1^3) r5+(b1^2) (r5^2)+(b1^3) (r5^2)+(b1^2) b2 (r5^2)-b1 r1 (r5^2)-b1 (r1^2) (r5^2)))/((b1-r1) r1 (b1-b5+r1-r5) (b1-b5-r1+r5) (-b1 r1-b1 b5 r1-b2 b5 r1+(r1^2)+b1 r5+(b1^2) r5+b1 b2 r5-r1 r5)),c30->-(((-2 (b1^2) (r1^2)-(b1^3) (r1^2)+b1 b5 (r1^2)-2 b1 b2 b5 (r1^2)+b2 (b5^2) (r1^2)+2 b1 (r1^3)-b5 (r1^3)+b1 (r1^4)+3 (b1^2) r1 r5+2 (b1^3) r1 r5+2 (b1^2) b2 r1 r5-b1 b5 r1 r5-3 b1 (r1^2) r5+b5 (r1^2) r5-2 b1 (r1^3) r5-(b1^2) (r5^2)-(b1^3) (r5^2)-(b1^2) b2 (r5^2)+b1 r1 (r5^2)+b1 (r1^2) (r5^2)) (-2 (b1^4) (r1^2)-(b1^5) (r1^2)+3 (b1^3) b5 (r1^2)-(b1^4) b5 (r1^2)-(b1^5) b5 (r1^2)-4 (b1^3) b2 b5 (r1^2)-(b1^4) b2 b5 (r1^2)-(b1^2) (b5^2) (r1^2)+3 (b1^3) (b5^2) (r1^2)+(b1^4) (b5^2) (r1^2)+6 (b1^2) b2 (b5^2) (r1^2)-(b1^3) b2 (b5^2) (r1^2)-2 (b1^2) (b2^2) (b5^2) (r1^2)-(b1^2) (b5^3) (r1^2)-2 b1 b2 (b5^3) (r1^2)+3 (b1^2) b2 (b5^3) (r1^2)+3 b1 (b2^2) (b5^3) (r1^2)-b1 b2 (b5^4) (r1^2)-(b2^2) (b5^4) (r1^2)+4 (b1^3) (r1^3)+(b1^4) (r1^3)-7 (b1^2) b5 (r1^3)+(b1^3) b5 (r1^3)+4 (b1^2) b2 b5 (r1^3)+2 b1 (b5^2) (r1^3)-5 (b1^2) (b5^2) (r1^3)-8 b1 b2 (b5^2) (r1^3)+b1 (b5^3) (r1^3)-(b1^2) (b5^3) (r1^3)+2 b2 (b5^3) (r1^3)-2 b1 b2 (b5^3) (r1^3)-(b2^2) (b5^3) (r1^3)+2 (b1^3) (r1^4)+5 b1 b5 (r1^4)+(b1^2) b5 (r1^4)+2 (b1^3) b5 (r1^4)+4 b1 b2 b5 (r1^4)+2 (b1^2) b2 b5 (r1^4)-(b5^2) (r1^4)+b1 (b5^2) (r1^4)-2 (b1^2) (b5^2) (r1^4)+2 b2 (b5^2) (r1^4)+b1 b2 (b5^2) (r1^4)+2 (b2^2) (b5^2) (r1^4)-b2 (b5^3) (r1^4)-4 b1 (r1^5)-2 (b1^2) (r1^5)-b5 (r1^5)-b1 b5 (r1^5)-4 b2 b5 (r1^5)+(b5^2) (r1^5)+(b5^3) (r1^5)+2 (r1^6)-b1 (r1^6)-b1 b5 (r1^6)-b2 b5 (r1^6)+(b5^2) (r1^6)+(r1^7)+4 (b1^4) r1 r5+5 (b1^5) r1 r5+(b1^6) r1 r5+4 (b1^4) b2 r1 r5+(b1^5) b2 r1 r5-6 (b1^3) b5 r1 r5-5 (b1^4) b5 r1 r5+(b1^5) b5 r1 r5-2 (b1^3) b2 b5 r1 r5+5 (b1^4) b2 b5 r1 r5+4 (b1^3) (b2^2) b5 r1 r5+2 (b1^2) (b5^2) r1 r5-(b1^3) (b5^2) r1 r5-3 (b1^4) (b5^2) r1 r5-4 (b1^2) b2 (b5^2) r1 r5-9 (b1^3) b2 (b5^2) r1 r5-6 (b1^2) (b2^2) (b5^2) r1 r5+(b1^2) (b5^3) r1 r5+(b1^3) (b5^3) r1 r5+2 b1 b2 (b5^3) r1 r5+3 (b1^2) b2 (b5^3) r1 r5+2 b1 (b2^2) (b5^3) r1 r5-7 (b1^3) (r1^2) r5-5 (b1^4) (r1^2) r5-4 (b1^3) b2 (r1^2) r5+14 (b1^2) b5 (r1^2) r5+9 (b1^3) b5 (r1^2) r5+6 (b1^2) b2 b5 (r1^2) r5-4 b1 (b5^2) (r1^2) r5+3 (b1^2) (b5^2) (r1^2) r5+3 (b1^3) (b5^2) (r1^2) r5+6 b1 b2 (b5^2) (r1^2) r5+6 (b1^2) b2 (b5^2) (r1^2) r5+3 b1 (b2^2) (b5^2) (r1^2) r5-b1 (b5^3) (r1^2) r5-2 b2 (b5^3) (r1^2) r5-5 (b1^2) (r1^3) r5-9 (b1^3) (r1^3) r5-2 (b1^4) (r1^3) r5-4 (b1^2) b2 (r1^3) r5-2 (b1^3) b2 (r1^3) r5-10 b1 b5 (r1^3) r5-3 (b1^2) b5 (r1^3) r5-2 (b1^3) b5 (r1^3) r5-14 b1 b2 b5 (r1^3) r5-8 (b1^2) b2 b5 (r1^3) r5-4 b1 (b2^2) b5 (r1^3) r5+2 (b5^2) (r1^3) r5-2 b1 (b5^2) (r1^3) r5+4 (b1^2) (b5^2) (r1^3) r5-2 b2 (b5^2) (r1^3) r5+3 b1 b2 (b5^2) (r1^3) r5-3 (b2^2) (b5^2) (r1^3) r5-b1 (b5^3) (r1^3) r5-b2 (b5^3) (r1^3) r5+15 b1 (r1^4) r5+9 (b1^2) (r1^4) r5+4 b1 b2 (r1^4) r5+2 b5 (r1^4) r5-b1 b5 (r1^4) r5+10 b2 b5 (r1^4) r5-3 b1 (b5^2) (r1^4) r5-7 (r1^5) r5+4 b1 (r1^5) r5+(b1^2) (r1^5) r5+b1 b2 (r1^5) r5+b1 b5 (r1^5) r5+3 b2 b5 (r1^5) r5-(b5^2) (r1^5) r5-4 (r1^6) r5-2 (b1^4) (r5^2)-4 (b1^5) (r5^2)-2 (b1^6) (r5^2)-4 (b1^4) b2 (r5^2)-4 (b1^5) b2 (r5^2)-2 (b1^4) (b2^2) (r5^2)+3 (b1^3) b5 (r5^2)+6 (b1^4) b5 (r5^2)+3 (b1^5) b5 (r5^2)+6 (b1^3) b2 b5 (r5^2)+6 (b1^4) b2 b5 (r5^2)+3 (b1^3) (b2^2) b5 (r5^2)-(b1^2) (b5^2) (r5^2)-2 (b1^3) (b5^2) (r5^2)-(b1^4) (b5^2) (r5^2)-2 (b1^2) b2 (b5^2) (r5^2)-2 (b1^3) b2 (b5^2) (r5^2)-(b1^2) (b2^2) (b5^2) (r5^2)+2 (b1^3) r1 (r5^2)+2 (b1^4) r1 (r5^2)+2 (b1^3) b2 r1 (r5^2)-7 (b1^2) b5 r1 (r5^2)-10 (b1^3) b5 r1 (r5^2)-3 (b1^4) b5 r1 (r5^2)-10 (b1^2) b2 b5 r1 (r5^2)-6 (b1^3) b2 b5 r1 (r5^2)-3 (b1^2) (b2^2) b5 r1 (r5^2)+2 b1 (b5^2) r1 (r5^2)+2 (b1^2) (b5^2) r1 (r5^2)+2 b1 b2 (b5^2) r1 (r5^2)+11 (b1^2) (r1^2) (r5^2)+14 (b1^3) (r1^2) (r5^2)+4 (b1^4) (r1^2) (r5^2)+12 (b1^2) b2 (r1^2) (r5^2)+7 (b1^3) b2 (r1^2) (r5^2)+2 (b1^2) (b2^2) (r1^2) (r5^2)+5 b1 b5 (r1^2) (r5^2)+3 (b1^2) b5 (r1^2) (r5^2)-2 (b1^3) b5 (r1^2) (r5^2)+12 b1 b2 b5 (r1^2) (r5^2)+3 (b1^2) b2 b5 (r1^2) (r5^2)+6 b1 (b2^2) b5 (r1^2) (r5^2)-(b5^2) (r1^2) (r5^2)+b1 (b5^2) (r1^2) (r5^2)+(b1^2) (b5^2) (r1^2) (r5^2)+2 b1 b2 (b5^2) (r1^2) (r5^2)+(b2^2) (b5^2) (r1^2) (r5^2)-20 b1 (r1^3) (r5^2)-12 (b1^2) (r1^3) (r5^2)-10 b1 b2 (r1^3) (r5^2)-b5 (r1^3) (r5^2)+b1 b5 (r1^3) (r5^2)+3 (b1^2) b5 (r1^3) (r5^2)-8 b2 b5 (r1^3) (r5^2)-(b5^2) (r1^3) (r5^2)+9 (r1^4) (r5^2)-6 b1 (r1^4) (r5^2)-2 (b1^2) (r1^4) (r5^2)-3 b1 b2 (r1^4) (r5^2)-b1 b5 (r1^4) (r5^2)-3 b2 b5 (r1^4) (r5^2)+6 (r1^5) (r5^2)+(b1^3) (r5^3)+2 (b1^4) (r5^3)+(b1^5) (r5^3)+2 (b1^3) b2 (r5^3)+2 (b1^4) b2 (r5^3)+(b1^3) (b2^2) (r5^3)-7 (b1^2) r1 (r5^3)-9 (b1^3) r1 (r5^3)-2 (b1^4) r1 (r5^3)-10 (b1^2) b2 r1 (r5^3)-5 (b1^3) b2 r1 (r5^3)-3 (b1^2) (b2^2) r1 (r5^3)-(b1^2) b5 r1 (r5^3)-(b1^3) b5 r1 (r5^3)-2 b1 b2 b5 r1 (r5^3)-3 (b1^2) b2 b5 r1 (r5^3)-2 b1 (b2^2) b5 r1 (r5^3)+11 b1 (r1^2) (r5^3)+7 (b1^2) (r1^2) (r5^3)-(b1^3) (r1^2) (r5^3)+8 b1 b2 (r1^2) (r5^3)+b1 b5 (r1^2) (r5^3)+2 b2 b5 (r1^2) (r5^3)-5 (r1^3) (r5^3)+4 b1 (r1^3) (r5^3)+2 (b1^2) (r1^3) (r5^3)+3 b1 b2 (r1^3) (r5^3)+b1 b5 (r1^3) (r5^3)+b2 b5 (r1^3) (r5^3)-4 (r1^4) (r5^3)+(b1^2) (r5^4)+2 (b1^3) (r5^4)+(b1^4) (r5^4)+2 (b1^2) b2 (r5^4)+2 (b1^3) b2 (r5^4)+(b1^2) (b2^2) (r5^4)-2 b1 r1 (r5^4)-2 (b1^2) r1 (r5^4)-2 b1 b2 r1 (r5^4)+(r1^2) (r5^4)-b1 (r1^2) (r5^4)-(b1^2) (r1^2) (r5^4)-b1 b2 (r1^2) (r5^4)+(r1^3) (r5^4)))/((r1^2) (-b1+r1) (b1-b5+r1-r5) (-b1+b5+r1-r5) (b5 r1-b1 r5) (-b1 r1-b1 b5 r1-b2 b5 r1+(r1^2)+b1 r5+(b1^2) r5+b1 b2 r5-r1 r5))),c32->-(((-2 (b1^2) (r1^2)-(b1^3) (r1^2)+b1 b5 (r1^2)-2 b1 b2 b5 (r1^2)+b2 (b5^2) (r1^2)+2 b1 (r1^3)-b5 (r1^3)+b1 (r1^4)+3 (b1^2) r1 r5+2 (b1^3) r1 r5+2 (b1^2) b2 r1 r5-b1 b5 r1 r5-3 b1 (r1^2) r5+b5 (r1^2) r5-2 b1 (r1^3) r5-(b1^2) (r5^2)-(b1^3) (r5^2)-(b1^2) b2 (r5^2)+b1 r1 (r5^2)+b1 (r1^2) (r5^2)) (-(b1^3) r1-(b1^4) r1-(b1^3) b2 r1+(b1^2) b5 r1-(b1^4) b5 r1-2 (b1^3) b2 b5 r1-(b1^2) (b2^2) b5 r1+(b1^2) (b5^2) r1+(b1^3) (b5^2) r1+b1 b2 (b5^2) r1+2 (b1^2) b2 (b5^2) r1+b1 (b2^2) (b5^2) r1+(b1^2) (r1^2)+(b1^3) (r1^2)+(b1^2) b2 (r1^2)-2 b1 b5 (r1^2)-2 (b1^2) b5 (r1^2)-2 b1 b2 b5 (r1^2)-b1 (b5^2) (r1^2)-(b1^2) (b5^2) (r1^2)-b2 (b5^2) (r1^2)-2 b1 b2 (b5^2) (r1^2)-(b2^2) (b5^2) (r1^2)+b1 (r1^3)+(b1^2) (r1^3)+b1 b2 (r1^3)+b5 (r1^3)+2 b2 b5 (r1^3)+2 b1 b2 b5 (r1^3)+(b2^2) b5 (r1^3)-b1 (b5^2) (r1^3)-2 b2 (b5^2) (r1^3)-(r1^4)-b1 (r1^4)-b2 (r1^4)+2 b5 (r1^4)+(b5^2) (r1^4)+b5 (r1^5)+(b1^3) r5+2 (b1^4) r5+(b1^5) r5+2 (b1^3) b2 r5+2 (b1^4) b2 r5+(b1^3) (b2^2) r5-(b1^2) b5 r5-2 (b1^3) b5 