(Analysing the qsmooth CY h'surfaces.)

These are notes from Berkeley, Apr 2006. I messed around with several
dozen of the famous 7555 hypersurfaces, trying to write the curve
contribution in the form
 PP*Qorb(1/r(a,r-a),r)/(1-t^r) + PPP/[1,1,1,r].
It's not clear whether this is the right form, or whether we should
simply do the product PP*Qorb and write it as Poly/[1,1,r,r]. At the
end I write the suite of functions that partly automates the analysis.

More simply, P_C = Num/[1,1,r,r], where Num is symmetric polynomial of
deg 2r+2 with support [3..2r-1]. The expectation is that an open range
of polynomials will occur. That is, there are r-1 free coefficients.

different curves of 1/3 give:

0
-t^5 - 2*t^4 - t^3
-t^5 + t^4 - t^3
-t^5 - t^3
-t^4
+t^4
2*t^5 + t^4 + 2*t^3
3*t^5 + 3*t^3
3*t^5 + 2*t^4 + 3*t^3
3*t^5 + 5*t^4 + 3*t^3
4*t^5 + t^4 + 4*t^3
4*t^5 + 2*t^4 + 4*t^3
5*t^5 + 5*t^3
5*t^5 + 4*t^4 + 5*t^3
6*t^5 + 5*t^4 + 6*t^3
16*t^5 + 31*t^4 + 16*t^3

different curves of 1/4 give:

-t^6 + t^5 - t^4
-t^6 - t^5 - t^4
-t^7 - t^6 - t^4 - t^3
-t^7 - 2*t^6 - 3*t^5 - 2*t^4 - t^3
-4*t^7 - 7*t^6 - 9*t^5 - 7*t^4 - 4*t^3
t^7 + 2*t^6 + 2*t^4 + t^3
t^7 + 3*t^6 + 2*t^5 + 3*t^4 + t^3
2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3
2*t^7 + 3*t^6 + 3*t^5 + 3*t^4 + 2*t^3
2*t^7 + 4*t^6 + 4*t^4 + 2*t^3
2*t^7 + 4*t^6 + 2*t^5 + 4*t^4 + 2*t^3
2*t^7 + 3*t^6 + t^5 + 3*t^4 + 2*t^3
2*t^7 + 3*t^6 + 3*t^5 + 3*t^4 + 2*t^3
2*t^7 + 4*t^6 + 3*t^5 + 4*t^4 + 2*t^3
2*t^7 + 5*t^6 + 2*t^5 + 5*t^4 + 2*t^3
3*t^7 + 3*t^6 + 3*t^5 + 3*t^4 + 3*t^3
3*t^7 + 6*t^6 + 5*t^5 + 6*t^4 + 3*t^3
5*t^7 + 9*t^6 + 10*t^5 + 9*t^4 + 5*t^3
8*t^7 + 15*t^6 + 17*t^5 + 15*t^4 + 8*t^3

different curves of 1/5(1,4) give:

2*t^8 + 2*t^6 + 2*t^4
3*t^8 + 3*t^6 + 3*t^4
-t^9 - t^8 - 3*t^7 - 2*t^6 - 3*t^5 - t^4 - t^3
-t^9 - 2*t^8 - 3*t^7 - 4*t^6 - 3*t^5 - 2*t^4 - t^3
t^9 + t^8 + 3*t^7 + t^6 + 3*t^5 + t^4 + t^3
t^9 + 3*t^8 + 3*t^7 + 2*t^6 + 3*t^5 + 3*t^4 + t^3
t^9 + 3*t^8 + 3*t^7 + 4*t^6 + 3*t^5 + 3*t^4 + t^3
2*t^9 + 4*t^8 + 6*t^7 + 2*t^6 + 6*t^5 + 4*t^4 + 2*t^3

curves of 1/5(2,3) give:
2*t^9 + 4*t^8 + 6*t^7 + 5*t^6 + 6*t^5 + 4*t^4 + 2*t^3
-5*t^9 - 2*t^8 - 6*t^7 - 8*t^6 - 6*t^5 - 2*t^4 - 5*t^3

curve of 1/6(1,5)
10*t^11 + 18*t^10 + 30*t^9 + 39*t^8 + 49*t^7 + 39*t^6 + 30*t^5 + 18*t^4 + 10*t^3

curves of 1/7(1,6) give
-4*t^13 + t^12 - 5*t^11 - 4*t^9 - 4*t^8 - 4*t^7 - 5*t^5 + t^4 - 4*t^3

1/17(1,16) gives
(-2*t^33 - 4*t^32 - 5*t^31 - 3*t^30 - 5*t^29 - 9*t^28 - 9*t^27 - 6*t^26 -
 10*t^25 - 12*t^24 - 13*t^23 - 11*t^22 - 13*t^21 - 15*t^20 - 19*t^19 -
 14*t^18 - 19*t^17 - 15*t^16 - 13*t^15 - 11*t^14 - 13*t^13 - 12*t^12 -
 10*t^11 - 6*t^10 - 9*t^9 - 9*t^8 - 5*t^7 - 3*t^6 - 5*t^5 - 4*t^4 -
 2*t^3)

(-2*t^33 - 5*t^32 - 5*t^31 - 3*t^30 - 6*t^29 - 9*t^28 - 9*t^27 - 7*t^26 -
 10*t^25 - 12*t^24 - 14*t^23 - 11*t^22 - 13*t^21 - 16*t^20 - 19*t^19 -
 14*t^18 - 19*t^17 - 16*t^16 - 13*t^15 - 11*t^14 - 14*t^13 - 12*t^12 -
 10*t^11 - 7*t^10 - 9*t^9 - 9*t^8 - 6*t^7 - 3*t^6 - 5*t^5 - 5*t^4 -
 2*t^3)

(-6*t^33 - 12*t^32 - 13*t^31 - 11*t^30 - 17*t^29 - 24*t^28 - 26*t^27 - 22*t^26
 - 29*t^25 - 35*t^24 - 39*t^23 - 34*t^22 - 40*t^21 - 46*t^20 - 53*t^19 -
 45*t^18 - 53*t^17 - 46*t^16 - 40*t^15 - 34*t^14 - 39*t^13 - 35*t^12 -
 29*t^11 - 22*t^10 - 26*t^9 - 24*t^8-17*t^7-11*t^6 - 13*t^5 - 12*t^4 -
 6*t^3)

K := FieldOfFractions(R);

A := [15,1,2,2,3,7];
 curve term is 1*Qorb(2,[1,1],2)/(1-t^2)
A := [15,1,1,2,4,7];
 curve term is 0
A := [16,2,2,3,4,5]; doesn't work
A := [16,1,3,3,4,5];
 curve term is 1*Qorb(3,[1,2],3)/(1-t^3)
 +3*t^3/Denom([1,1,1,3])
A := [16,1,2,4,4,5];
 curve term is 0
A := [16,1,2,3,5,5];
 curve term is 1*Qorb(5,[2,3],5)/(1-t^5)
 +(t^3+t^5)/Denom([1,1,1,5]);
A := [16,1,1,4,5,5];
 curve term is Qorb(5,[1,4],5)/(1-t^5)+
 (2*t^3+2*t^4+2*t^5)/Denom([1,1,1,5])
A := [16,1,2,3,4,6];
 curve terms are 1*Qorb(2,[1,1],2)/(1-t^2)
 and t^3/Denom([1,1,1,3]);
A := [16,1,1,4,4,6];
 curve term is (2*t^3+2*t^4)/Denom([1,1,1,4]);

P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[7,2,2,3]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-
1*Qorb(2,[1,1],2)/(1-t^2)
;

P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[7,1,2,4],[4,1,1,2]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-
0*Qorb(2,[1,1],2)/(1-t^2)
;

P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[4,2,3,3],[4,2,3,3],[4,2,3,3],[4,2,3,3]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-
4*Qorb(2,[1,1],2)/(1-t^2)
-
1*Qorb(3,[1,2],3)/(1-t^3)
-
2*t^3/Denom([1,1,1,3])
;

P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[5,3,3,4]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-
1*Qorb(3,[1,2],3)/(1-t^3)
-
3*t^3/Denom([1,1,1,3])
;

P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[5,2,4,4],[4,1,1,2],[4,1,1,2],[4,1,1,2],[4,1,1,2]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-0
;

P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[3,2,2,2]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-
1*Qorb(5,[2,3],5)/(1-t^5)
-
(t^3+t^5)/Denom([1,1,1,5])
;

P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-
1*Qorb(5,[1,4],5)/(1-t^5)
-
(2*t^3+2*t^4+2*t^5)/Denom([1,1,1,5])
;