r5-(b1^4) b5 r5-2 (b1^2) b2 b5 r5-2 (b1^3) b2 b5 r5-(b1^2) (b2^2) b5 r5+2 b1 b5 r1 r5+4 (b1^2) b5 r1 r5+2 (b1^3) b5 r1 r5+4 b1 b2 b5 r1 r5+4 (b1^2) b2 b5 r1 r5+2 b1 (b2^2) b5 r1 r5-3 b1 (r1^2) r5-2 (b1^2) (r1^2) r5-4 b1 b2 (r1^2) r5-2 (b1^2) b2 (r1^2) r5-b1 (b2^2) (r1^2) r5-b5 (r1^2) r5+2 (b1^2) b5 (r1^2) r5-2 b2 b5 (r1^2) r5+2 b1 b2 b5 (r1^2) r5-(b2^2) b5 (r1^2) r5+2 (r1^3) r5+2 b2 (r1^3) r5-2 b5 (r1^3) r5-2 b1 b5 (r1^3) r5-b1 (r1^4) r5-b5 (r1^4) r5-(b1^2) (r5^2)-2 (b1^3) (r5^2)-(b1^4) (r5^2)-2 (b1^2) b2 (r5^2)-2 (b1^3) b2 (r5^2)-(b1^2) (b2^2) (r5^2)+2 b1 r1 (r5^2)+(b1^2) r1 (r5^2)-(b1^3) r1 (r5^2)+3 b1 b2 r1 (r5^2)+b1 (b2^2) r1 (r5^2)-(r1^2) (r5^2)+b1 (r1^2) (r5^2)+(b1^2) (r1^2) (r5^2)-b2 (r1^2) (r5^2)+b1 (r1^3) (r5^2)))/((r1^2) (-b1+r1) (b1-b5+r1-r5) (-b1+b5+r1-r5) (-b1 r1-b1 b5 r1-b2 b5 r1+(r1^2)+b1 r5+(b1^2) r5+b1 b2 r5-r1 r5))),c103->((b1-r1) (b1+r1) (b1-b5-r1+r5) (-b1 r1-b1 b5 r1-b2 b5 r1+(r1^2)+b5 (r1^2)+b1 r5+(b1^2) r5+b1 b2 r5-r1 r5-b1 r1 r5) (b1 r1+(b1^2) r1+b2 b5 r1-(r1^2)-(r1^3)-b1 r5-(b1^2) r5-b1 b2 r5+r1 r5+(r1^2) r5) (-(b1^3) (r1^2)-(b1^4) (r1^2)+(b1^2) b5 (r1^2)-(b1^4) b5 (r1^2)-2 (b1^2) b2 b5 (r1^2)-(b1^3) b2 b5 (r1^2)+(b1^2) (b5^2) (r1^2)+(b1^3) (b5^2) (r1^2)+2 b1 b2 (b5^2) (r1^2)-b1 (b2^2) (b5^2) (r1^2)+b1 b2 (b5^3) (r1^2)+(b2^2) (b5^3) (r1^2)+(b1^2) (r1^3)-(b1^4) (r1^3)-2 b1 b5 (r1^3)-(b1^2) b5 (r1^3)+(b1^3) b5 (r1^3)-(b1^2) b2 b5 (r1^3)-b1 (b5^2) (r1^3)-(b1^2) (b5^2) (r1^3)-2 b2 (b5^2) (r1^3)-b1 b2 (b5^2) (r1^3)-(b2^2) (b5^2) (r1^3)+b1 (r1^4)+2 (b1^2) (r1^4)+(b1^3) (r1^4)+b5 (r1^4)+2 b2 b5 (r1^4)+b1 b2 b5 (r1^4)-b1 (b5^2) (r1^4)-b2 (b5^2) (r1^4)-(r1^5)+(b1^2) (r1^5)+b5 (r1^5)-b1 b5 (r1^5)+b2 b5 (r1^5)+(b5^2) (r1^5)-(r1^6)-b1 (r1^6)+b5 (r1^6)+2 (b1^3) r1 r5+3 (b1^4) r1 r5+(b1^5) r1 r5+2 (b1^3) b2 r1 r5+(b1^4) b2 r1 r5-2 (b1^2) b5 r1 r5-2 (b1^3) b5 r1 r5+2 (b1^3) b2 b5 r1 r5+2 (b1^2) (b2^2) b5 r1 r5-(b1^2) (b5^2) r1 r5-(b1^3) (b5^2) r1 r5-2 b1 b2 (b5^2) r1 r5-3 (b1^2) b2 (b5^2) r1 r5-2 b1 (b2^2) (b5^2) r1 r5-(b1^2) (r1^2) r5+(b1^3) (r1^2) r5+2 (b1^4) (r1^2) r5+(b1^3) b2 (r1^2) r5+4 b1 b5 (r1^2) r5+4 (b1^2) b5 (r1^2) r5+4 b1 b2 b5 (r1^2) r5+4 (b1^2) b2 b5 (r1^2) r5+2 b1 (b2^2) b5 (r1^2) r5+b1 (b5^2) (r1^2) r5+(b1^2) (b5^2) (r1^2) r5+2 b2 (b5^2) (r1^2) r5+b1 b2 (b5^2) (r1^2) r5+(b2^2) (b5^2) (r1^2) r5-4 b1 (r1^3) r5-6 (b1^2) (r1^3) r5-3 (b1^3) (r1^3) r5-2 b1 b2 (r1^3) r5-(b1^2) b2 (r1^3) r5-2 b5 (r1^3) r5+2 (b1^2) b5 (r1^3) r5-4 b2 b5 (r1^3) r5+b1 (b5^2) (r1^3) r5+b2 (b5^2) (r1^3) r5+3 (r1^4) r5-b1 (r1^4) r5-2 (b1^2) (r1^4) r5-b1 b2 (r1^4) r5-2 b5 (r1^4) r5-2 b2 b5 (r1^4) r5-(b5^2) (r1^4) r5+3 (r1^5) r5+2 b1 (r1^5) r5-2 b5 (r1^5) r5-(b1^3) (r5^2)-2 (b1^4) (r5^2)-(b1^5) (r5^2)-2 (b1^3) b2 (r5^2)-2 (b1^4) b2 (r5^2)-(b1^3) (b2^2) (r5^2)+(b1^2) b5 (r5^2)+2 (b1^3) b5 (r5^2)+(b1^4) b5 (r5^2)+2 (b1^2) b2 b5 (r5^2)+2 (b1^3) b2 b5 (r5^2)+(b1^2) (b2^2) b5 (r5^2)-(b1^2) r1 (r5^2)-3 (b1^3) r1 (r5^2)-2 (b1^4) r1 (r5^2)-2 (b1^2) b2 r1 (r5^2)-3 (b1^3) b2 r1 (r5^2)-(b1^2) (b2^2) r1 (r5^2)-2 b1 b5 r1 (r5^2)-3 (b1^2) b5 r1 (r5^2)-(b1^3) b5 r1 (r5^2)-4 b1 b2 b5 r1 (r5^2)-3 (b1^2) b2 b5 r1 (r5^2)-2 b1 (b2^2) b5 r1 (r5^2)+5 b1 (r1^2) (r5^2)+6 (b1^2) (r1^2) (r5^2)+2 (b1^3) (r1^2) (r5^2)+4 b1 b2 (r1^2) (r5^2)+(b1^2) b2 (r1^2) (r5^2)+b5 (r1^2) (r5^2)-2 (b1^2) b5 (r1^2) (r5^2)+2 b2 b5 (r1^2) (r5^2)-b1 b2 b5 (r1^2) (r5^2)-3 (r1^3) (r5^2)+2 b1 (r1^3) (r5^2)+2 (b1^2) (r1^3) (r5^2)+2 b1 b2 (r1^3) (r5^2)+b5 (r1^3) (r5^2)+b1 b5 (r1^3) (r5^2)+b2 b5 (r1^3) (r5^2)-3 (r1^4) (r5^2)-b1 (r1^4) (r5^2)+b5 (r1^4) (r5^2)+(b1^2) (r5^3)+2 (b1^3) (r5^3)+(b1^4) (r5^3)+2 (b1^2) b2 (r5^3)+2 (b1^3) b2 (r5^3)+(b1^2) (b2^2) (r5^3)-2 b1 r1 (r5^3)-2 (b1^2) r1 (r5^3)-2 b1 b2 r1 (r5^3)+(r1^2) (r5^3)-b1 (r1^2) (r5^3)-(b1^2) (r1^2) (r5^3)-b1 b2 (r1^2) (r5^3)+(r1^3) (r5^3)))/((((-b5 r1+b1 r5))^2) (((2 b1 r1+(b1^2) r1+b1 b5 r1+2 b2 b5 r1-2 (r1^2)-b5 (r1^2)-(r1^3)-2 b1 r5-2 (b1^2) r5-2 b1 b2 r5+2 r1 r5+b1 r1 r5+(r1^2) r5))^2) (((2 (b1^2) (r1^2)+(b1^3) (r1^2)-b1 b5 (r1^2)+2 b1 b2 b5 (r1^2)-b2 (b5^2) (r1^2)-2 b1 (r1^3)+b5 (r1^3)-b1 (r1^4)-3 (b1^2) r1 r5-2 (b1^3) r1 r5-2 (b1^2) b2 r1 r5+b1 b5 r1 r5+3 b1 (r1^2) r5-b5 (r1^2) r5+2 b1 (r1^3) r5+(b1^2) (r5^2)+(b1^3) (r5^2)+(b1^2) b2 (r5^2)-b1 r1 (r5^2)-b1 (r1^2) (r5^2)))^2)),c110->-(((b1-r1) r1 (b1+r1) (b1-b5-r1+r5) (b1 r1+(b1^2) r1+b2 b5 r1-(r1^2)-(r1^3)-b1 r5-(b1^2) r5-b1 b2 r5+r1 r5+(r1^2) r5))/((-b5 r1+b1 r5) (2 b1 r1+(b1^2) r1+b1 b5 r1+2 b2 b5 r1-2 (r1^2)-b5 (r1^2)-(r1^3)-2 b1 r5-2 (b1^2) r5-2 b1 b2 r5+2 r1 r5+b1 r1 r5+(r1^2) r5) (2 (b1^2) (r1^2)+(b1^3) (r1^2)-b1 b5 (r1^2)+2 b1 b2 b5 (r1^2)-b2 (b5^2) (r1^2)-2 b1 (r1^3)+b5 (r1^3)-b1 (r1^4)-3 (b1^2) r1 r5-2 (b1^3) r1 r5-2 (b1^2) b2 r1 r5+b1 b5 r1 r5+3 b1 (r1^2) r5-b5 (r1^2) r5+2 b1 (r1^3) r5+(b1^2) (r5^2)+(b1^3) (r5^2)+(b1^2) b2 (r5^2)-b1 r1 (r5^2)-b1 (r1^2) (r5^2)))),c135->(4 (b1^5) (r1^2)+4 (b1^6) (r1^2)+(b1^7) (r1^2)-6 (b1^4) b5 (r1^2)-3 (b1^5) b5 (r1^2)+8 (b1^4) b2 b5 (r1^2)+4 (b1^5) b2 b5 (r1^2)+2 (b1^3) (b5^2) (r1^2)-2 (b1^4) (b5^2) (r1^2)-(b1^5) (b5^2) (r1^2)-12 (b1^3) b2 (b5^2) (r1^2)-3 (b1^4) b2 (b5^2) (r1^2)+4 (b1^3) (b2^2) (b5^2) (r1^2)+(b1^3) (b5^3) (r1^2)+4 (b1^2) b2 (b5^3) (r1^2)-2 (b1^3) b2 (b5^3) (r1^2)-6 (b1^2) (b2^2) (b5^3) (r1^2)+(b1^2) b2 (b5^4) (r1^2)+2 b1 (b2^2) (b5^4) (r1^2)-8 (b1^4) (r1^3)-4 (b1^5) (r1^3)+12 (b1^3) b5 (r1^3)+3 (b1^4) b5 (r1^3)-8 (b1^3) b2 b5 (r1^3)-5 (b1^2) (b5^2) (r1^3)+(b1^3) (b5^2) (r1^3)+12 (b1^2) b2 (b5^2) (r1^3)-2 (b1^2) (b5^3) (r1^3)-(b1^3) (b5^3) (r1^3)-6 b1 b2 (b5^3) (r1^3)-(b1^2) b2 (b5^3) (r1^3)-b1 b2 (b5^4) (r1^3)-(b2^2) (b5^4) (r1^3)+4 (b1^3) (r1^4)-4 (b1^4) (r1^4)-2 (b1^5) (r1^4)-6 (b1^2) b5 (r1^4)+5 (b1^3) b5 (r1^4)+(b1^4) b5 (r1^4)-4 (b1^3) b2 b5 (r1^4)+4 b1 (b5^2) (r1^4)+5 (b1^2) b2 (b5^2) (r1^4)+2 b1 (b5^3) (r1^4)+2 (b1^2) (b5^3) (r1^4)+2 b2 (b5^3) (r1^4)-b1 b2 (b5^3) (r1^4)+b2 (b5^4) (r1^4)-4 (b1^2) (r1^5)-(b1^4) (r1^5)-9 (b1^2) b5 (r1^5)-2 (b1^3) b5 (r1^5)-8 b1 b2 b5 (r1^5)-4 (b1^2) b2 b5 (r1^5)-(b5^2) (r1^5)+5 b1 (b5^2) (r1^5)+2 (b1^2) (b5^2) (r1^5)-4 b1 b2 (b5^2) (r1^5)-4 (b2^2) (b5^2) (r1^5)-(b5^3) (r1^5)+b1 (b5^3) (r1^5)+4 b2 (b5^3) (r1^5)+8 b1 (r1^6)+4 (b1^2) (r1^6)+(b1^3) (r1^6)+6 b1 b5 (r1^6)+8 b2 b5 (r1^6)-4 (b5^2) (r1^6)+b1 (b5^2) (r1^6)+2 b2 (b5^2) (r1^6)-2 (b5^3) (r1^6)-4 (r1^7)+4 b1 (r1^7)+2 (b1^2) (r1^7)-2 b5 (r1^7)+2 b1 b5 (r1^7)+4 b2 b5 (r1^7)-2 (b5^2) (r1^7)-4 (r1^8)-b5 (r1^8)-(r1^9)-8 (b1^5) r1 r5-12 (b1^6) r1 r5-4 (b1^7) r1 r5-8 (b1^5) b2 r1 r5-4 (b1^6) b2 r1 r5+12 (b1^4) b5 r1 r5+15 (b1^5) b5 r1 r5+3 (b1^6) b5 r1 r5+4 (b1^4) b2 b5 r1 r5-5 (b1^5) b2 b5 r1 r5-8 (b1^4) (b2^2) b5 r1 r5-4 (b1^3) (b5^2) r1 r5-2 (b1^4) (b5^2) r1 r5+2 (b1^5) (b5^2) r1 r5+8 (b1^3) b2 (b5^2) r1 r5+14 (b1^4) b2 (b5^2) r1 r5+12 (b1^3) (b2^2) (b5^2) r1 r5-(b1^3) (b5^3) r1 r5-(b1^4) (b5^3) r1 r5-4 (b1^2) b2 (b5^3) r1 r5-5 (b1^3) b2 (b5^3) r1 r5-4 (b1^2) (b2^2) (b5^3) r1 r5+16 (b1^4) (r1^2) r5+14 (b1^5) (r1^2) r5+(b1^6) (r1^2) r5+8 (b1^4) b2 (r1^2) r5-23 (b1^3) b5 (r1^2) r5-18 (b1^4) b5 (r1^2) r5-(b1^5) b5 (r1^2) r5-4 (b1^3) b2 b5 (r1^2) r5+2 (b1^4) b2 b5 (r1^2) r5+10 (b1^2) (b5^2) (r1^2) r5+8 (b1^3) (b5^2) (r1^2) r5+(b1^4) (b5^2) (r1^2) r5-4 (b1^2) b2 (b5^2) (r1^2) r5-2 (b1^3) b2 (b5^2) (r1^2) r5+2 (b1^2) (b5^3) (r1^2) r5+2 (b1^3) (b5^3) (r1^2) r5+6 b1 b2 (b5^3) (r1^2) r5+6 (b1^2) b2 (b5^3) (r1^2) r5+3 b1 (b2^2) (b5^3) (r1^2) r5-8 (b1^3) (r1^3) r5+6 (b1^4) (r1^3) r5+6 (b1^5) (r1^3) r5+4 (b1^4) b2 (r1^3) r5+9 (b1^2) b5 (r1^3) r5-11 (b1^3) b5 (r1^3) r5-7 (b1^4) b5 (r1^3) r5-(b1^3) b2 b5 (r1^3) r5-8 b1 (b5^2) (r1^3) r5-4 (b1^2) (b5^2) (r1^3) r5-2 (b1^3) (b5^2) (r1^3) r5-6 b1 b2 (b5^2) (r1^3) r5-8 (b1^2) b2 (b5^2) (r1^3) r5-2 b1 (b5^3) (r1^3) r5-(b1^2) (b5^3) (r1^3) r5-2 b2 (b5^3) (r1^3) r5-(b2^2) (b5^3) (r1^3) r5+14 (b1^2) (r1^4) r5+9 (b1^3) (r1^4) r5+4 (b1^4) (r1^4) r5+8 (b1^2) b2 (r1^4) r5+4 (b1^3) b2 (r1^4) r5+3 b1 b5 (r1^4) r5+21 (b1^2) b5 (r1^4) r5+6 (b1^3) b5 (r1^4) r5+20 b1 b2 b5 (r1^4) r5+17 (b1^2) b2 b5 (r1^4) r5+8 b1 (b2^2) b5 (r1^4) r5+2 (b5^2) (r1^4) r5-9 b1 (b5^2) (r1^4) r5-5 (b1^2) (b5^2) (r1^4) r5+2 b2 (b5^2) (r1^4) r5-b1 b2 (b5^2) (r1^4) r5+6 (b2^2) (b5^2) (r1^4) r5+(b5^3) (r1^4) r5-2 b1 (b5^3) (r1^4) r5-3 b2 (b5^3) (r1^4) r5-28 b1 (r1^5) r5-17 (b1^2) (r1^5) r5-2 (b1^3) (r1^5) r5-8 b1 b2 (r1^5) r5-b5 (r1^5) r5-12 b1 b5 (r1^5) r5+(b1^2) b5 (r1^5) r5-20 b2 b5 (r1^5) r5-2 b1 b2 b5 (r1^5) r5+7 (b5^2) (r1^5) r5-3 b2 (b5^2) (r1^5) r5+2 (b5^3) (r1^5) r5+14 (r1^6) r5-15 b1 (r1^6) r5-9 (b1^2) (r1^6) r5-4 b1 b2 (r1^6) r5+5 b5 (r1^6) r5-5 b1 b5 (r1^6) r5-11 b2 b5 (r1^6) r5+4 (b5^2) (r1^6) r5+15 (r1^7) r5+3 b5 (r1^7) r5+4 (r1^8) r5+4 (b1^5) (r5^2)+8 (b1^6) (r5^2)+4 (b1^7) (r5^2)+8 (b1^5) b2 (r5^2)+8 (b1^6) b2 (r5^2)+4 (b1^5) (b2^2) (r5^2)-6 (b1^4) b5 (r5^2)-12 (b1^5) b5 (r5^2)-6 (b1^6) b5 (r5^2)-12 (b1^4) b2 b5 (r5^2)-12 (b1^5) b2 b5 (r5^2)-6 (b1^4) (b2^2) b5 (r5^2)+2 (b1^3) (b5^2) (r5^2)+4 (b1^4) (b5^2) (r5^2)+2 (b1^5) (b5^2) (r5^2)+4 (b1^3) b2 (b5^2) (r5^2)+4 (b1^4) b2 (b5^2) (r5^2)+2 (b1^3) (b2^2) (b5^2) (r5^2)-8 (b1^4) r1 (r5^2)-10 (b1^5) r1 (r5^2)-2 (b1^6) r1 (r5^2)-8 (b1^4) b2 r1 (r5^2)-2 (b1^5) b2 r1 (r5^2)+10 (b1^3) b5 r1 (r5^2)+13 (b1^4) b5 r1 (r5^2)+3 (b1^5) b5 r1 (r5^2)+10 (b1^3) b2 b5 r1 (r5^2)+3 (b1^4) b2 b5 r1 (r5^2)-5 (b1^2) (b5^2) r1 (r5^2)-9 (b1^3) (b5^2) r1 (r5^2)-4 (b1^4) (b5^2) r1 (r5^2)-8 (b1^2) b2 (b5^2) r1 (r5^2)-7 (b1^3) b2 (b5^2) r1 (r5^2)-3 (b1^2) (b2^2) (b5^2) r1 (r5^2)+5 (b1^3) (r1^2) (r5^2)+2 (b1^4) (r1^2) (r5^2)-2 (b1^5) (r1^2) (r5^2)-4 (b1^4) b2 (r1^2) (r5^2)+9 (b1^3) b5 (r1^2) (r5^2)+9 (b1^4) b5 (r1^2) (r5^2)+6 (b1^2) b2 b5 (r1^2) (r5^2)+13 (b1^3) b2 b5 (r1^2) (r5^2)+4 b1 (b5^2) (r1^2) (r5^2)+4 (b1^2) (b5^2) (r1^2) (r5^2)-(b1^3) (b5^2) (r1^2) (r5^2)+6 b1 b2 (b5^2) (r1^2) (r5^2)+3 b1 (b2^2) (b5^2) (r1^2) (r5^2)-20 (b1^2) (r1^3) (r5^2)-26 (b1^3) (r1^3) (r5^2)-8 (b1^4) (r1^3) (r5^2)-20 (b1^2) b2 (r1^3) (r5^2)-13 (b1^3) b2 (r1^3) (r5^2)-4 (b1^2) (b2^2) (r1^3) (r5^2)-6 b1 b5 (r1^3) (r5^2)-14 (b1^2) b5 (r1^3) (r5^2)-7 (b1^3) b5 (r1^3) (r5^2)-20 b1 b2 b5 (r1^3) (r5^2)-16 (b1^2) b2 b5 (r1^3) (r5^2)-12 b1 (b2^2) b5 (r1^3) (r5^2)-(b5^2) (r1^3) (r5^2)+4 b1 (b5^2) (r1^3) (r5^2)+6 (b1^2) (b5^2) (r1^3) (r5^2)-2 b2 (b5^2) (r1^3) (r5^2)+2 b1 b2 (b5^2) (r1^3) (r5^2)-2 (b2^2) (b5^2) (r1^3) (r5^2)+37 b1 (r1^4) (r5^2)+25 (b1^2) (r1^4) (r5^2)-2 (b1^3) (r1^4) (r5^2)+20 b1 b2 (r1^4) (r5^2)+2 b5 (r1^4) (r5^2)+8 b1 b5 (r1^4) (r5^2)+16 b2 b5 (r1^4) (r5^2)+2 b1 b2 b5 (r1^4) (r5^2)-3 (b5^2) (r1^4) (r5^2)-b1 (b5^2) (r1^4) (r5^2)+b2 (b5^2) (r1^4) (r5^2)-18 (r1^5) (r5^2)+22 b1 (r1^5) (r5^2)+16 (b1^2) (r1^5) (r5^2)+11 b1 b2 (r1^5) (r5^2)-4 b5 (r1^5) (r5^2)+4 b1 b5 (r1^5) (r5^2)+10 b2 b5 (r1^5) (r5^2)-2 (b5^2) (r1^5) (r5^2)-21 (r1^6) (r5^2)-3 b5 (r1^6) (r5^2)-6 (r1^7) (r5^2)+(b1^3) b5 (r5^3)+2 (b1^4) b5 (r5^3)+(b1^5) b5 (r5^3)+2 (b1^3) b2 b5 (r5^3)+2 (b1^4) b2 b5 (r5^3)+(b1^3) (b2^2) b5 (r5^3)-2 (b1^3) r1 (r5^3)-6 (b1^4) r1 (r5^3)-4 (b1^5) r1 (r5^3)-2 (b1^3) b2 r1 (r5^3)-4 (b1^4) b2 r1 (r5^3)-3 (b1^2) b5 r1 (r5^3)-3 (b1^3) b5 r1 (r5^3)-6 (b1^2) b2 b5 r1 (r5^3)-3 (b1^3) b2 b5 r1 (r5^3)-3 (b1^2) (b2^2) b5 r1 (r5^3)+14 (b1^2) (r1^2) (r5^3)+24 (b1^3) (r1^2) (r5^3)+9 (b1^4) (r1^2) (r5^3)+18 (b1^2) b2 (r1^2) (r5^3)+13 (b1^3) b2 (r1^2) (r5^3)+6 (b1^2) (b2^2) (r1^2) (r5^3)+3 b1 b5 (r1^2) (r5^3)+2 (b1^2) b5 (r1^2) (r5^3)+8 b1 b2 b5 (r1^2) (r5^3)+6 (b1^2) b2 b5 (r1^2) (r5^3)+4 b1 (b2^2) b5 (r1^2) (r5^3)-22 b1 (r1^3) (r5^3)-16 (b1^2) (r1^3) (r5^3)+4 (b1^3) (r1^3) (r5^3)-16 b1 b2 (r1^3) (r5^3)+(b1^2) b2 (r1^3) (r5^3)-b5 (r1^3) (r5^3)-2 b1 b5 (r1^3) (r5^3)-(b1^2) b5 (r1^3) (r5^3)-4 b2 b5 (r1^3) (r5^3)+10 (r1^4) (r5^3)-15 b1 (r1^4) (r5^3)-13 (b1^2) (r1^4) (r5^3)-10 b1 b2 (r1^4) (r5^3)+b5 (r1^4) (r5^3)-b1 b5 (r1^4) (r5^3)-3 b2 b5 (r1^4) (r5^3)+13 (r1^5) (r5^3)+b5 (r1^5) (r5^3)+4 (r1^6) (r5^3)+(b1^3) (r5^4)+2 (b1^4) (r5^4)+(b1^5) (r5^4)+2 (b1^3) b2 (r5^4)+2 (b1^4) b2 (r5^4)+(b1^3) (b2^2) (r5^4)-4 (b1^2) r1 (r5^4)-7 (b1^3) r1 (r5^4)-3 (b1^4) r1 (r5^4)-6 (b1^2) b2 r1 (r5^4)-5 (b1^3) b2 r1 (r5^4)-2 (b1^2) (b2^2) r1 (r5^4)+5 b1 (r1^2) (r5^4)+4 (b1^2) (r1^2) (r5^4)-(b1^3) (r1^2) (r5^4)+4 b1 b2 (r1^2) (r5^4)-(b1^2) b2 (r1^2) (r5^4)-2 (r1^3) (r5^4)+4 b1 (r1^3) (r5^4)+4 (b1^2) (r1^3) (r5^4)+3 b1 b2 (r1^3) (r5^4)-3 (r1^4) (r5^4)-(r1^5) (r5^4))/((b5 r1-b1 r5) (-2 b1 r1-(b1^2) r1-b1 b5 r1-2 b2 b5 r1+2 (r1^2)+b5 (r1^2)+(r1^3)+2 b1 r5+2 (b1^2) r5+2 b1 b2 r5-2 r1 r5-b1 r1 r5-(r1^2) r5) (-2 (b1^2) (r1^2)-(b1^3) (r1^2)+b1 b5 (r1^2)-2 b1 b2 b5 (r1^2)+b2 (b5^2) (r1^2)+2 b1 (r1^3)-b5 (r1^3)+b1 (r1^4)+3 (b1^2) r1 r5+2 (b1^3) r1 r5+2 (b1^2) b2 r1 r5-b1 b5 r1 r5-3 b1 (r1^2) r5+b5 (r1^2) r5-2 b1 (r1^3) r5-(b1^2) (r5^2)-(b1^3) (r5^2)-(b1^2) b2 (r5^2)+b1 r1 (r5^2)+b1 (r1^2) (r5^2))),c137->-((-2 (b1^4) b5 (r1^2)-3 (b1^5) b5 (r1^2)-(b1^6) b5 (r1^2)-2 (b1^4) b2 b5 (r1^2)-(b1^5) b2 b5 (r1^2)+2 (b1^3) (b5^2) (r1^2)+2 (b1^4) (b5^2) (r1^2)-2 (b1^4) b2 (b5^2) (r1^2)-2 (b1^3) (b2^2) (b5^2) (r1^2)+(b1^3) (b5^3) (r1^2)+(b1^4) (b5^3) (r1^2)+2 (b1^2) b2 (b5^3) (r1^2)+3 (b1^3) b2 (b5^3) (r1^2)+2 (b1^2) (b2^2) (b5^3) (r1^2)-2 (b1^5) (r1^3)-(b1^6) (r1^3)+4 (b1^3) b5 (r1^3)+7 (b1^4) b5 (r1^3)+(b1^5) b5 (r1^3)+2 (b1^3) b2 b5 (r1^3)-2 (b1^4) b2 b5 (r1^3)-3 (b1^2) (b5^2) (r1^3)-2 (b1^3) (b5^2) (r1^3)+2 (b1^4) (b5^2) (r1^3)+(b1^2) b2 (b5^2) (r1^3)+5 (b1^3) b2 (b5^2) (r1^3)-(b1^2) (b5^3) (r1^3)-(b1^3) (b5^3) (r1^3)-b1 b2 (b5^3) (r1^3)-(b1^2) b2 (b5^3) (r1^3)+b1 (b2^2) (b5^3) (r1^3)+4 (b1^4) (r1^4)+(b1^5) (r1^4)-2 (b1^2) b5 (r1^4)-(b1^3) b5 (r1^4)+4 (b1^4) b5 (r1^4)+3 (b1^3) b2 b5 (r1^4)-5 (b1^2) (b5^2) (r1^4)-4 (b1^3) (b5^2) (r1^4)+(b1^2) b2 (b5^2) (r1^4)-(b1^2) (b5^3) (r1^4)-b2 (b5^3) (r1^4)-3 b1 b2 (b5^3) (r1^4)+(b2^2) (b5^3) (r1^4)+2 (b1^4) (r1^5)-7 (b1^2) b5 (r1^5)-4 (b1^3) b5 (r1^5)+2 b1 b2 b5 (r1^5)+3 (b1^2) b2 b5 (r1^5)+(b5^2) (r1^5)+4 b1 (b5^2) (r1^5)-2 (b1^2) (b5^2) (r1^5)-b2 (b5^2) (r1^5)-3 b1 b2 (b5^2) (r1^5)+2 (b2^2) (b5^2) (r1^5)+b1 (b5^3) (r1^5)-b2 (b5^3) (r1^5)-4 (b1^2) (r1^6)-2 (b1^3) (r1^6)+4 b1 b5 (r1^6)-3 (b1^2) b5 (r1^6)-2 b2 b5 (r1^6)-2 b1 b2 b5 (r1^6)+(b5^2) (r1^6)+4 b1 (b5^2) (r1^6)-b2 (b5^2) (r1^6)+2 b1 (r1^7)-(b1^2) (r1^7)+3 b1 b5 (r1^7)-b2 b5 (r1^7)+b1 (r1^8)+2 (b1^5) r1 r5+3 (b1^6) r1 r5+(b1^7) r1 r5+2 (b1^5) b2 r1 r5+(b1^6) b2 r1 r5+2 (b1^5) b5 r1 r5+2 (b1^6) b5 r1 r5+4 (b1^4) b2 b5 r1 r5+6 (b1^5) b2 b5 r1 r5+4 (b1^4) (b2^2) b5 r1 r5-2 (b1^3) (b5^2) r1 r5-5 (b1^4) (b5^2) r1 r5-3 (b1^5) (b5^2) r1 r5-6 (b1^3) b2 (b5^2) r1 r5-7 (b1^4) b2 (b5^2) r1 r5-4 (b1^3) (b2^2) (b5^2) r1 r5-4 (b1^4) (r1^2) r5-(b1^5) (r1^2) r5+2 (b1^6) (r1^2) r5-2 (b1^4) b2 (r1^2) r5+2 (b1^5) b2 (r1^2) r5-2 (b1^3) b5 (r1^2) r5-10 (b1^4) b5 (r1^2) r5-6 (b1^5) b5 (r1^2) r5-6 (b1^3) b2 b5 (r1^2) r5-6 (b1^4) b2 b5 (r1^2) r5+3 (b1^2) (b5^2) (r1^2) r5+5 (b1^3) (b5^2) (r1^2) r5+(b1^4) (b5^2) (r1^2) r5+2 (b1^2) b2 (b5^2) (r1^2) r5-2 (b1^3) b2 (b5^2) (r1^2) r5-3 (b1^2) (b2^2) (b5^2) (r1^2) r5+2 (b1^3) (r1^3) r5-11 (b1^4) (r1^3) r5-7 (b1^5) (r1^3) r5-3 (b1^4) b2 (r1^3) r5+4 (b1^2) b5 (r1^3) r5+12 (b1^3) b5 (r1^3) r5-8 (b1^3) b2 b5 (r1^3) r5+5 (b1^2) (b5^2) (r1^3) r5+7 (b1^3) (b5^2) (r1^3) r5+3 b1 b2 (b5^2) (r1^3) r5+8 (b1^2) b2 (b5^2) (r1^3) r5-3 b1 (b2^2) (b5^2) (r1^3) r5+7 (b1^3) (r1^4) r5-2 (b1^4) (r1^4) r5-2 (b1^2) b2 (r1^4) r5-3 (b1^3) b2 (r1^4) r5-2 b1 b5 (r1^4) r5+6 (b1^2) b5 (r1^4) r5+12 (b1^3) b5 (r1^4) r5-2 b1 b2 b5 (r1^4) r5-4 b1 (b2^2) b5 (r1^4) r5-(b5^2) (r1^4) r5-4 b1 (b5^2) (r1^4) r5-(b1^2) (b5^2) (r1^4) r5+b2 (b5^2) (r1^4) r5+6 b1 b2 (b5^2) (r1^4) r5-2 (b2^2) (b5^2) (r1^4) r5+8 (b1^2) (r1^5) r5+9 (b1^3) (r1^5) r5+2 b1 b2 (r1^5) r5+2 (b1^2) b2 (r1^5) r5-10 b1 b5 (r1^5) r5-2 (b1^2) b5 (r1^5) r5+4 b2 b5 (r1^5) r5+6 b1 b2 b5 (r1^5) r5-(b5^2) (r1^5) r5-4 b1 (b5^2) (r1^5) r5+b2 (b5^2) (r1^5) r5-6 b1 (r1^6) r5+b1 b2 (r1^6) r5-6 b1 b5 (r1^6) r5+2 b2 b5 (r1^6) r5-3 b1 (r1^7) r5-2 (b1^5) (r5^2)-4 (b1^6) (r5^2)-2 (b1^7) (r5^2)-4 (b1^5) b2 (r5^2)-4 (b1^6) b2 (r5^2)-2 (b1^5) (b2^2) (r5^2)+2 (b1^4) b5 (r5^2)+4 (b1^5) b5 (r5^2)+2 (b1^6) b5 (r5^2)+4 (b1^4) b2 b5 (r5^2)+4 (b1^5) b2 b5 (r5^2)+2 (b1^4) (b2^2) b5 (r5^2)+5 (b1^4) r1 (r5^2)+6 (b1^5) r1 (r5^2)+(b1^6) r1 (r5^2)+5 (b1^4) b2 r1 (r5^2)+(b1^5) b2 r1 (r5^2)-2 (b1^3) b5 r1 (r5^2)+2 (b1^5) b5 r1 (r5^2)+(b1^3) b2 b5 r1 (r5^2)+5 (b1^4) b2 b5 r1 (r5^2)+3 (b1^3) (b2^2) b5 r1 (r5^2)-4 (b1^3) (r1^2) (r5^2)+5 (b1^4) (r1^2) (r5^2)+7 (b1^5) (r1^2) (r5^2)+7 (b1^4) b2 (r1^2) (r5^2)-2 (b1^2) b5 (r1^2) (r5^2)-11 (b1^3) b5 (r1^2) (r5^2)-7 (b1^4) b5 (r1^2) (r5^2)-3 (b1^2) b2 b5 (r1^2) (r5^2)-4 (b1^3) b2 b5 (r1^2) (r5^2)+3 (b1^2) (b2^2) b5 (r1^2) (r5^2)+(b1^2) (r1^3) (r5^2)-10 (b1^3) (r1^3) (r5^2)-4 (b1^4) (r1^3) (r5^2)+3 (b1^2) b2 (r1^3) (r5^2)+3 (b1^3) b2 (r1^3) (r5^2)+2 (b1^2) (b2^2) (r1^3) (r5^2)+2 b1 b5 (r1^3) (r5^2)+(b1^2) b5 (r1^3) (r5^2)-5 (b1^3) b5 (r1^3) (r5^2)-6 (b1^2) b2 b5 (r1^3) (r5^2)+4 b1 (b2^2) b5 (r1^3) (r5^2)-3 (b1^2) (r1^4) (r5^2)-8 (b1^3) (r1^4) (r5^2)-4 b1 b2 (r1^4) (r5^2)-5 (b1^2) b2 (r1^4) (r5^2)+6 b1 b5 (r1^4) (r5^2)+5 (b1^2) b5 (r1^4) (r5^2)-2 b2 b5 (r1^4) (r5^2)-4 b1 b2 b5 (r1^4) (r5^2)+6 b1 (r1^5) (r5^2)+3 (b1^2) (r1^5) (r5^2)-2 b1 b2 (r1^5) (r5^2)+3 b1 b5 (r1^5) (r5^2)-b2 b5 (r1^5) (r5^2)+3 b1 (r1^6) (r5^2)-(b1^4) (r5^3)-2 (b1^5) (r5^3)-(b1^6) (r5^3)-2 (b1^4) b2 (r5^3)-2 (b1^5) b2 (r5^3)-(b1^4) (b2^2) (r5^3)+2 (b1^3) r1 (r5^3)+2 (b1^4) r1 (r5^3)+(b1^3) b2 r1 (r5^3)-(b1^4) b2 r1 (r5^3)-(b1^3) (b2^2) r1 (r5^3)-(b1^2) (r1^2) (r5^3)+3 (b1^3) (r1^2) (r5^3)+3 (b1^4) (r1^2) (r5^3)-(b1^2) b2 (r1^2) (r5^3)+(b1^3) b2 (r1^2) (r5^3)-2 (b1^2) (b2^2) (r1^2) (r5^3)-(b1^2) (r1^3) (r5^3)+(b1^3) (r1^3) (r5^3)+2 b1 b2 (r1^3) (r5^3)+3 (b1^2) b2 (r1^3) (r5^3)-2 b1 (r1^4) (r5^3)-2 (b1^2) (r1^4) (r5^3)+b1 b2 (r1^4) (r5^3)-b1 (r1^5) (r5^3))/((b5 r1-b1 r5) (-2 b1 r1-(b1^2) r1-b1 b5 r1-2 b2 b5 r1+2 (r1^2)+b5 (r1^2)+(r1^3)+2 b1 r5+2 (b1^2) r5+2 b1 b2 r5-2 r1 r5-b1 r1 r5-(r1^2) r5) (-2 (b1^2) (r1^2)-(b1^3) (r1^2)+b1 b5 (r1^2)-2 b1 b2 b5 (r1^2)+b2 (b5^2) (r1^2)+2 b1 (r1^3)-b5 (r1^3)+b1 (r1^4)+3 (b1^2) r1 r5+2 (b1^3) r1 r5+2 (b1^2) b2 r1 r5-b1 b5 r1 r5-3 b1 (r1^2) r5+b5 (r1^2) r5-2 b1 (r1^3) r5-(b1^2) (r5^2)-(b1^3) (r5^2)-(b1^2) b2 (r5^2)+b1 r1 (r5^2)+b1 (r1^2) (r5^2)))),c142->((r1^2) (-b1+r1) (-b1+b5+r1-r5) (-(b1^2)+b1 b5+(r1^2)-r1 r5) (-2 (b1^2) (r1^2)-(b1^3) (r1^2)+b1 b5 (r1^2)-(b1^2) b5 (r1^2)-2 b1 b2 b5 (r1^2)+b1 (b5^2) (r1^2)+b2 (b5^2) (r1^2)+2 b1 (r1^3)-b5 (r1^3)+b1 (r1^4)-b5 (r1^4)+3 (b1^2) r1 r5+3 (b1^3) r1 r5+2 (b1^2) b2 r1 r5-b1 b5 r1 r5-(b1^2) b5 r1 r5-3 b1 (r1^2) r5+b5 (r1^2) r5-b1 (r1^3) r5+b5 (r1^3) r5-(b1^2) (r5^2)-(b1^3) (r5^2)-(b1^2) b2 (r5^2)+b1 r1 (r5^2)))/((b5 r1-b1 r5) (-2 b1 r1-(b1^2) r1-b1 b5 r1-2 b2 b5 r1+2 (r1^2)+b5 (r1^2)+(r1^3)+2 b1 r5+2 (b1^2) r5+2 b1 b2 r5-2 r1 r5-b1 r1 r5-(r1^2) r5) (((-2 (b1^2) (r1^2)-(b1^3) (r1^2)+b1 b5 (r1^2)-2 b1 b2 b5 (r1^2)+b2 (b5^2) (r1^2)+2 b1 (r1^3)-b5 (r1^3)+b1 (r1^4)+3 (b1^2) r1 r5+2 (b1^3) r1 r5+2 (b1^2) b2 r1 r5-b1 b5 r1 r5-3 b1 (r1^2) r5+b5 (r1^2) r5-2 b1 (r1^3) r5-(b1^2) (r5^2)-(b1^3) (r5^2)-(b1^2) b2 (r5^2)+b1 r1 (r5^2)+b1 (r1^2) (r5^2)))^2)),c123->((b1-b5) (((b1-r1))^2) (r1^2) (b1+r1) (r1-r5) (b1-b5-r1+r5) (b1 r1+(b1^2) r1+b2 b5 r1-(r1^2)-(r1^3)-b1 r5-(b1^2) r5-b1 b2 r5+r1 r5+(r1^2) r5))/((-b5 r1+b1 r5) (2 b1 r1+(b1^2) r1+b1 b5 r1+2 b2 b5 r1-2 (r1^2)-b5 (r1^2)-(r1^3)-2 b1 r5-2 (b1^2) r5-2 b1 b2 r5+2 r1 r5+b1 r1 r5+(r1^2) r5) (((2 (b1^2) (r1^2)+(b1^3) (r1^2)-b1 b5 (r1^2)+2 b1 b2 b5 (r1^2)-b2 (b5^2) (r1^2)-2 b1 (r1^3)+b5 (r1^3)-b1 (r1^4)-3 (b1^2) r1 r5-2 (b1^3) r1 r5-2 (b1^2) b2 r1 r5+b1 b5 r1 r5+3 b1 (r1^2) r5-b5 (r1^2) r5+2 b1 (r1^3) r5+(b1^2) (r5^2)+(b1^3) (r5^2)+(b1^2) b2 (r5^2)-b1 r1 (r5^2)-b1 (r1^2) (r5^2)))^2))};