P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[6,1,2,3]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-
1*Qorb(2,[1,1],2)/(1-t^2)
-
t^3/Denom([1,1,1,3]);

P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[6,1,1,4]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-
(2*t^3+2*t^4)/Denom([1,1,1,4]);

A := [16,1,2,2,4,7];
 curve term is 4*Qorb(2,[1,1],2)/(1-t^2);
A := [16,1,2,2,3,8];
 curve term is 2*Qorb(2,[1,1],2)/(1-t^2);
A := [16,1,1,3,3,8];
 curve term is 1*Qorb(3,[1,2],3)/(1-t^3)+4*t^3/Denom([1,1,1,3]);
A := [16,1,1,2,4,8];
 curve term is 0!
A := [17,2,3,3,4,5];
 curve terms are 1*Qorb(3,[1,2],3)/(1-t^3)
 -2*t^3/Denom([1,1,1,3])
 +1*Qorb(2,[1,1],2)/(1-t^2)
A := [17,2,2,3,5,5];
 curve terms are 1*Qorb(5,[2,3],5)/(1-t^5)
 -(5*t^3-3*t^4+5*t^5)/Denom([1,1,1,5])
 +1*Qorb(2,[1,1],2)/(1-t^2)
A := [17,1,2,4,5,5];
 curve term is 1*Qorb(5,[1,4],5)/(1-t^5)-(t^3-2*t^4+t^5)/Denom([1,1,1,5])
A := [17,1,2,3,5,6];
 curve term is -t^3/Denom([1,1,1,3]) only!
A := [17,2,2,3,3,7];
 curve terms are 1*Qorb(3,[1,2],3)/(1-t^3) - 3*t^3/Denom([1,1,1,3])
 and 1*Qorb(2,[1,1],2)/(1-t^2);

A := [16,1,2,2,4,7];
P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[7,1,2,4]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
 -4*Qorb(2,[1,1],2)/(1-t^2);

A := [16,1,2,2,3,8];
P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[3,2,2,2]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
 -2*Qorb(2,[1,1],2)/(1-t^2);

A := [16,1,1,3,3,8];
P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
 -1*Qorb(3,[1,2],3)/(1-t^3)
 -4*t^3/Denom([1,1,1,3]);

A := [16,1,1,2,4,8];
P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[4,1,1,2],[4,1,1,2]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B];

A := [17,2,3,3,4,5];
P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[5,3,3,4],[4,2,3,3]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-
1*Qorb(3,[1,2],3)/(1-t^3)
+
2*t^3/Denom([1,1,1,3])
-
1*Qorb(2,[1,1],2)/(1-t^2)
;

A := [17,2,2,3,5,5];
P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[3,2,2,2]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-
1*Qorb(5,[2,3],5)/(1-t^5)
+
(5*t^3-3*t^4+5*t^5)/Denom([1,1,1,5])
-
1*Qorb(2,[1,1],2)/(1-t^2)
;

A := [17,1,2,3,5,6];
P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[6,1,2,3],[5,1,1,3]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
+
t^3/Denom([1,1,1,3])
;

A := [17,2,2,3,3,7];
P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[7,2,2,3]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-
1*Qorb(3,[1,2],3)/(1-t^3)
+
3*t^3/Denom([1,1,1,3])
-
1*Qorb(2,[1,1],2)/(1-t^2)
;

A := [17,1,3,4,4,5]; not qsmooth at 5 -- a triumph for my algorithm
A := [17,2,2,3,5,5];
 curve of 1/2 gives Q2
 curve of 1/5(2,3) gives Q523+(-5*t^5+3*t^4-5*t^3)/Denom([1,1,1,5]);
A := [17,1,2,4,5,5];
 curve terms is 1*Qorb(5,[1,4],5)/(1-t^5)
 -(t^5 - 2*t^4 + t^3)/Denom([1,1,1,5]);
A := [17,1,2,3,5,6];
 curve term is -1*t^3/Denom([1,1,1,3]);
A := [17,2,2,3,3,7];
 curve terms are 1*Qorb(3,[1,2],3)/(1-t^3)
 -3*t^3/Denom([1,1,1,3])
 +1*Qorb(2,[1,1],2)/(1-t^2);
A := [17,1,2,3,4,7];
 B := [[7,1,2,4],[4,2,3,3],[3,1,1,1]];
 curve terms is 1*Qorb(2,[1,1],2)/(1-t^2);
A := [17,1,2,3,3,8];
 B := [[8,2,3,3]];
 curve terms are 1*Qorb(3,[1,2],3)/(1-t^3)
 -t^3/Denom([1,1,1,3])
 -2*Qorb(2,[1,1],2)/(1-t^2);
A := [17,1,1,2,5,8];
 B := [[5,1,1,3],[8,1,2,5]];
 curve term is 1*Qorb(2,[1,1],2)/(1-t^2)
A := [18,2,3,3,5,5];
 B := [[3,2,2,2],[3,2,2,2],[3,2,2,2],[3,2,2,2],[3,2,2,2],[3,2,2,2]];
 curve term is 1*Qorb(5,[2,3],5)/(1-t^5)
 +(4*t^5 - 3*t^4 + 4*t^3)/Denom([1,1,1,5]);
A := [18,2,3,3,4,6];
 B := [[4,2,3,3]];
 curve terms are 3*Qorb(3,[1,2],3)/(1-t^3)
 and 2*Qorb(2,[1,1],2)/(1-t^2);
A := [18,2,2,3,5,6];
 curve terms are 3*Qorb(2,[1,1],2)/(1-t^2)
 and -3*t^3/Denom([1,1,1,3]);
A := [18,1,3,3,5,6];
 curve term is 3*Qorb(3,[1,2],3)/(1-t^3)
 +3*t^3/Denom([1,1,1,3]);
A := [18,1,2,3,6,6];
 curve terms all 0!

A := [17,1,2,4,5,5];
P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[4,1,1,2]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-
1*Qorb(5,[1,4],5)/(1-t^5)
+
(t^5 - 2*t^4 + t^3)/Denom([1,1,1,5]);

A := [17,1,2,3,5,6];
P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[6,1,2,3],[5,1,1,3]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
+
1*t^3/Denom([1,1,1,3]);

A := [17,2,2,3,3,7];
P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[7,2,2,3]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-
1*Qorb(3,[1,2],3)/(1-t^3)
+
3*t^3/Denom([1,1,1,3])
-
1*Qorb(2,[1,1],2)/(1-t^2);

A := [18,2,2,3,5,6];
P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[5,1,2,2]]; // this is false
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-
3*Qorb(2,[1,1],2)/(1-t^2)
+
3*t^3/Denom([1,1,1,3]);
-
3*Qorb(3,[1,2],3)/(1-t^3)
+
3*t^3/Denom([1,1,1,3]);
-
3*Qorb(2,[1,1],2)/(1-t^2)
;

A := [18,1,2,3,6,6];
P := (1-t^A[1])/Denom(A[2..6]);
A := [18,1,3,3,5,6];
P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[5,1,1,3]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-
3*Qorb(3,[1,2],3)/(1-t^3)
-
3*t^3/Denom([1,1,1,3]);
-
0*Qorb(2,[1,1],2)/(1-t^2)
;

A := [18,1,2,3,6,6];
P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[6,1,2,3],[6,1,2,3],[6,1,2,3]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-
0*Qorb(3,[1,2],3)/(1-t^3)
-
0*Qorb(2,[1,1],2)/(1-t^2)
;

A := [18,1,1,2,6,8];
 curve term is 0!
A := [18,2,2,2,3,9];
 curve terms are 9*Qorb(2,[1,1],2)/(1-t^2)
 and -2*t^3/Denom([1,1,1,3]);
A := [18,1,2,3,3,9];
 curve term is 2*Qorb(3,[1,2],3)/(1-t^3)
 +t^3/Denom([1,1,1,3]);
A := [18,1,2,2,4,9];
 curve term 4*Qorb(2,[1,1],2)/(1-t^2)
A := [19,2,3,4,5,5];
 curve term is +1*Qorb(5,[2,3],5)/(1-t^5)
 -(2*t^3+2*t^5)/Denom([1,1,1,5]);
A := [19,1,3,4,5,6];
 curve terms are 1*Qorb(3,[1,2],3)/(1-t^3)
 +2*t^3/Denom([1,1,1,3]);
A := [19,1,2,4,5,7];
 curve term 0!

A := [18,1,1,2,6,8];
P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[8,1,1,6]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B];

A := [18,2,2,2,3,9];
P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-
9*Qorb(2,[1,1],2)/(1-t^2)
+
2*t^3/Denom([1,1,1,3]);

A := [18,1,2,3,3,9];
P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-
2*Qorb(3,[1,2],3)/(1-t^3)
-
t^3/Denom([1,1,1,3]);

A := [18,1,2,2,4,9];
P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[4,1,1,2]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-
4*Qorb(2,[1,1],2)/(1-t^2)

A := [19,2,3,4,5,5];

A := [19,2,3,4,5,5];
P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[3,2,2,2],[4,1,1,2]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-
1*Qorb(5,[2,3],5)/(1-t^5)
+
(2*t^3+2*t^5)/Denom([1,1,1,5]);

A := [19,1,3,4,5,6];
P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[6,3,4,5],[5,1,1,3],[4,1,1,2]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-
1*Qorb(3,[1,2],3)/(1-t^3)
-
2*t^3/Denom([1,1,1,3]);

A := [19,1,2,4,5,7];
P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[7,1,2,4],[5,1,2,2],[4,1,1,2]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-
0*Qorb(3,[1,2],3)/(1-t^3)
-
0*Qorb(2,[1,1],2)/(1-t^2)
;

A := [24,1,2,4,5,12];
 curve terms 0!
A := [24,1,2,3,6,12];
 curve terms 0!
A := [24,1,1,4,6,12];
 curve term is -2*Qorb(2,[1,1],2)/(1-t^2) (note the minus!)
A := [24,1,1,2,8,12];
 curve term is 0!
A := [25,3,4,5,6,7];
 curve terms are 1*Qorb(3,[1,2],3)/(1-t^3)
 +2*t^3/Denom([1,1,1,3])
 and 1*Qorb(2,[1,1],2)/(1-t^2);
A := [25,2,4,5,7,7];
 curve terms are 1*Qorb(7,[2,5],7)/(1-t^7)
 +(-3*t^7 + 4*t^6 - 6*t^5 + 4*t^4 - 3*t^3)/Denom([1,1,1,7])
 and 1*Qorb(2,[1,1],2)/(1-t^2);
A := [25,1,4,6,7,7];
 curve term is 1*Qorb(7,[1,6],7)/(1-t^7)
 +(2*t^7 + 2*t^6 - t^5 + 2*t^4 + 2*t^3)/Denom([1,1,1,7]);
A := [25,1,5,5,6,8];
 curve term 3*Qorb(2,[1,1],2)/(1-t^2)

A := [24,1,2,4,5,12];
P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[5,1,2,2],[4,1,1,2],[4,1,1,2]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B];

K := FieldOfFractions(R);
A := [24,1,2,3,6,12];
P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B:=[[6,1,2,3],[6,1,2,3]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B];

A := [24,1,1,4,6,12];
P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[6,1,1,4],[6,1,1,4],[4,1,1,2],[4,1,1,2]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
+
2*Qorb(2,[1,1],2)/(1-t^2)

A := [24,1,1,2,8,12];
P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[4,1,1,2]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B];

A := [25,3,4,5,6,7];
P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[7,3,5,6],[6,3,4,5],[4,2,3,3]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-
1*Qorb(3,[1,2],3)/(1-t^3)
-
2*t^3/Denom([1,1,1,3])
-
1*Qorb(2,[1,1],2)/(1-t^2);

A := [25,2,4,5,7,7];
P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[4,2,3,3]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-
1*Qorb(7,[2,5],7)/(1-t^7)
-
(-3*t^7 + 4*t^6 - 6*t^5 + 4*t^4 - 3*t^3)/Denom([1,1,1,7])
-
1*Qorb(2,[1,1],2)/(1-t^2);

A := [25,1,4,6,7,7];
P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[6,4,7,7],[4,2,3,3]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-
1*Qorb(7,[1,6],7)/(1-t^7)
-
(2*t^7 + 2*t^6 - t^5 + 2*t^4 + 2*t^3)/Denom([1,1,1,7]);

A := [25,1,5,5,6,8];
P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[8,5,5,6],[6,2,5,5],[5,1,1,3],[5,1,1,3],[5,1,1,3],[5,1,1,3],[5,1,1,3]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-
3*Qorb(2,[1,1],2)/(1-t^2)
-
0*Qorb(3,[1,2],3)/(1-t^3)
-
0*Qorb(2,[1,1],2)/(1-t^2)
;

A := [39,1,4,5,10,19];
 B := [[19,4,5,10],[10,1,4,5],[4,1,1,2]];
 curve of 1/5 gives term (1/t+t)*Qorb(5,[1,4],5)/(1-t^5)
 -(t^3+t^5)/Denom([1,1,1,5])
 curve of 1/2 gives 0

A := [39,1,3,6,10,19];
 B := [[19,3,6,10],[10,1,3,6],[6,1,1,4]];
 curve of 1/3 gives term 6*t^3/Denom([1,1,1,3])

A := [39,1,2,4,13,19]; curve of 1/2 gives term 0
A := [40,4,8,9,9,10];
 B := [5 times [4,1,1,2]];
 curve of 1/2 gives term -2*Qorb(2,[1,1],2)/(1-t^2)
 curve of 1/9 gives term 1*Qorb(9,[1,8],9)/(1-t^9)
 (-2*t^9 - 2*t^8 - 2*t^7 + 2*t^6 - 2*t^5 - 2*t^4 - 2*t^3)/Denom([1,1,1,9])

A := [40,5,7,8,10,10];
 B := [[10,5,7,8],[10,5,7,8],[10,5,7,8],[10,5,7,8],[7,1,3,3]];
 curve of 1/5(2,3) gives term 4*(1/t+1+t)*Qorb(5,[2,3],5)/(1-t^5)
 -2*(t^3+t^5)/Denom([1,1,1,5])
 curve of 1/2 gives -3*Qorb(2,[1,1],2)/(1-t^2)

A := [40,3,8,9,10,10];
P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
 := [[10,3,8,9],[10,3,8,9],[10,3,8,9],[10,3,8,9],[9,1,1,7]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-
4*(1/t+1+t)*Qorb(5,[2,3],5)/(1-t^5)
-
-2*(t^3+t^5)/Denom([1,1,1,5])

A := [40,5,7,8,10,10];
P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[10,5,7,8],[10,5,7,8],[10,5,7,8],[10,5,7,8],[7,1,3,3]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-
4*(1/t+1+t)*Qorb(5,[2,3],5)/(1-t^5)
-
-2*(t^3+t^5)/Denom([1,1,1,5])
-
-3*Qorb(2,[1,1],2)/(1-t^2)

A := [40,4,8,9,9,10];
P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[4,1,1,2],[4,1,1,2],[4,1,1,2],[4,1,1,2],[4,1,1,2]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-
-2*Qorb(2,[1,1],2)/(1-t^2)
-
1*Qorb(9,[1,8],9)/(1-t^9)
-
(-2*t^9 - 2*t^8 - 2*t^7 + 2*t^6 - 2*t^5 - 2*t^4 - 2*t^3)/Denom([1,1,1,9])
;

A := [39,1,2,4,13,19];
P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[19,2,4,13],[4,1,1,2]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]

A := [39,1,3,3,13,19];
P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[19,3,3,13],[3,1,1,1],[3,1,1,1],[3,1,1,1],[3,1,1,1],[3,1,1,1],
 [3,1,1,1],[3,1,1,1],[3,1,1,1],[3,1,1,1],[3,1,1,1],[3,1,1,1],[3,1,1,1],[3,1,1,1]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]

A := [39,1,3,6,10,19];
P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[19,3,6,10],[10,1,3,6],[6,1,1,4]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-
0*Qorb(3,[1,2],3)/(1-t^3)
-
6*t^3/Denom([1,1,1,3])
;

A := [39,1,4,5,10,19];
P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[19,4,5,10],[10,1,4,5],[4,1,1,2]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-
-1*Qorb(2,[1,1],2)/(1-t^2)
-
(1/t+t)*Qorb(5,[1,4],5)/(1-t^5)
+
(t^3+t^5)/Denom([1,1,1,5])
;