Factor[Eqs/.Solution1212]  (* verification and building quadrics; takes awhile *)
QuadricsWithTwoLines=Factor[Quadrics15/.Solution1212];
Variables[QuadricsWithTwoLines]

We write down the result to a file for future reference. These are called 1212 because one can check that this is the structure of the singular fiber, generically. This is done by picking random values of the 5 variables, we skip this computation.

fname ="Desktop/MumfordFPP2021/quadricsSecondTry1212";
OpenWrite[fname];
Write[fname, QuadricsWithTwoLines];
Close[fname];

Table[ByteCount[QuadricsWithTwoLines[[k]]],{k,1,9}]

LineInP5Solved=Factor[LineInP5/.Solution1212];
Line2InP5Solved=Factor[Line2InP5/.Solution1212];
JacMat=Table[D[{QuadricsWithTwoLines[[1]],QuadricsWithTwoLines[[2]],QuadricsWithTwoLines[[3]],QuadricsWithTwoLines[[4]]}/.{u0->1,u1->1,v1->1},vs[[k]]],{k,2,5}]/.LineInP5Solved;
DetJacMat=Factor[Det[JacMat]];
TwoPoints=Numerator[DetJacMat][[5]];
 (* this is easily seen to be the correct factor *)
disctp=Factor[Coefficient[TwoPoints,t]^2 - 4 Coefficient[TwoPoints,t^2](TwoPoints/.t->0)  ]
disctpsqfree=(b1-r1) (b1+r1) (b1-b5+r1-r5) (b1-b5-r1+r5) (2 b1 r1+(b1^2) r1+b1 b5 r1+2 b2 b5 r1-2 (r1^2)+b5 (r1^2)-(r1^3)-2 b1 r5-2 (b1^2) r5-2 b1 b2 r5+2 r1 r5-b1 r1 r5+(r1^2) r5) (2 b1 r1+(b1^2) r1+b1 b5 r1+2 b2 b5 r1-2 (r1^2)-b5 (r1^2)-(r1^3)-2 b1 r5-2 (b1^2) r5-2 b1 b2 r5+2 r1 r5+b1 r1 r5+(r1^2) r5) ;

The idea is to parametrize a quadric r^2 = disc in the variables r and b2. This will allow us to have the singular points in the special fiber defined over the field of parameters.

b2tot5=Factor[Solve[(( 2 b1 r1+(b1^2) r1+b1 b5 r1+2 b2 b5 r1-2 (r1^2)+b5 (r1^2)-(r1^3)-2 b1 r5-2 (b1^2) r5-2 b1 b2 r5+2 r1 r5-b1 r1 r5+(r1^2) r5) t5)^2 == disctpsqfree,b2]][[2]]
Factor[disctpsqfree/.b2tot5]

This simple change of variables reduces the size of the equations.

btot=Solve[{t1==b1-r1,t2==b1+r1,t3==b1-b5+r1-r5,t4== b1-b5-r1+r5},{b1,b5,r1,r5}][[1]]
Factor[disctpsqfree/.b2tot5/.btot]
Solution1212better=Factor[Table[Solution1212[[k]][[1]]-> (Solution1212[[k]][[2]]/.Join[btot,Factor[b2tot5/.btot]]),{k,1,Length[Solution1212]}]];
QuadricsBetter=Factor[Quadrics15/.Solution1212better];
QuadricsGood=Numerator[Factor[QuadricsBetter]];
Table[ByteCount[QuadricsGood[[k]]],{k,1,9}]

fname ="Desktop/MumfordFPP2021/QuadricsGood1212";
OpenWrite[fname];
Write[fname, QuadricsGood];
Close[fname];

We will now compute four points of the intersection of the components of the special fiber. We will be looking for the following singularities at these points.
On the first line we will have a smooth point and an A_1 singular point. On the second line we will have two A_2 singularities. So we don’t need to worry about which points on the lines lie on which conics. It is hoped that A_2 singularities will occur at singular loci of the A_1 singularities locus.

(* we start by finding the points *)
LineInP5Good=Factor[LineInP5/.Solution1212better];
Line2InP5Good=Factor[Line2InP5/.Solution1212better];
JacMat=Factor[Table[D[{QuadricsBetter[[1]],QuadricsBetter[[2]],QuadricsBetter[[3]],QuadricsBetter[[4]]}/.{u0->1,u1->1,v1->1},vs[[k]]],{k,2,5}]/.LineInP5Good];
DetJacMat=Factor[Det[JacMat]];
sols=Solve[{t^2+t≠ 0,DetJacMat==0},t];
pt1=Factor[Join[{u0->1,u1->1},LineInP5Good/.sols[[1]]]];
pt2=Factor[Join[{u0->1,u1->1},LineInP5Good/.sols[[2]]]];

JacMat=Factor[Table[D[{QuadricsBetter[[1]],QuadricsBetter[[2]],QuadricsBetter[[3]],QuadricsBetter[[4]]}/.{u0->1,u1->1,v1->1},vs[[k]]],{k,2,5}]/.Line2InP5Good];
DetJacMat=Factor[Det[JacMat]];
sols=Solve[{t^2+t≠ 0,DetJacMat==0},t];
pt3=Factor[Join[{u0->1,u1->1},Line2InP5Good/.sols[[1]]]];
pt4=Factor[Join[{u0->1,u1->1},Line2InP5Good/.sols[[2]]]];

Now we compute the condition of the intersection of quadrics being singular at points pt1, pt3, pt4. This takes some work. We compute minors of the Jacobian matrix and then work to figure out the relevant factors. The idea is to pick a relatively small size minor, and then for each factor see if vanishing of it leads to a matrix of correct rank. We also check that the points pt1, pt2, pt3, pt4 do not lie in a P^3.

(* clear denominators of pts *)
pt1vs=vs/.pt1;
For[i=1,i≤6,i++,
den=Denominator[pt1vs[[i]]];
pt1vs = Factor[pt1vs den];
];
pt1=Join[{u0->1,u1->1},Table[vs[[k]]-> pt1vs[[k]],{k,1,6}]];


pt2vs=vs/.pt2;
For[i=1,i≤6,i++,
den=Denominator[pt2vs[[i]]];
pt2vs = Factor[pt2vs den];
];
pt2=Join[{u0->1,u1->1},Table[vs[[k]]-> pt2vs[[k]],{k,1,6}]];

pt3vs=vs/.pt3;
For[i=1,i≤6,i++,
den=Denominator[pt3vs[[i]]];
pt3vs = Factor[pt3vs den];
];
pt3=Join[{u0->1,u1->1},Table[vs[[k]]-> pt3vs[[k]],{k,1,6}]];

pt4vs=vs/.pt4;
For[i=1,i≤6,i++,
den=Denominator[pt4vs[[i]]];
pt4vs = Factor[pt4vs den];
];
pt4=Join[{u0->1,u1->1},Table[vs[[k]]-> pt4vs[[k]],{k,1,6}]];


(* get to it *) 
Print["get to it"];
FiveVars={t1,t2,t3,t4,t5};
RandomCut=Table[0== Sum[(RandomInteger[100]+1)FiveVars[[m]],{m,1,5}],{k,1,4}]

PointsWithNodes={pt1,pt3,pt4};
PossibleFactors={};
For[numpt=1,numpt≤3,numpt++,
JacMat=Table[D[QuadricsGood[[k]],uvs[[l]]]/.PointsWithNodes[[numpt]],{k,1,9},{l,1,8}];
minjac=Flatten[Minors[JacMat,4]];
Print["computed matrix and its minors"];

smallpoly=0;
bc=10^10;
For[k=1,k≤Length[minjac],k++,
If[Mod[k,100]==5,Print[k," ",bc]];
If[ByteCount[minjac[[k]]]<bc,
If[0≠ minjac[[k]]/.Table[FiveVars[[m]]-> 10+RandomInteger[30],{m,1,5}],
bc=ByteCount[minjac[[k]]];
smallpoly=minjac[[k]];
If[bc<13000000,k=Length[minjac]+1];
];
];
];

Print["starting to compute factors of small polynomial"];(* this takes time *)
factors=Factor[smallpoly];
Print["computed factors of a small polynomial"];
Print[{Length[factors],Table[ByteCount[factors[[k]]],{k,1,Length[factors]}]}];


factorsgood={};
nonzero=Factor[(v2/.pt1)(v1/.pt2)-(v2/.pt2)(v1/.pt1)];
nonzero=nonzero  Factor[(v2/.pt3)(v1/.pt4)-(v2/.pt4)(v1/.pt3)];
For[k=1,k≤Length[factors],k++,
fi=NSolve[Join[{0≠ nonzero,factors[[k]]==0},RandomCut],200];
If[Length[fi]>0,
If[MatrixRank[JacMat/.fi[[1]]]== 3,
If[MatrixRank[(vs/.{pt1,pt2,pt3,pt4})/.fi[[1]]]== 4,
AppendTo[factorsgood,factors[[k]]];
];];
];
];
Print["done with checking factors"];
Print[Length[factorsgood]];
AppendTo[PossibleFactors,factorsgood];
];

Table[ByteCount[PossibleFactors[[1]][[k]]],{k,1,Length[PossibleFactors[[1]]]}]
Table[ByteCount[PossibleFactors[[2]][[k]]],{k,1,Length[PossibleFactors[[2]]]}]
Table[ByteCount[PossibleFactors[[3]][[k]]],{k,1,Length[PossibleFactors[[3]]]}]
GoodFactors={PossibleFactors[[1]][[1]] ,PossibleFactors[[2]][[1]],PossibleFactors[[3]][[1]]};
 (* In some runs, some small factors randomly get into it.*)

fname ="Desktop/MumfordFPP2021/GoodFactors1212";
OpenWrite[fname];
Write[fname, GoodFactors];
Close[fname];

We can simplify the GoodFactors by making a further variable change. In fact, we can even solve for two of the new variables s1 and s5. We can express s2 as a root of a cubic polynomial with coefficients in C[s3,s4].

GoodFactorsNew=Numerator[Factor[GoodFactors/.{t1->s1,t2->s2,t3->s3 s5/s1,t4->s4 s5/s2,t5->s5}]];
FiveVarsNew={s1,s2,s3,s4,s5};
slss1=Factor[Solve[{s1≠0,GoodFactorsNew[[1]]+GoodFactorsNew[[3]]==0},s1]][[1]]  (* lucky for us, it's linear in s1 *)
GoodFactorsNewLeft=Numerator[Factor[{GoodFactorsNew[[2]],GoodFactorsNew[[3]]}/.slss1]];
GoodFactorsNewLeft[[1]]=GoodFactorsNewLeft[[1]]/((s3-s4) s4 (s5^10) );
GoodFactorsNewLeft[[2]]=GoodFactorsNewLeft[[2]]/((s3-s4) s4 (-3-s3-s4+s3 s4) (s5^10) );
Exponent[GoodFactorsNewLeft[[1]],FiveVarsNew]
Exponent[GoodFactorsNewLeft[[2]],FiveVarsNew]

(* we can use two quadratic equations to solve for s5 *)
Factor[Coefficient[GoodFactorsNewLeft[[1]],s5^2]]
Factor[Coefficient[GoodFactorsNewLeft[[2]],s5^2]]
Factor[GoodFactorsNewLeft[[1]](-2+2 s3+3 s2 s3+4 s2 (s3^2)+s2 (s3^3)+2 s4+8 s3 s4-2 s2 s3 s4-2 (s3^2) s4-4 s2 (s3^2) s4-2 s2 (s3^3) s4-2 s3 (s4^2)-s2 s3 (s4^2)-6 (s3^2) (s4^2)+s2 (s3^3) (s4^2)) 
+ GoodFactorsNewLeft[[2]](6-9 s2 s3+2 (s3^2)+3 s2 (s3^2)+5 s2 (s3^3)+s2 (s3^4)-14 s3 s4-15 s2 s3 s4+8 (s3^2) s4+13 s2 (s3^2) s4-2 (s3^3) s4+3 s2 (s3^3) s4-s2 (s3^4) s4+2 (s4^2)+8 s3 (s4^2)-7 s2 s3 (s4^2)+22 (s3^2) (s4^2)+13 s2 (s3^2) (s4^2)-8 (s3^3) (s4^2)-5 s2 (s3^3) (s4^2)-s2 (s3^4) (s4^2)-2 s3 (s4^3)-s2 s3 (s4^3)-8 (s3^2) (s4^3)+3 s2 (s3^2) (s4^3)-14 (s3^3) (s4^3)-3 s2 (s3^3) (s4^3)+s2 (s3^4) (s4^3))];
solss5=Factor[Solve[0==%,s5][[1]]];
LastOne=PolynomialGCD[Numerator[Factor[GoodFactorsNewLeft[[1]]/.solss5]],Numerator[Factor[GoodFactorsNewLeft[[2]]/.solss5]]];
Length[LastOne]

It turned out that the last option is the only viable one.

LastOneGood=Factor[LastOne[[9]]]

This is a reality check -- it works.

ThreeToFive={solss5[[1]],s1-> ((s1/.slss1)/.solss5[[1]])};
sm={s3->1+RandomInteger[20],s4->1+RandomInteger[20]};
sm=Join[sm,NSolve[0==LastOneGood/.sm,2000][[1]]];
sm=Join[sm,ThreeToFive/.sm];
tsm={t1->s1,t2->s2,t3->s3 s5/s1,t4->s4 s5/s2,t5->s5}/.sm;
qsm=QuadricsGood/.tsm;
pt3sm=pt3/.tsm;
pt1sm=pt1/.tsm;
pt4sm=pt4/.tsm;
pt2sm=pt2/.tsm;
MatrixRank[Table[D[qsm[[k]],uvs[[m]]]/.pt1sm,{k,1,9},{m,1,8}]]
MatrixRank[Table[D[qsm[[k]],uvs[[m]]]/.pt2sm,{k,1,9},{m,1,8}]]
MatrixRank[Table[D[qsm[[k]],uvs[[m]]]/.pt3sm,{k,1,9},{m,1,8}]]
MatrixRank[Table[D[qsm[[k]],uvs[[m]]]/.pt4sm,{k,1,9},{m,1,8}]]

fname ="Desktop/MumfordFPP2021/LastOneGood";
OpenWrite[fname];
Write[fname,LastOneGood];
Close[fname];

The plan now is to put conditions on having a worse than A1 singularity at pt3 and pt4.

A Magma computation has shown that modulo 79 the following 6 collections of s1,...,s5,t1,..,t5 
give the desired singularities
 [ 61, 14, 47, 52, 1, 61, 14, 72, 15, 1]
[ 8, 15, 65, 27, 12, 8, 15, 58, 69, 12 ]   
[ 6, 19, 32, 14, 28, 6, 19, 44, 4, 28 ]
[ 4, 44, 14, 32, 28, 4, 44, 19, 6,  28 ]
[ 69, 58, 27, 65, 12, 69, 58, 15,  8, 12]
[ 15, 72, 52, 47, 1, 15, 72, 14, 61, 1 ]
We will extend these to modulo powers of 79 and then find the relevant algebraic numbers.