A := [1,2,4,6,7,20];
 curve of 1/2 gives Qorb(2,[1,1],2)/(1-t^2);
A := [1,2,4,6,11,24];
 curve of 1/2 gives 2*Qorb(2,[1,1],2)/(1-t^2);
A := [1,2,4,6,13,26];
 curve of 1/2 gives Qorb(2,[1,1],2)/(1-t^2);
A := [1,2,4,7,7,21];
 curve of 1/2 gives Qorb(2,[1,1],2)/(1-t^2);
A := [1,2,4,7,10,24];
 curve of 1/2 gives Qorb(2,[1,1],2)/(1-t^2);
A := [1,2,4,7,14,28];
 curve of 1/2 gives Qorb(2,[1,1],2)/(1-t^2);
A := [1,2,4,8,13,28];
 curve of 1/2 gives Qorb(2,[1,1],2)/(1-t^2);
A := [1,2,4,9,12,28];
 curve of 1/3 gives -1*Qorb(3,[1,2],3)/(1-t^3)
 -t^3/Denom([1,1,1,3]);
 curve of 1/2 gives 0
A := [1,2,4,9,15,31];
 curve of 1/3 gives -2*Qorb(3,[1,2],3)/(1-t^3)
 -t^3/Denom([1,1,1,3]);
 curve of 1/2 gives 0
A := [1,2,4,10,13,30];
 curve of 1/2 gives Qorb(2,[1,1],2)/(1-t^2);
A := [1,2,4,10,17,34];
 curve of 1/2 gives Qorb(2,[1,1],2)/(1-t^2);
A := [1,2,4,11,17,35];
 curve of 1/2 gives 0
A := [1,2,4,12,17,36];
 curve of 1/2 gives 0
A := [1,2,4,13,19,39];
 curve of 1/2 gives 0
A := [1,2,4,13,20,40];
 curve of 1/2 gives 0
A := [1,2,4,14,21,42];
 curve of 1/2 gives Qorb(2,[1,1],2)/(1-t^2);

A := [1,2,4,14,21,42];
P := (1-t^A[6])/Denom(A[1..5]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[7,1,2,4],[4,1,1,2]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-1*Qorb(2,[1,1],2)/(1-t^2)
+(t^4-t^5+t^6)/Denom([1,1,1,7]);

A := [1,2,4,13,20,40];
P := (1-t^A[6])/Denom(A[1..5]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[13,2,4,7],[4,1,1,2],[4,1,1,2]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-0*Qorb(2,[1,1],2)/(1-t^2);

A := [1,2,4,13,19,39];
P := (1-t^A[6])/Denom(A[1..5]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[19,2,4,13],[4,1,1,2]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-0*Qorb(2,[1,1],2)/(1-t^2);

A := [1,2,4,12,17,36];
P := (1-t^A[6])/Denom(A[1..5]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[17,1,4,12],[4,1,1,2],[4,1,1,2],[4,1,1,2]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-0*Qorb(2,[1,1],2)/(1-t^2);

A := [1,2,4,11,17,35];
P := (1-t^A[6])/Denom(A[1..5]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[17,2,4,11],[11,1,4,6],[4,1,1,2]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-0*Qorb(2,[1,1],2)/(1-t^2);

A := [1,2,4,10,17,34];
P := (1-t^A[6])/Denom(A[1..5]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[10,1,2,7],[4,1,1,2]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-1*Qorb(2,[1,1],2)/(1-t^2);

A := [1,2,4,10,13,30];
P := (1-t^A[6])/Denom(A[1..5]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[13,1,2,10],[4,1,1,2]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-1*Qorb(2,[1,1],2)/(1-t^2);

A := [1,2,4,9,15,31];
P := (1-t^A[6])/Denom(A[1..5]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[15,2,4,9],[9,1,2,6],[4,1,1,2]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-0*Qorb(2,[1,1],2)/(1-t^2)
+2*Qorb(3,[1,2],3)/(1-t^3)
+t^3/Denom([1,1,1,3]);

A := [1,2,4,9,12,28];
P := (1-t^A[6])/Denom(A[1..5]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[12,1,2,9],[9,2,3,4],[4,1,1,2],[4,1,1,2]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-0*Qorb(2,[1,1],2)/(1-t^2)
+1*Qorb(3,[1,2],3)/(1-t^3)
+t^3/Denom([1,1,1,3]);

A := [1,2,4,6,7,20];
A := [1,2,4,8,13,28];
P := (1-t^A[6])/Denom(A[1..5]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[13,1,4,8],[8,1,2,5],[4,1,1,2],[4,1,1,2],[4,1,1,2]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-1*Qorb(2,[1,1],2)/(1-t^2);

A := [1,2,4,6,7,20];
A := [1,2,4,7,14,28];
P := (1-t^A[6])/Denom(A[1..5]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[7,1,2,4],[7,1,2,4]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-1*Qorb(2,[1,1],2)/(1-t^2);

A := [1,2,4,6,7,20];
A := [1,2,4,7,10,24];
P := (1-t^A[6])/Denom(A[1..5]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[10,1,2,7],[7,1,2,4]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-1*Qorb(2,[1,1],2)/(1-t^2);

A := [1,2,4,6,7,20];

A := [1,2,4,7,7,21];
P := (1-t^A[6])/Denom(A[1..5]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[7,1,2,4],[7,1,2,4],[7,1,2,4],[4,2,3,3]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-1*Qorb(2,[1,1],2)/(1-t^2);

A := [1,2,4,6,7,20];
A := [1,2,4,6,13,26];
P := (1-t^A[6])/Denom(A[1..5]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[6,1,1,4],[4,1,1,2]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-1*Qorb(2,[1,1],2)/(1-t^2);

A := [1,2,4,6,7,20];
A := [1,2,4,6,11,24];
P := (1-t^A[6])/Denom(A[1..5]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[11,1,4,6]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-2*Qorb(2,[1,1],2)/(1-t^2);

A := [1,2,4,6,7,20];
P := (1-t^A[6])/Denom(A[1..5]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[7,1,2,4],[6,1,1,4]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-Qorb(2,[1,1],2)/(1-t^2);

A := [1,16,50,133,200,400];
 B := [[133,16,50,67],[50,1,16,33],[50,1,16,33],[8,1,2,5]];
 curve of 1/2 gives term Qorb(2,[1,1],2)/(1-t^2);

A := [1,16,51,60,128,256];
 B := [[60,1,8,51],[16,1,3,12],[16,1,3,12],[51,9,16,26]];
 curve of 1/4 gives -Qorb(4,[1,3],4)/(1-t^4);
 curve of 1/3 gives -Qorb(3,[1,2],3)/(1-t^3)+2*t^3/Denom([1,1,1,3])

A := [1,16,51,68,120,256];
 This one is pretty amazing.
 B := [[120,1,51,68],[51,16,17,18],[68,1,16,51],[8,1,3,4]];
 curve of 1/3 gives -t^3/Denom([1,1,1,3])
 curve of 1/4 give -(5+9*t+5*t^2)*(1+t^2)/t^2*Qorb(4,[1,3],4)/(1-t^4)
 (still slightly anomalous)
 curve of 1/17(1,16) gives regular term
 +(-3*t^15-2*t^14-t^13-2*t^11-2*t^10-3*t^9
 +2*t^8-3*t^7-2*t^6-2*t^5-t^3-2*t^2-3*t)/t^8*Qorb(17,[1,16],17)/(1-t^17)
 -(-2*t^17-3*t^16+2*t^14-3*t^13-3*t^12
 +2*t^10-3*t^8-3*t^7+2*t^6-3*t^4-2*t^3)/Denom([1,1,1,17]);

A := [1,16,51,68,136,272];
 B := [[51,1,16,34],[8,1,3,4],[68,1,16,51],[68,1,16,51]];
 curve of 1/4 gives term
 -(-t^4+t^5-t^6)/t^5*Qorb(4,[1,3],4)/(1-t^4)+8*(t^3+t^4)/Denom([1,1,1,4]);
 curve of 1/17(1,16) gives
 -(-5*t^14-4*t^13-2*t^12+t^11-4*t^10-4*t^9-5*t^8+3*t^7-5*t^6
 -4*t^5-4*t^4+t^3-2*t^2-4*t-5)/t^7*Qorb(17,[1,16],17)/(1-t^17)
 +t^3*(-6*t^14-6*t^13-t^12+2*t^11-6*t^10-7*t^9-2*t^8+4*t^7
 -2*t^6-7*t^5-6*t^4+2*t^3-t^2-6*t-6)/Denom([1,1,1,17])

A := [1,16,51,136,204,408];
 B := [[16,1,3,12],[8,1,3,4],[51,1,16,34],[51,1,16,34],[68,1,16,51]];
 curve of 1/4 gives term
 -(t^4-t^5+t^6)/t^5*Qorb(4,[1,3],4)/(1-t^4)-4*(t^3+t^4)/Denom([1,1,1,4]);
 ? or minus that
 curve of 1/17 gives
 +(-4*t^14-2*t^13-t^12-t^11-2*t^10-2*t^9-4*t^8+t^7-4*t^6
 -2*t^5-2*t^4-t^3-t^2-2*t-4)/t^7*Qorb(17,[1,16],17)/(1-t^17)
 -(-2*t^14-2*t^13-t^12+2*t^11-2*t^10-4*t^9+3*t^7-4*t^5
 -2*t^4+2*t^3-t^2-2*t-2)*t^3/Denom([1,1,1,17])

A := [1,16,52,61,130,260];
 curve of 1/2 gives -Q2
A := [1,16,52,69,86,224];
 curve of 1/2 gives -Q2
A := [1,16,52,69,122,260];
 curve of 1/2 gives -Q2
A := [1,16,52,69,138,276];
 curve of 1/2 gives 0
A := [1,16,67,84,168,336];
 curve of 1/4 gives Q4 + (1/t+t)*Q4
A := [1,16,68,85,102,272];
 curve of 1/2 gives -2*Q2
 curve of 1/17(1,16) gives -Z1/t^7*Q17 +Z2/Denom([1,1,1,17])
 where Z1 is -2*t^14+t^13-3*t^12+3*t^11-5*t^10+5*t^9-8*t^8+6*t^7
 -8*t^6+5*t^5-5*t^4+3*t^3-3*t^2+t-2
 and Z2 is 2*t^16-2*t^15+4*t^14-4*t^13+5*t^12-5*t^11+7*t^10
 -5*t^9+5*t^8-4*t^7+4*t^6-2*t^5+2*t^4

A := [1,16,68,85,170,340];
 curve of 1/2 gives -1*Q2
 curve of 1/17(1,16) gives -Z1/t^7*Q17 +Z2/Denom([1,1,1,17])
 where Z1 is -t^14-t^13-2*t^12+2*t^11-3*t^10+3*t^9-6*t^8+3*t^7
 -6*t^6+3*t^5-3*t^4+2*t^3-2*t^2-t-1
 and Z2 is t^17+2*t^16+3*t^14-3*t^13+4*t^12-2*t^11+5*t^10
 -2*t^9+4*t^8-3*t^7 + 3*t^6 + 2*t^4 + t^3

A := [1,16,70,88,105,280];
 curve of 1/2 gives 0
A := [1,16,84,117,134,352];
 curve of 1/2 gives -Q2
 curve of 1/3 gives -Q3+t^3/Denom([1,1,1,3])
A := [1,16,86,120,137,360];
 curve of 1/2 gives -Q2
A := [1,16,100,133,150,400];
 curve of 1/2 gives 0
A := [1,16,102,136,153,408];
 curve of 1/2 gives -Q2
 curve of 1/17(1,16) gives -Z1/t^7*Q17 +Z2/Denom([1,1,1,17])
 where Z1 is -3*t^14+t^13-3*t^12+t^11-4*t^10+3*t^9-6*t^8+3*t^7
 -6*t^6+3*t^5-4*t^4+t^3-3*t^2+t-3
 and Z2 is t^17+t^16-t^15+2*t^14-t^13+2*t^12-2*t^11+5*t^10
 -2*t^9+2*t^8-t^7+2*t^6-t^5+t^4+t^3
A := [1,17,19,57,77,171]; no curve
A := [1,17,22,31,70,141];
 curve of 1/2 gives -3*Q2
A := [1,16,52,61,130,260];
 B:=[[61,1,8,52],[26,1,9,16],[16,1,2,13],[4,1,1,2]];
 line of half give term -Qorb(2,[1,1],2)/(1-t^2)

A := [1,16,51,136,204,408];
B := [[16,1,3,12],[8,1,3,4],[51,1,16,34],[51,1,16,34],[68,1,16,51]];
P := (1-t^A[6])/Denom(A[1..5]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
+(-4*t^14-2*t^13-t^12-t^11-2*t^10-2*t^9-4*t^8+t^7-4*t^6
 -2*t^5-2*t^4-t^3-t^2-2*t-4)/t^7*Qorb(17,[1,16],17)/(1-t^17)
-(-2*t^14-2*t^13-t^12+2*t^11-2*t^10-4*t^9+3*t^7-4*t^5
 -2*t^4+2*t^3-t^2-2*t-2)*t^3/Denom([1,1,1,17])
-(t^4-t^5+t^6)/t^5*Qorb(4,[1,3],4)/(1-t^4)-4*(t^3+t^4)/Denom([1,1,1,4]);

A := [1,16,51,68,136,272];
P := (1-t^A[6])/Denom(A[1..5]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
B := [[51,1,16,34],[8,1,3,4],[68,1,16,51],[68,1,16,51]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
+(-5*t^14-4*t^13-2*t^12+t^11-4*t^10-4*t^9-5*t^8+3*t^7-5*t^6
 -4*t^5-4*t^4+t^3-2*t^2-4*t-5)/t^7*Qorb(17,[1,16],17)/(1-t^17)
-t^3*(-6*t^14-6*t^13-t^12+2*t^11-6*t^10-7*t^9-2*t^8+4*t^7
-2*t^6-7*t^5-6*t^4+2*t^3-t^2-6*t-6)/Denom([1,1,1,17])
+(-t^4+t^5-t^6)/t^5*Qorb(4,[1,3],4)/(1-t^4)
-8*(t^3+t^4)/Denom([1,1,1,4]);

A := [1,16,51,68,120,256];
P := (1-t^A[6])/Denom(A[1..5]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[120,1,51,68],[51,16,17,18],[68,1,16,51],[8,1,3,4]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
 +t^3/Denom([1,1,1,3])
 +(5+9*t+5*t^2)*(1+t^2)/t^2*Qorb(4,[1,3],4)/(1-t^4)
 +(-3*t^15-2*t^14-t^13-2*t^11-2*t^10-3*t^9
+2*t^8-3*t^7-2*t^6-2*t^5-t^3-2*t^2-3*t)/t^8*Qorb(17,[1,16],17)/(1-t^17)
-(-2*t^17-3*t^16+2*t^14-3*t^13-3*t^12
 +2*t^10-3*t^8-3*t^7+2*t^6-3*t^4-2*t^3)/Denom([1,1,1,17]);

A := [1,16,51,60,128,256];
P := (1-t^A[6])/Denom(A[1..5]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[60,1,8,51],[16,1,3,12],[16,1,3,12],[51,9,16,26]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
 +Qorb(3,[1,2],3)/(1-t^3)-2*t^3/Denom([1,1,1,3])
 +Qorb(4,[1,3],4)/(1-t^4);

A := [1,16,50,133,200,400];
P := (1-t^A[6])/Denom(A[1..5]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[133,16,50,67],[50,1,16,33],[50,1,16,33],[8,1,2,5]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
 -Qorb(2,[1,1],2)/(1-t^2);

A := [1,16,43,120,180,360];
P := (1-t^A[6])/Denom(A[1..5]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[60,1,16,43],[43,1,8,34],[16,1,4,11],[8,1,3,4]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
 +(2+3*t+2*t^2)*(1+t^2)/t^2*Qorb(4,[1,3],4)/(1-t^4);

A := [1,16,43,52,60,172];
P := (1-t^A[6])/Denom(A[1..5]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
B := [[60,1,16,43],[52,1,8,43],[16,1,4,11]];
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
+(2+3*t+4*t^2+3*t^3+2*t^4)/t^2*Qorb(4,[1,3],4)/(1-t^4);
// or
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
+(2+3*t+2*t^2)*(1+t^2)/t^2*Qorb(4,[1,3],4)/(1-t^4);

A := [39,1,3,4,12,19];
 B := [[19,3,4,12],[12,1,4,7],[3,1,1,1]];
 curve of 1/4 gives term -(1/t+t)*Qorb(4,[1,3],4)/(1-t^4)
 -(t^3+t^4)/Denom([1,1,1,4]);
 curve of 1/3 gives 2*t^3/Denom([1,1,1,3])

A := [39,1,3,6,10,19];

A := [360,1,16,43,120,180];
 B := [[60,1,16,43],[43,1,8,34],[16,1,4,11],[8,1,3,4]];
 the curve of 1/4 give term (1/t+t)*Qorb(4,[1,3],4)/(1-t^4)
 +2*(t^3+t^4)/Denom([1,1,1,4]);

A := [224,1,16,43,52,112];
 B := [[52,1,8,43],[43,1,16,26],[16,1,4,11],[16,1,4,11]];
 the curve of 1/4 give term (1/t+t)*Qorb(4,[1,3],4)/(1-t^4)
 +2*(t^3+t^4)/Denom([1,1,1,4]);