The condition of having a worse than A_1 singularity at the point appears to be the following. Solve for v5, v6 and slice u0->1,v1->1 to reduce to four variables u1,v2,v3,v4. Pick a remaining equation which is singular at the point. This should be a hyperplane in the space of all equations, so we can try to take the space spanned by the two smaller ones. For this equation check that the restriction of the Hessian matrix to the tangent three-dimensional space is degenerate. 

Given a 4x4 Hessian matrix H and 4x1 vector V of partials, a sufficient condition for the former to be singular on the kernel of the latter is to have rank at most two for the Hessian. This is very likely the correct condition in our case. However, in reality we should be looking at only one determinant, not 10. Specifically, we may want to say that the coefficient at t of the determinant of H + t V^*V is identically zero.

H={{h11,h12,h13,h14},{h12,h22,h23,h24},{h13,h23,h33,h34},{h14,h24,h34,h44}};
W={W1,W2,W3,W4};
tbz=Factor[Coefficient[Expand[Det[H+t Table[W[[i]]W[[j]],{i,1,4},{j,1,4}]]],t]]

Eqs=QuadricsGood/.Solve[{QuadricsGood[[4]]==0,QuadricsGood[[6]]==0},{v5,v6}][[1]];
Eqs=Numerator[Factor[Eqs]];
Table[ByteCount[Eqs[[k]]],{k,1,Length[Eqs]}]

We will look at the extension of the first solution to mod 79^2, 79^3 etc. Each step requires a fair bit of work, but it doesn’t take long to get to 79^100.

magmas234= {{14,47,52},{15, 65, 27},{ 19, 32, 14},{44, 14, 32},{ 58, 27, 65},{ 72, 52, 47}};
betters234={};
For[n=1,n≤6,n++,
Print[n];
{s2old,s3old,s4old}=magmas234[[n]];

deg=1;

For[deg=1,deg≤100,deg++,
(* Now let's compute the values of s1 and s2 *)

ToMeasure={{0,0,0},{1,0,0},{0,1,0},{0,0,1},{67,66,8}};
TheValues={};
For[mm=1,mm≤Length[ToMeasure],mm++,
spoint={s2->s2old+ 79^deg ToMeasure[[mm]][[1]],s3->s3old+ 79^deg ToMeasure[[mm]][[2]],s4->s4old+ 79^deg ToMeasure[[mm]][[3]]} ;

spoint=Join[spoint,Solve[s5==(s5/.solss5/.spoint),Modulus->79^(deg+1)][[1]]];
spoint=Join[spoint,Solve[s1==(s1/.slss1/.spoint),Modulus->79^(deg+1)][[1]]];
spoint=Join[spoint,Solve[{t1==  s1,t2== s2,t3== s3 s5/s1,t4== s4 s5/s2,t5== s5}/.spoint,{t1,t2,t3,t4,t5},Modulus->79^(deg+1)][[1]]];

pt3spoint=Table[uvs[[k]]-> Mod[uvs[[k]]/.pt3/.spoint,79^(deg+1)],{k,1,8}];
Eqs1=PolynomialMod[({Eqs[[2]],Eqs[[3]]}/.spoint)/.{u0->1,v1->(v1/.pt3spoint)},79^(deg+1)]; (* taking a slice *)
Wptspoint=Mod[{D[Eqs1[[1]],u1],D[Eqs1[[1]],v2],D[Eqs1[[1]],v3],D[Eqs1[[1]],v4]}/.pt3spoint,79^(deg+1)] ;(* finding W *)
EqsSing= PolynomialMod[Wptspoint[[1]]Eqs1[[2]]-(D[Eqs1[[2]],u1]/.pt3spoint)Eqs1[[1]],79^(deg+1)];
Hptspoint=Mod[Table[D[D[EqsSing,{u1,v2,v3,v4}[[k]]],{u1,v2,v3,v4}[[l]]],{k,1,4},{l,1,4}]/.pt3spoint,79^(deg+1)];
SubHW=Join[Flatten[Table[H[[i]][[j]]->Hptspoint[[i]][[j]],{j,1,4},{i,1,j}]],Table[W[[k]]-> Wptspoint[[k]],{k,1,4}]];
tbz3spoint=Mod[tbz/.SubHW,79^(deg+1)];
If[Mod[tbz3spoint,79^deg]≠0,Print["pt3 warning"]];

pt4spoint=Table[uvs[[k]]-> Mod[uvs[[k]]/.pt4/.spoint,79^(deg+1)],{k,1,8}];
Eqs1=PolynomialMod[({Eqs[[2]],Eqs[[3]]}/.spoint)/.{u0->1,v1->(v1/.pt4spoint)},79^(deg+1)]; (* taking a slice *)
Wptspoint=Mod[{D[Eqs1[[1]],u1],D[Eqs1[[1]],v2],D[Eqs1[[1]],v3],D[Eqs1[[1]],v4]}/.pt4spoint,79^(deg+1)] ;(* finding W *)
EqsSing= PolynomialMod[Wptspoint[[1]]Eqs1[[2]]-(D[Eqs1[[2]],u1]/.pt4spoint)Eqs1[[1]],79^(deg+1)];
Hptspoint=Mod[Table[D[D[EqsSing,{u1,v2,v3,v4}[[k]]],{u1,v2,v3,v4}[[l]]],{k,1,4},{l,1,4}]/.pt4spoint,79^(deg+1)];
SubHW=Join[Flatten[Table[H[[i]][[j]]->Hptspoint[[i]][[j]],{j,1,4},{i,1,j}]],Table[W[[k]]-> Wptspoint[[k]],{k,1,4}]];
tbz4spoint=Mod[tbz/.SubHW,79^(deg+1)];
If[Mod[tbz4spoint,79^deg]≠0,Print["pt4 warning"]];

LOGspoint=Mod[LastOneGood/.spoint,79^(deg+1)];
If[Mod[LOGspoint/.spoint,79^deg]≠0,Print["Warning LOG"]];

AppendTo[TheValues,{tbz3spoint,tbz4spoint,LOGspoint}];
];
slsa1a2a3=Solve[Table[0==Factor[(a1 TheValues[[2]][[k]]+a2  TheValues[[3]][[k]]+ a3 TheValues[[4]][[k]] - (a1+a2+a3-1)TheValues[[1]][[k]])/79^deg],{k,1,3}],Modulus->79][[1]];
{s2old,s3old,s4old}+=79^deg{a1,a2,a3}/.slsa1a2a3;

];
AppendTo[betters234,{s2old,s3old,s4old}];
];

We will now recognize these as algebraic numbers.  A bit of experimentation indicates that we should be looking at degree 12 polynomials.

acc=101;
smallpolys={};
For[n=1,n≤6,n++,
smallpolysn={};
For[i=1,i≤3,i++,
whatis=betters234[[n]][[i]];
thebasis=Append[Table[Mod[whatis^k,79^acc],{k,0,12}],79^acc];
smalllincomb=LatticeReduce[NullSpace[{thebasis}]][[1]];
smallpoly=Sum[smalllincomb[[k]] t^(k-1),{k,1,13}];
AppendTo[smallpolysn,smallpoly];
];
AppendTo[smallpolys,smallpolysn];
];
smallpolys 
Length[DeleteDuplicates[Table[smallpolys[[k]]/(smallpolys[[k]]/.t->0),{k,1,6}]]](* these are basically all the same *)
smallpolys=smallpolys[[1]];

We will now see which of the combinations of the roots of the above polynomials satisfy LastOneGood. We get 12 possibilities. For whatever reason the Magma method only picked up single roots of the polynomial for s2 in Z_79, of which there happen to be 6.

root2=RootReduce[Solve[smallpolys[[1]]==0]];
root3=RootReduce[Solve[smallpolys[[2]]==0]];
root4=RootReduce[Solve[smallpolys[[3]]==0]];
s234combos={};
For[i=1,i≤12,i++,For[j=1,j≤12,j++,
For[k=1,k≤12,k++,
If[0== Chop[LastOneGood/.N[{s2->(t/.root2[[i]]),s3->(t/.root3[[j]]),s4->(t/.root4[[k]])},500]],
AppendTo[s234combos,{t/.root2[[i]],t/.root3[[j]],t/.root4[[k]]}];
Print[Length[s234combos]];
];
]]];

s234combos
(* we will now compute the t1,t2,t3,t4,t5; approximate approach works best; then we get the quadrics *)

QuadricsGoodExactAll={};
For[num=1,num≤12,num++,
Print[num];
spoint=N[{s2-> s234combos[[num]] [[1]],s3-> s234combos[[num]] [[2]],s4-> s234combos[[num]] [[3]]},2000];
spoint=Join[spoint,{s5->(s5/.solss5/.spoint)}];
spoint=Join[spoint,{s1->(s1/.slss1/.spoint)}];
spoint=Join[spoint,{t1->  ( s1/.spoint),t2-> (s2/.spoint),t3-> s3 s5/s1/.spoint,t4-> s4 s5/s2/.spoint,t5-> s5/.spoint}];
tgood=Table[FiveVars[[k]]->RootApproximant[FiveVars[[k]]/.spoint,12],{k,1,5}];
QuadricsGoodExact={};

qg1=QuadricsGood[[1]]/.spoint;
qg1=qg1/Coefficient[qg1,v3^2];
AppendTo[QuadricsGoodExact,Sum[RootApproximant[Coefficient[qg1,Product[uvs[[k]]^(as[[k]]/.deg28[[l]]),{k,1,8}]],12] Product[uvs[[k]]^(as[[k]]/.deg28[[l]]),{k,1,8}],{l,1,Length[deg28]}]];
qg1=QuadricsGood[[2]]/.spoint;
qg1=qg1/Coefficient[qg1,v3 v4];
AppendTo[QuadricsGoodExact,Sum[RootApproximant[Coefficient[qg1,Product[uvs[[k]]^(as[[k]]/.deg28[[l]]),{k,1,8}]],12] Product[uvs[[k]]^(as[[k]]/.deg28[[l]]),{k,1,8}],{l,1,Length[deg28]}]];
qg1=QuadricsGood[[3]]/.spoint;
qg1=qg1/Coefficient[qg1,v4^2];
AppendTo[QuadricsGoodExact,Sum[RootApproximant[Coefficient[qg1,Product[uvs[[k]]^(as[[k]]/.deg28[[l]]),{k,1,8}]],12] Product[uvs[[k]]^(as[[k]]/.deg28[[l]]),{k,1,8}],{l,1,Length[deg28]}]];

qg1=QuadricsGood[[4]]/.spoint;
qg1=qg1/Coefficient[qg1,v3 v6];
AppendTo[QuadricsGoodExact,Sum[RootApproximant[Coefficient[qg1,Product[uvs[[k]]^(as[[k]]/.deg29[[l]]),{k,1,8}]],12] Product[uvs[[k]]^(as[[k]]/.deg29[[l]]),{k,1,8}],{l,1,Length[deg29]}]];
qg1=QuadricsGood[[5]]/.spoint;
qg1=qg1/Coefficient[qg1,v3 v5];
AppendTo[QuadricsGoodExact,Sum[RootApproximant[Coefficient[qg1,Product[uvs[[k]]^(as[[k]]/.deg29[[l]]),{k,1,8}]],12] Product[uvs[[k]]^(as[[k]]/.deg29[[l]]),{k,1,8}],{l,1,Length[deg29]}]];
qg1=QuadricsGood[[6]]/.spoint;
qg1=qg1/Coefficient[qg1,v4 v5];
AppendTo[QuadricsGoodExact,Sum[RootApproximant[Coefficient[qg1,Product[uvs[[k]]^(as[[k]]/.deg29[[l]]),{k,1,8}]],12] Product[uvs[[k]]^(as[[k]]/.deg29[[l]]),{k,1,8}],{l,1,Length[deg29]}]];

qg1=QuadricsGood[[7]]/.spoint;
qg1=qg1/Coefficient[qg1,v5^2];
AppendTo[QuadricsGoodExact,Sum[RootApproximant[Coefficient[qg1,Product[uvs[[k]]^(as[[k]]/.deg210[[l]]),{k,1,8}]],12] Product[uvs[[k]]^(as[[k]]/.deg210[[l]]),{k,1,8}],{l,1,Length[deg210]}]];
qg1=QuadricsGood[[8]]/.spoint;
qg1=qg1/Coefficient[qg1,v5 v6];
AppendTo[QuadricsGoodExact,Sum[RootApproximant[Coefficient[qg1,Product[uvs[[k]]^(as[[k]]/.deg210[[l]]),{k,1,8}]],12] Product[uvs[[k]]^(as[[k]]/.deg210[[l]]),{k,1,8}],{l,1,Length[deg210]}]];
qg1=QuadricsGood[[9]]/.spoint;
qg1=qg1/Coefficient[qg1,v6^2];
AppendTo[QuadricsGoodExact,Sum[RootApproximant[Coefficient[qg1,Product[uvs[[k]]^(as[[k]]/.deg210[[l]]),{k,1,8}]],12] Product[uvs[[k]]^(as[[k]]/.deg210[[l]]),{k,1,8}],{l,1,Length[deg210]}]];