A := [172,1,16,43,52,60];
 B := [[60,1,16,43],[52,1,8,43],[16,1,4,11]];
 the curve of 1/4 give term (1/t+t)*Qorb(4,[1,3],4)/(1-t^4)
 +2*(t^3+t^4)/Denom([1,1,1,4]);

A := [25,4,4,5,5,7];
 B := [[7,4,5,5],[5,2,4,4],[5,2,4,4],[5,2,4,4],[5,2,4,4],[5,2,4,4]];
 curve term is 1*Qorb(4,[1,3],4)/(1-t^4)
 +(3*t^3+3*t^4)/Denom([1,1,1,4]);

A := [19,3,3,4,4,5];
 B := [[5,3,3,4]];
 curve of 1/3 gives 1*Qorb(3,[1,2],3)/(1-t^3)
 +3*t^3/Denom([1,1,1,3])
 and curve of 1/4 gives 1*Qorb(4,[1,3],4)/(1-t^4)
 -(2*t^3+2*t^4)/Denom([1,1,1,4])

A := [17,1,1,3,4,8];
 B := [[3,1,1,1],[8,1,3,4]];
 line of 1/4 gives (1/t+1+t)*Qorb(4,[1,3],4)/(1-t^4)
 +(2*t^3 + 2*t^4)/Denom([1,1,1,4]);

A := [18,1,3,4,4,6];
 B := [[3,1,1,1],[3,1,1,1],[3,1,1,1]];
 curve term is (-1/t+1-t)*Qorb(4,[1,3],4)/(1-t^4);

A := [18,2,3,4,4,5];
 B := [[5,2,4,4]];
 line of 1/4 and curve of 1/2
 line gives 1*Qorb(4,[1,3],4)/(1-t^4)
 curve gives 2*Qorb(2,[1,1],2)/(1-t^2)

A := [16,2,3,3,4,4];
 B := [[4,2,3,3],[4,2,3,3],[4,2,3,3],[4,2,3,3]];
 curve terms are 4*Qorb(2,[1,1],2)/(1-t^2)
 + 1*Qorb(3,[1,2],3)/(1-t^3) + 2*t^3/Denom([1,1,1,3])

A := [16,1,3,4,4,4];
 B := [[3,1,1,1]];
 curve term is 4*Qorb(4,[1,3],4)/(1-t^4)
 +(2*t^3+2*t^4)/Denom([1,1,1,4])

A := [17,1,1,3,4,8];
 B := [[3,1,1,1],[8,1,3,4]];
 curve of 1/4 gives (1/t+1+t)*Qorb(4,[1,3],4)/(1-t^4)
 + (2*t^3 + 2*t^4)/Denom([1,1,1,4]);

P := (1-t^A[1])/Denom(A[2..6]);
PPP := S!(P*(1-t)^4); a := Coefficient(PPP,1); b := Coefficient(PPP,2);
PI := (1+a*t+b*t^2+a*t^3+t^4)/(1-t)^4;
P-PI-&+[K | Qorb(b[1],b[2..4],0) : b in B]
-4*Qorb(4,[1,3],4)/(1-t^4)
-(2*t^3+2*t^4)/Denom([1,1,1,4]);

[[a,b,c] : a in [-2..2], b in [-2..2], c in [-2..2] |
X eq (a/t+b+a*t)*Qorb(4,[1,3],4)/(1-t^4)+(c*t^3+c*t^4)/Denom([1,1,1,4])];

A := [2,19,22,45,66,154];
 curve of 1/3 gives -Q3-t^3/Denom([1,1,1,3]);
A := [2,19,24,32,75,152];
 curve of 1/2 gives 0
 curve of 1/3 gives -2*Q3-4*t^3/Denom([1,1,1,3]);
A := [2,19,24,64,107,216];
 curve of 1/2 gives -4*Q2
A := [2,19,24,88,131,264];
 curve of 1/2 gives -2*Q2
A := [2,19,25,30,74,150];
 curve of 1/2 gives 3*Q2
A := [2,19,28,35,49,133];
 curve of 1/2 gives 0
 curve of 1/7(2,5) gives (1/t^2 - 2 + t^2)*Q725
 +(-4*t^7 + 4*t^6 - 7*t^5 + 4*t^4 - 4*t^3)/Denom([1,1,1,7])
A := [2,19,28,35,84,168];
 curve of 1/7(2,5) gives -*Q725
 +(-2*t^7 + 2*t^6 - 3*t^5 + 2*t^4 - 2*t^3)/Denom([1,1,1,7])
A := [2,19,28,98,147,294];
 curve of 1/7(2,5) gives (2/t^2+1/t-1+t+2*t^2)*Q725
 +(-3*t^7 + 5*t^6 - 7*t^5 + 5*t^4 - 3*t^3)/Denom([1,1,1,7])
A := [2,19,44,109,155,329];
A := [2,19,65,86,153,325];
A := [2,19,65,86,172,344];
A := [2,19,65,151,218,455];
A := [2,19,67,90,159,337];
A := [2,19,82,204,307,614];
A := [2,19,84,105,210,420];
 curve of 1/2 gives Q2
 curve of 1/21(2,19) gives big challenge
 A1/t^9*Q21219 + B1/Denom([1,1,1,21])
 where
> A1;
t^18 + t^17 + 2*t^16 + 2*t^15 + t^14 + t^13 + 2*t^12 + 2*t^11 - t^10 - 2*t^9 -
 t^8 + 2*t^7 + 2*t^6 + t^5 + t^4 + 2*t^3 + 2*t^2 + t + 1
> B1;
-3*t^21 - 2*t^19 - 2*t^18 - 4*t^16 + 3*t^15 - 8*t^14 + 7*t^13 - 8*t^12 +
7*t^11 - 8*t^10 + 3*t^9 - 4*t^8 - 2*t^6 - 2*t^5 - 3*t^3

A := [2,19,103,246,370,740];
 curve of 1/2 gives 8*Q2
A := [2,19,122,284,427,854];
 curve of 1/2 gives 15*Q2
A := [2,19,124,166,311,622];
 curve of 1/2 gives 2*Q2
A := [2,20,23,35,60,140];
 curve of 1/2 gives Q2
 curve of 1/5(2,3) gives -2*Q523 -(t^3-t^4+t^5)/Denom([1,1,1,5])
A := [2,20,23,45,70,160];
 curve of 1/2 gives 2*Q2
 curve of 1/5(2,3) gives (-1-t-t^2)*Q523 -(t^3-t^4+t^5)/Denom([1,1,1,5])

A := [2,20,23,70,95,210];
A := [2,20,23,70,115,230];
A := [2,20,27,32,79,160];
A := [2,20,27,69,91,209];
 curve of 1/2 gives 3*Q2, curve of 1/3 gives -4*Q3-5*t^3/Denom([1,1,1,3])
A := [2,20,32,37,89,180];
A := [2,20,32,43,95,192];
A := [2,20,32,53,105,212];
A := [2,20,33,45,100,200];
A := [2,20,33,55,90,200];
A := [2,20,33,55,110,220];
A := [2,20,33,110,165,330];
A := [2,20,43,65,130,260];
A := [2,20,45,66,67,200];
A := [2,20,64,75,159,320];
A := [10,21,135,322,478,966];
A := [10,21,156,364,541,1092];
A := [10,21,165,392,588,1176];
A := [10,22,23,88,143,286];
A := [10,22,33,45,55,165];
A := [10,22,42,63,115,252];
A := [10,22,53,85,148,318];
A := [10,22,53,85,170,340];
A := [10,22,53,138,201,424];
A := [10,22,85,106,117,340];
A := [10,23,38,109,157,337];
A := [10,23,76,195,281,585];
A := [10,23,76,195,304,608];
A := [10,23,99,264,396,792];
A := [10,23,122,150,305,610];
A := [10,23,152,195,228,608];
A := [10,24,39,47,60,180];
A := [10,24,78,95,183,390];
A := [10,24,126,155,315,630];
A := [10,25,27,38,62,162];
A := [10,25,27,38,100,200];
 B := [[25,2,10,13],[25,2,10,13],[10,5,7,8],[10,5,7,8],
 [38,24,25,27],[27,10,19,25];

General task: given A := [a1,..a5,d]
 (I) write routine to find B as a function of A;
 (II) write routine for P,PPP,PI,P-PI (done)

A := [10,25,27,38,100,200];
 B := [[25,2,10,13],[25,2,10,13],[10,5,7,8],[10,5,7,8],
 [38,24,25,27],[27,10,19,25]];
 curve of 1/5(2,3) gives
 2*(1/t+t)*Qorb(5,[2,3],5)/(1-t^5)
 +(-2*t^3+2*t^4-2*t^5)/Denom([1,1,1,5]);

Q2 := Qorb(2,[1,1],2)/(1-t^2);
Q3 := Qorb(3,[1,2],3)/(1-t^3);
Q4 := Qorb(4,[1,3],4)/(1-t^4);

A := [1,3,13,17,21,55];
 B := [[13,1,4,8],[17,1,3,13],[21,1,3,17]];
 curve of 1/3 gives 2*t^3/Denom([1,1,1,3])
 A := [1,3,13,17,31,65];
 B := [[3,1,1,1],[17,1,3,13],[31,1,13,17]];
 no curve
A := [1,3,13,17,34,68];
 no curve
A := [1,3,13,18,22,57];
 curve of 1/2 but X(A) is 0
A := [1,3,13,30,44,91];
 curve of 1/2 gives -1*Q2,
 curve of 1/3 gives 2*t^3/Denom([1,1,1,3])
A := [1,3,13,31,48,96];
 no curve
A := [1,3,14,21,38,77];
 curve of 1/3 gives 4*t^3/Denom([1,1,1,3])
A := [1,3,14,33,48,99];
 curve of 1/2 gives 0
 curve of 1/3 gives 3*t^3/Denom([1,1,1,3])
A := [1,3,14,35,52,105];
 curve of 1/2 gives 0
 curve of 1/3 gives 4*t^3/Denom([1,1,1,3])
A := [1,3,15,19,23,61];
 curve of 1/3 gives -Q3+4*t^3/Denom([1,1,1,3]);
A := [1,3,15,19,37,75];
 no curve
A := [1,3,15,19,38,76];
 curve of 1/3 gives -Q3+5*t^3/Denom([1,1,1,3]);
A := [1,3,15,20,36,75];
 curve of 1/3 gives 2*t^3/Denom([1,1,1,3])
 curve of 1/4 gives (1/t+t)*Q4; Q4 := Qorb(4,[1,3],4)/(1-t^4);
A := [1,3,15,34,52,105];
 curve of 1/3 gives -4*t^3/Denom([1,1,1,3]);
A := [1,3,15,35,51,105];
 curve of 1/3 gives Q3+2*t^3/Denom([1,1,1,3]);
A := [1,3,16,20,24,64];
 curve of 1/2 gives -1*Q2,
 curve of 1/3 gives 2*t^3/Denom([1,1,1,3])
A := [1,3,16,20,40,80];
 curve of 1/4 gives (3/t+1+3*t)*Q4 + 2*(t^3+t^4)/Denom([1,1,1,4])
A := [1,3,16,24,28,72];
 curve of 1/4 gives (1/t-1+t)*Q4
A := [1,3,16,24,44,88];
 curve of 1/3 gives 3*t^3/Denom([1,1,1,3])
 curve of 1/4 gives (1/t-1+t)*Q4
A := [1,3,16,40,60,120];
 curve of 1/4 gives ..

 Write routine to find B as a function of A;
 I explain how to do a systematic study of basket of points and curves
 Example A := [1,16,51,136,204,408];
**Relevant**: r = hcf >= 2 of a nonempty subset of ai.
This is calculated by the routine

Relevant := [r : r in [2..A[5]] |
 r eq GCD([Integers() | A[i] : i in [1..5] | A[i] mod r eq 0])];

For r in Relevant, calculate [a mod r : a in A];
Amod := [a mod r : a in A];
 let J = subset of ai == 0 mod r. J is nonempty by definition of relevant.
Case division based on J:

1. J = [i]. "Coordinate point on X"
 This means that r = ai does not divide d or any aj. By qsmooth,
 r divides(d-aj) for some j. The point Pi in X is a 1/r(ak,al,am)
 point where ak,al,am are the 3 remaining a in A.
 Action: Put it in B.

Test for this case by Boolean
 // if
 #[a : a in Amod[1..5] | a eq 0] eq 1 and Amod[6] ne 0
 // then ..
The following magma function constructs the orbifold point:
Amod := [a mod r : a in A];
if #[a : a in Amod[1..5] | a eq 0] eq 1 and Amod[6] ne 0 then
Insert(Exclude(Exclude(Amod[1..5],Amod[6]),0),1,r); end if;
 // Operator on Sequences. Given r = 16 and the sequence
 // Amod = [1,0,3,8,12,8], take Amod[1..5], exclude Amod[6]
 // exclude 0 and insert r in the 1st position to get [16,1,3,12].

2. J = [i,d]. "Coordinate point not on X"
 This means that r = ai divides d and no aj. The pure power
 xi^(d/ai) in f, so Pi notin X.
 No action.

Test for this case by Boolean
 // if
 #[a : a in Amod[1..5] | a eq 0] eq 1 and Amod[6] eq 0
 // then .. (do nothing)

3. J = [i,j]. "Coordinate line is a line of transverse orbifold points"
 This means that r = hcf(ai,aj) does not divide d or any of the 3
 remaining ak,al,am in A. By qsmooth, one of these, say ak == d mod r,
 and the line Lij is a line of transverse 1/r(al,am).
 Action: Put it in C

Test for this case by Boolean
 // if
 #[a : a in Amod[1..5] | a eq 0] eq 2 and Amod[6] ne 0
 // then .. (put line Lij in C)

4. J = [i,j,d]. "Coordinate line meets X in orbifold points"
 This means that r = hcf(ai,aj) divides d and no ak. There are a
 number of 1/r(ak,al,am) points on the open line Lij, where ak,al,am
 are the 3 remaining a in A. The set is S^0(d) in PP(ai,aj).
 Action: Add this number of 1/r(ak,al,am) points to B

Test for this case by Boolean
 // if
 #[a : a in Amod[1..5] | a eq 0] eq 2 and Amod[6] eq 0
 // then .. (calculate number of S^0(d) in PP(ai,aj) and add that
 // number of 1/r(ak.al,am) points to B)

5. J = [i,j,k,d]. "Coordinate plane meets X in curve of transverse
 orbifold point"
 Action: Add curve C(d) in PP(ai,aj,ak) of 1/r(al,am) to C.

Test for this case by Boolean
 // if
 #[a : a in Amod[1..5] | a eq 0] eq 3 and Amod[6] eq 0
 // then .. (put curve C(d) in PP(ai,aj,ak) of 1/r(al,am) in C)

> A := [1,16,51,136,204,408];
> Relevant := [r : r in [2..A[5]] | r eq GCD([Integers()
 | A[i] : i in [1..5] | A[i] mod r eq 0])];
// that is, I want r to divide a subset, and to be its GCD
> R;
[ 4, 8, 16, 17, 51, 68, 136, 204 ]
[[a mod r : a in A] : r in Relevant];
[
 [ 1, 0, 3, 0, 0, 0 ], // case 5, J is [2,4,5] and [6]
 [ 1, 0, 3, 0, 4, 0 ], // case 4, J is [2,4] and [6]
 [ 1, 0, 3, 8, 12, 8 ], // case 2, J is [2]
 [ 1, 16, 0, 0, 0, 0 ], // case 5, J is [3,4,5] and [6]
 [ 1, 16, 0, 34, 0, 0 ], // case 4, J is [3,5] and [6]
 [ 1, 16, 51, 0, 0, 0 ], // case 4, J is [4,5] and [6]
 [ 1, 16, 51, 0, 68, 0 ], // case 2, J is [4] and [6]
 [ 1, 16, 51, 136, 0, 0 ] // case 2, J is [5] and [6]
]

function OldBask(A);
 B := [];
 Relevant := [r : r in [2..A[5]] | r eq GCD([Integers()
 | A[i] : i in [1..5] | A[i] mod r eq 0])];
 for r in Relevant do
 Amod := [a mod r : a in A ];
 case #[a : a in Amod[1..5] | a eq 0]:
 when 0:; // no sing, do nothing
 when 4:; //error
 when 5:; //error
 when 1:
 if Amod[6] ne 0
 then Append(~B, Insert(Exclude(Exclude(Amod[1..5],Amod[6]),0),1,r));
 end if; // If ai divides d then Pi not on X
 when 2:
 if Amod[6] eq 0 then
 // calculate the number S^0(d) in PP(ai,aj)
 Num := Floor(A[6]/LCM([a : a in A[1..5] | a mod r eq 0]));
 for i in [1..Num] do
 Append(~B, Insert(Exclude(Exclude(Amod[1..5],0),0),1,r));
 end for;
 end if; // If r does not divides d then Lij is line of 1/r
 when 3:; // necessarily r divides d and curve of 1/r
 end case;
end for;
return B; // also return C
end function;