AppendTo[QuadricsGoodExactAll,QuadricsGoodExact];
];

The setup suggests that it may be possible to make a variable change so that the quadrics are defined over a smaller field. The logical approach to it would be to try to have the singular points (or more generally the four points pt1,...,pt4 have nice coordinates.

fname ="Desktop/MumfordFPP2021/quadricsovernumberfieldofdeg12";
OpenWrite[fname];
Write[fname, QuadricsGoodExactAll];
Close[fname];

num=1;
spoint=N[{s2-> s234combos[[num]] [[1]],s3-> s234combos[[num]] [[2]],s4-> s234combos[[num]] [[3]]},5000];
spoint=Join[spoint,{s5->(s5/.solss5/.spoint)}];
spoint=Join[spoint,{s1->(s1/.slss1/.spoint)}];
spoint=Join[spoint,{t1->  ( s1/.spoint),t2-> (s2/.spoint),t3-> s3 s5/s1/.spoint,t4-> s4 s5/s2/.spoint,t5-> s5/.spoint}];

pt1sm=vs/.pt1/.spoint;
pt1sm=RootApproximant[pt1sm/pt1sm[[1]],12]
pt2sm=vs/.pt2/.spoint;
pt2sm=RootApproximant[pt2sm/pt2sm[[1]],12]
pt3sm=vs/.pt3/.spoint;
pt3sm=RootApproximant[pt3sm/pt3sm[[1]],12]
pt4sm=vs/.pt4/.spoint;
pt4sm=RootApproximant[pt4sm/pt4sm[[1]],12]
Quadrics=QuadricsGoodExactAll[[1]];

ws={w1,w2,w3,w4,w5,w6};
uws={u0,u1,w1,w2,w3,w4,w5,w6};VarChange={v1->w1,v2-> l1 w2+l2 u1 w1, v3-> l3 w3 + l4 w4 + l5 u0^2 w1, v4-> l6 w3 + l7 w4 + l8 u0^2 w1, v5-> l9 w5 + l10 w6 + l11 u0 w2 + l12 u0 u1 w1,v6-> l13 w5 + l14 w6 + l15 u0 w2 + l16 u0 u1 w1}/.{u0->1,u1->1};
prec=2000;
MatrixVarChange=N[Table[Coefficient[vs[[k]]/.VarChange,ws[[l]]],{k,1,6},{l,1,6}],prec];
(*
standardpoint1={1,0,0,0,0,0};
standardpoint2={1,1,1,1,1,1};
standardpoint3={1,*,1,0,1,0};
standardpoint3={1,*,*,*,0,0};
*)
sls1=NSolve[Table[0==( MatrixVarChange.N[pt1sm,prec])[[k]],{k,2,6}]];
MatrixVarChange1=(MatrixVarChange/.sls1[[1]]);
newpt=MatrixVarChange1.N[pt2sm,prec];
sls2=NSolve[{newpt[[1]]==newpt[[3]],newpt[[1]]==newpt[[4]],newpt[[1]]==newpt[[5]],newpt[[1]]==newpt[[6]],newpt[[1]]==newpt[[2]]}];
MatrixVarChange2=(MatrixVarChange1/.sls2[[1]]);
newpt=MatrixVarChange2.N[pt3sm,prec];
sls3=NSolve[{newpt[[1]]==newpt[[3]],0==newpt[[4]],newpt[[1]]==newpt[[5]],0==newpt[[6]]}];
MatrixVarChange3=(MatrixVarChange2/.sls3[[1]]);
newpt=MatrixVarChange3.N[pt4sm,prec];
sls4=NSolve[{newpt[[5]]==0,0==newpt[[6]]}];
MatrixVarChange4=(MatrixVarChange3/.sls4[[1]]);

RootApproximant[Chop[N[MatrixVarChange4.pt1sm,500]]]
RootApproximant[Chop[N[MatrixVarChange4.pt2sm,500]]]
RootApproximant[Chop[N[MatrixVarChange4.pt3sm,500]]]
RootApproximant[Chop[N[MatrixVarChange4.pt4sm,500]]]

It is easier to make a change with high accuracy and then identify the coefficients.

ws={w1,w2,w3,w4,w5,w6};
uws={u0,u1,w1,w2,w3,w4,w5,w6};
InvVC=Inverse[MatrixVarChange4];
VtoW={v1->w1,v2-> l1 w2+l2 u1 w1, v3-> l3 w3 + l4 w4 + l5 u0^2 w1, v4-> l6 w3 + l7 w4 + l8 u0^2 w1, v5-> l9 w5 + l10 w6 + l11 u0 w2 + l12 u0 u1 w1,v6-> l13 w5 + l14 w6 + l15 u0 w2 + l16 u0 u1 w1}/.{l1->InvVC[[2]][[2]],l2->InvVC[[2]][[1]],l3->InvVC[[3]][[3]],l4->InvVC[[3]][[4]],l5->InvVC[[3]][[1]],l6->InvVC[[4]][[3]],l7->InvVC[[4]][[4]],l8->InvVC[[4]][[1]],l9->InvVC[[5]][[5]],l10->InvVC[[5]][[6]],l11->InvVC[[5]][[2]],l12->InvVC[[5]][[1]],l13->InvVC[[6]][[5]],l14->InvVC[[6]][[6]],l15->InvVC[[6]][[2]],l16->InvVC[[6]][[1]]};

QuadricsW={}; (* let's compute new quadrics *)

qtry=(a1 Quadrics[[1]] + a2 Quadrics[[2]] + a3 Quadrics[[3]])/.VtoW;
sls=Solve[{1==Coefficient[qtry,w3^2],0==Coefficient[qtry,w3 w4],0==Coefficient[qtry,w4^2]}][[1]];
AppendTo[QuadricsW,
Sum[RootApproximant[Chop[Coefficient[qtry/.sls,Product[uws[[k]]^(as[[k]]/.deg28[[l]]),{k,1,8}]]]] Product[uws[[k]]^(as[[k]]/.deg28[[l]]),{k,1,8}],{l,1,Length[deg28]}]];
qtry=(a1 Quadrics[[1]] + a2 Quadrics[[2]] + a3 Quadrics[[3]])/.VtoW;
sls=Solve[{0==Coefficient[qtry,w3^2],1==Coefficient[qtry,w3 w4],0==Coefficient[qtry,w4^2]}][[1]];
AppendTo[QuadricsW,
Sum[RootApproximant[Chop[Coefficient[qtry/.sls,Product[uws[[k]]^(as[[k]]/.deg28[[l]]),{k,1,8}]]]] Product[uws[[k]]^(as[[k]]/.deg28[[l]]),{k,1,8}],{l,1,Length[deg28]}]];
qtry=(a1 Quadrics[[1]] + a2 Quadrics[[2]] + a3 Quadrics[[3]])/.VtoW;
sls=Solve[{0==Coefficient[qtry,w3^2],0==Coefficient[qtry,w3 w4],1==Coefficient[qtry,w4^2]}][[1]];
AppendTo[QuadricsW,
Sum[RootApproximant[Chop[Coefficient[qtry/.sls,Product[uws[[k]]^(as[[k]]/.deg28[[l]]),{k,1,8}]]]] Product[uws[[k]]^(as[[k]]/.deg28[[l]]),{k,1,8}],{l,1,Length[deg28]}]];

qtry=(a1 Quadrics[[4]] + a2 Quadrics[[5]] + a3 Quadrics[[6]])/.VtoW;
sls=Solve[{1==Coefficient[qtry,w3 w6],0==Coefficient[qtry,w3 w5],0==Coefficient[qtry,w4 w5]}][[1]];
AppendTo[QuadricsW,
Sum[RootApproximant[Chop[Coefficient[qtry/.sls,Product[uws[[k]]^(as[[k]]/.deg29[[l]]),{k,1,8}]]]] Product[uws[[k]]^(as[[k]]/.deg29[[l]]),{k,1,8}],{l,1,Length[deg29]}]];
qtry=(a1 Quadrics[[4]] + a2 Quadrics[[5]] + a3 Quadrics[[6]])/.VtoW;
sls=Solve[{0==Coefficient[qtry,w3 w6],1==Coefficient[qtry,w3 w5],0==Coefficient[qtry,w4 w5]}][[1]];
AppendTo[QuadricsW,
Sum[RootApproximant[Chop[Coefficient[qtry/.sls,Product[uws[[k]]^(as[[k]]/.deg29[[l]]),{k,1,8}]]]] Product[uws[[k]]^(as[[k]]/.deg29[[l]]),{k,1,8}],{l,1,Length[deg29]}]];
qtry=(a1 Quadrics[[4]] + a2 Quadrics[[5]] + a3 Quadrics[[6]])/.VtoW;
sls=Solve[{0==Coefficient[qtry,w3 w6],0==Coefficient[qtry,w3 w5],1==Coefficient[qtry,w4 w5]}][[1]];
AppendTo[QuadricsW,
Sum[RootApproximant[Chop[Coefficient[qtry/.sls,Product[uws[[k]]^(as[[k]]/.deg29[[l]]),{k,1,8}]]]] Product[uws[[k]]^(as[[k]]/.deg29[[l]]),{k,1,8}],{l,1,Length[deg29]}]];

qtry=(a1 Quadrics[[7]] + a2 Quadrics[[8]] + a3 Quadrics[[9]])/.VtoW;
sls=Solve[{1==Coefficient[qtry,w5^2],0==Coefficient[qtry,w5 w6],0==Coefficient[qtry,w6^2]}][[1]];
AppendTo[QuadricsW,
Sum[RootApproximant[Chop[Coefficient[qtry/.sls,Product[uws[[k]]^(as[[k]]/.deg210[[l]]),{k,1,8}]]]] Product[uws[[k]]^(as[[k]]/.deg210[[l]]),{k,1,8}],{l,1,Length[deg210]}]];
qtry=(a1 Quadrics[[7]] + a2 Quadrics[[8]] + a3 Quadrics[[9]])/.VtoW;
sls=Solve[{0==Coefficient[qtry,w5^2],1==Coefficient[qtry,w5 w6],0==Coefficient[qtry,w6^2]}][[1]];
AppendTo[QuadricsW,
Sum[RootApproximant[Chop[Coefficient[qtry/.sls,Product[uws[[k]]^(as[[k]]/.deg210[[l]]),{k,1,8}]]]] Product[uws[[k]]^(as[[k]]/.deg210[[l]]),{k,1,8}],{l,1,Length[deg210]}]];
qtry=(a1 Quadrics[[7]] + a2 Quadrics[[8]] + a3 Quadrics[[9]])/.VtoW;
sls=Solve[{0==Coefficient[qtry,w5^2],0==Coefficient[qtry,w5 w6],1==Coefficient[qtry,w6^2]}][[1]];
AppendTo[QuadricsW,
Sum[RootApproximant[Chop[Coefficient[qtry/.sls,Product[uws[[k]]^(as[[k]]/.deg210[[l]]),{k,1,8}]]]] Product[uws[[k]]^(as[[k]]/.deg210[[l]]),{k,1,8}],{l,1,Length[deg210]}]];

QuadricsW

fname ="Desktop/MumfordFPP2021/QuadricsW";
OpenWrite[fname];
Write[fname, QuadricsW];
Close[fname];