AA := [[1,4,4,5,10,24], [1,4,4,6,9,24], [1,4,4,7,9,25], [1,4,4,7,12,28], [1,4,4,7,16,32], [1,4,4,9,14,32], [1,4,4,9,18,36], [1,4,4,13,18,40], [1,4,4,17,26,52], [1,4,5,5,5,20], [1,4,5,5,10,25], [1,4,5,5,14,29], [1,4,5,6,8,24], [1,4,5,7,8,25], [1,4,5,8,18,36], [1,4,5,10,10,30], [1,4,5,10,16,36], [1,4,5,10,20,40], [1,4,5,15,20,45], [1,4,5,15,24,49], [1,4,5,20,30,60]];

AA := [[1,3,4,4,4,16],[1,3,4,4,8,20],[1,3,4,4,9,21],[1,3,4,4,12,24],[1,3,4,5,12,25],[1,3,4,6,7,21],[1,3,4,6,10,24],[1,3,4,6,14,28],[1,3,4,8,8,24],[1,3,4,8,12,28],[1,3,4,8,13,29],[1,3,4,8,15,31],[1,3,4,8,16,32],[1,3,4,12,16,36]];

AA := [[1,3,17,22,26,69],[1,3,17,39,60,120],[1,3,18,23,27,72],[1,3,18,25,46,93],[1,3,18,41,63,126],[1,3,18,43,64,129],[1,3,19,45,68,136],[1,3,21,28,52,105],[1,3,21,29,33,87],[1,3,21,29,54,108],[1,3,21,49,73,147],[1,3,21,50,75,150],[1,3,22,51,77,154],[1,3,23,31,35,93],[1,3,23,54,81,162],[1,3,24,32,36,96],[1,3,24,32,60,120],[1,4,4,5,10,24],[1,4,4,6,9,24],[1,4,4,7,9,25],[1,4,4,7,12,28],[1,4,4,7,16,32],[1,4,4,9,14,32],[1,4,4,9,18,36],[1,4,4,13,18,40],[1,4,4,17,26,52],[1,4,5,5,5,20],[1,4,5,5,10,25],[1,4,5,5,14,29],[1,4,5,6,8,24]];

AA := [[7,34,36,84,127,288],[7,36,143,336,486,1008],[7,36,150,350,507,1050],[7,36,179,408,630,1260],[7,36,186,422,651,1302],[7,36,251,588,882,1764],[7,38,54,144,243,486],[7,38,60,210,315,630],[7,41,165,385,557,1155],[7,41,206,467,721,1442],[7,43,51,53,154,308],[7,50,58,60,175,350],[7,59,60,84,210,420],[8,8,9,10,13,48],[8,8,9,11,28,64],[8,9,10,18,35,80],[8,9,10,18,45,90],[8,9,10,23,50,100],[8,9,10,27,27,81],[8,9,10,27,46,100],[8,9,10,27,54,108],[8,9,10,45,63,135],[8,9,11,16,28,72],[8,9,11,16,36,80],[8,9,11,16,44,88],[8,9,12,23,52,104],[8,9,12,25,54,108],[8,9,13,15,15,60]];

// Given a qsmooth CY hypersurface Y(d) in PP(a1,.. a5)
// in the form A := [a1,..a5,d], parse it into the form
// PI (initial term)
// plus sum Porb(1/r(a,b,c)) with 1/r(a,b,c) in PointBasket
// plus sum Ar/[1,1,r,r] where Ar is a symmetric polynomial
// of symmetric degree 2r+2 with support in [3..2r-1].

function P(A)
return (1-t^A[6])/Denom(A[1..5]);
end function;

function PI(A) // Initial term P_I, only for CY 3folds
n1 := #[i : i in A | i eq 1];
n2 := #[i : i in A | i eq 2];
return (1 + (n1-4)*t + (n2+Binomial(n1-3,2))*t^2 + (n1-4)*t^3 + t^4)/(1-t)^4;
end function;

function Bask(A);
 B := []; C := [];
 Relevant := [r : r in [2..A[5]] | r eq GCD([Integers()
 | A[i] : i in [1..5] | A[i] mod r eq 0])];
 for r in Relevant do
 Amod := [a mod r : a in A ];
 case [#[a : a in Amod[1..5] | a eq 0],#[a : a in Amod[6..6] | a eq 0]]:
 // when 0:; // no sing, do nothing when 4:; //error when 5:; //error
 when [1,0]:
 Append(~B, Insert(Exclude(Exclude(Amod[1..5],Amod[6]),0),1,r));
 // end this case: If ai divides d then Pi not on X
 when [2,1]:
 // calculate the number S^0(d) in PP(ai,aj)
 Num := Floor(A[6]/LCM([a : a in A[1..5] | a mod r eq 0]));
 for i in [1..Num] do
 Append(~B, Insert(Exclude(Exclude(Amod[1..5],0),0),1,r));
 end for;
 // end if; If r does not divides d then Lij is line of 1/r
 when [2,0]:
 Append(~C, Insert(Exclude(Exclude(Exclude(Amod[1..5],0),0),Amod[6]),1,r));
 when [3,1]:; // necessarily r divides d and curve of 1/r
 Append(~C, Insert(Exclude(Exclude(Exclude(Amod[1..5],0),0),0),1,r));
 end case;
end for;
return B, C;
end function;

function PointTerms(B)
return &+[K | Qorb(b[1],b[2..4],0) : b in B];
end function;

function X(A)
return P(A)-PI(A)-PointTerms(Bask(A));
end function;

function PC(A) // The curve terms P_C
B,C := Bask(A);
YY := PartialFractionDecomposition(X(A)/t^3*(1-t)^4);
return [t^3/(1-t)^4*&+[K|y[3]/y[1]^y[2] : y in YY
 | IsDivisibleBy(1-t^r,y[1])] where r is c[1] : c in C];
end function;

for A in AA[1..100] do B := Bask(A);
 sing := [b : b in B | #b ne 4];
 if sing ne [] then
 A; sing[1];
 end if;
end for;

> BB:=[A : A in AAA | (([8,3,5] in C) or ([8,5,3] in C)) where B,C is Bask(A)];

[X(A) eq &+[ K | y : y in PC(A)] : A in BB];








The same but with the basket sorted for easier searching



function P(A)
return (1-t^A[6])/Denom(A[1..5]);
end function;

function PI(A) // Initial term P_I, only for CY 3folds
n1 := #[i : i in A | i eq 1];
n2 := #[i : i in A | i eq 2];
return (1 + (n1-4)*t + (n2+Binomial(n1-3,2))*t^2 + (n1-4)*t^3 + t^4)/(1-t)^4;
end function;

function Bask(A);
 B := []; C := [];
 Relevant := [r : r in [2..A[5]] | r eq GCD([Integers()
 | A[i] : i in [1..5] | A[i] mod r eq 0])];
 for r in Relevant do
 Amod := [a mod r : a in A ];
 case [#[a : a in Amod[1..5] | a eq 0],#[a : a in Amod[6..6] | a eq 0]]:
 // when 0:; // no sing, do nothing when 4:; //error when 5:; //error
 when [1,0]:
 Append(~B, Insert(Sort(Exclude(Exclude(Amod[1..5],Amod[6]),0)),1,r));
 // end this case: If ai divides d then Pi not on X
 when [2,1]:
 // calculate the number S^0(d) in PP(ai,aj)
 Num := Floor(A[6]/LCM([a : a in A[1..5] | a mod r eq 0]));
 for i in [1..Num] do
 Append(~B, Insert(Sort(Exclude(Exclude(Amod[1..5],0),0)),1,r));
 end for;
 // end if; If r does not divides d then Lij is line of 1/r
 when [2,0]:
 Append(~C, Insert(Exclude(Exclude(Exclude(Amod[1..5],0),0),Amod[6]),1,r));
 when [3,1]:; // necessarily r divides d and curve of 1/r
 Append(~C, Insert(Exclude(Exclude(Exclude(Amod[1..5],0),0),0),1,r));
 end case;
end for;
return B, C;
end function;

function PointTerms(B)
return &+[K | Qorb(b[1],b[2..4],0) : b in B];
end function;

function X(A)
return P(A)-PI(A)-PointTerms(Bask(A));
end function;

function PC(A) // The curve terms P_C
B,C := Bask(A);
YY := PartialFractionDecomposition(X(A)/t^3*(1-t)^4);
return [t^3/(1-t)^4*&+[K|y[3]/y[1]^y[2] : y in YY
 | IsDivisibleBy(1-t^r,y[1])] where r is c[1] : c in C];
end function;