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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 206 "This file contains the co
mputations relevant to the paper \"On the classification of non-self-d
ual modular categories\" by Hong and Rowell. Thus covers the case with
 Galois group (01),(34), b=0, a=-g and y=f." }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 37 "with(LinearAlgebra): with(Groebner): " }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 109 "Using the Galois argument assuming that \+
(01)(34), the orthogonality relations imply z=(d-1) and h1=-1/2(d+1).
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "S1:=subs(\{x=1,y=f,a=-g,b=0\},
Matrix(5,5,[[1,d,f,g,g],[d,x,y,a,a],[f,y,z,b,b],[g,a,b,h1+I*h2,h1-I*h2
],[g,a,b,h1-I*h2,h1+I*h2]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#S1
G-%'RTABLEG6%\"*g%HM:-%'MATRIXG6#7'7'\"\"\"%\"dG%\"fG%\"gGF17'F/F.F0,$
F1!\"\"F37'F0F0%\"zG\"\"!F77'F1F3F7,&%#h1GF.*&%#h2GF.^#F.F.F.,&F:F.*&^
#F4F.F<F.F.7'F1F3F7F>F9%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 77 "C:=Matrix(5,5,[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0
,0,0,0,1],[0,0,0,1,0]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG-%'R
TABLEG6%\"*C&HM:-%'MATRIXG6#7'7'\"\"\"\"\"!F/F/F/7'F/F.F/F/F/7'F/F/F.F
/F/7'F/F/F/F/F.7'F/F/F/F.F/%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 100 "These relations describe the condition that S^2 is proportiion
al to the \"charge conjugation matrix.\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 46 "Srels:=factor(convert(evalm(S1^2-K^2*C),set));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&SrelsG<+\"\"!*&%\"fG\"\"\",(F)F)%\"
dGF)%\"zGF)F)*&,(F)!\"\"*&\"\"#F)%#h1GF)F/F+F)F)%\"gGF),$F-F/,(*&F1F)F
+F)F)*$)F(F1F)F)*&F1F))F3F1F)F/,(*&F1F)F8F)F)*$)F,F1F)F)*$)%\"KGF1F)F/
,(*&F1F)F:F)F)*&F1F))F2F1F)F)*&F1F))%#h2GF1F)F/,**&F1F)F:F)F)*&F1F)FEF
)F)*&F1F)FGF)F)F?F/,,F)F)*$)F+F1F)F)F7F)*&F1F)F:F)F)F?F/" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "factor(Basis(`union`(Srels,\{1+d+z,
-1-2*h1+d\}),lexdeg([K,d,f,g,z],[h2,h1])));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7',(*&\"\"#\"\"\"%#h1GF'F'F&F'%\"zGF',(F'!\"\"*&F&F'F(F
'F+%\"dGF',(*$)%\"gGF&F'F'*$)F(F&F'F'*$)%#h2GF&F'F+,,F&F'*&\"\"%F'F(F'
F'*&F&F'F5F'F+*&F&F'F3F'F'*$)%\"fGF&F'F'*&,&%\"KGF'*&F&F'F6F'F+F',&FAF
'*&F&F'F6F'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "s1rules:
=solve(\{2+z+2*h1, -1-2*h1+d\},\{z,h1\});" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%(s1rulesG<$/%#h1G,&#\"\"\"\"\"#!\"\"*&F+F,%\"dGF*F*/%
\"zG,&F.F,F*F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "S:=subs(s
1rules,S1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG-%'RTABLEG6%\"*7'
QM:-%'MATRIXG6#7'7'\"\"\"%\"dG%\"fG%\"gGF17'F/F.F0,$F1!\"\"F37'F0F0,&F
/F4F.F4\"\"!F77'F1F3F7,(*&\"\"#F4F/F.F.#F.F;F4*&%#h2GF.^#F.F.F.,(*&F;F
4F/F.F.#F.F;F4*&^#F4F.F>F.F.7'F1F3F7F@F9%'MatrixG" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 119 "These relations come down to a single relation am
ong the d,f and g (the other allow elimination of K, h1 and h2 and z).
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "orthrel:=\{2*d+f^2-2*g^
2\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(orthrelG<#,(*&\"\"#\"\"\"%
\"dGF)F)*$)%\"fGF(F)F)*&F(F))%\"gGF(F)!\"\"" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 316 "Observe that if d is an integer, then d=1 since (01) i
nterchanges d and 1/d. this in turn implies that f is an integer divid
ing (d+1)=2, hence f=1 or 2, so that g^2=(2d+f^2)/2=(2+1)/2 or (2+4)/2
.  It must be the latter, since g^2 is an algebraic integer.  So g=sqr
t(3), d=1 and f=2 is the only way d is an integer.  " }}{PARA 0 "" 0 "
" {TEXT -1 152 "Note that since (01) interchanges g and -g/d, g cannot
 be an integer, and if f is an integer f=f/d so that d=1. We must cons
ider this case eventually.  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 154 "
We use the various symmetries of the N_\{i,j\}^k to write down the fus
ion matrices in terms of just 14 variables.  Note that M4 is just the \+
transpose of M3." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "M1:=Mat
rix([[0,1,0,0,0],[1,n1,n2,n3,n3],[0,n2,n4,n5,n5],[0,n3,n5,n6,n7],[0,n3
,n5,n7,n6]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M1G-%'RTABLEG6%\"*
w'QM:-%'MATRIXG6#7'7'\"\"!\"\"\"F.F.F.7'F/%#n1G%#n2G%#n3GF37'F.F2%#n4G
%#n5GF67'F.F3F6%#n6G%#n7G7'F.F3F6F9F8%'MatrixG" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 94 "M2:=Matrix([[0,0,1,0,0],[0,n2,n4,n5,n5],[1,n4,
n8,n9,n9],[0,n5,n9,n10,n11],[0,n5,n9,n11,n10]]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#M2G-%'RTABLEG6%\"*S(QM:-%'MATRIXG6#7'7'\"\"!F.\"\"\"
F.F.7'F.%#n2G%#n4G%#n5GF37'F/F2%#n8G%#n9GF67'F.F3F6%$n10G%$n11G7'F.F3F
6F9F8%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "M3:=Matri
x([[0,0,0,0,1],[0,n3,n5,n7,n6],[0,n5,n9,n11,n10],[1,n6,n10,n12,n13],[0
,n7,n11,n14,n12]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M3G-%'RTABLE
G6%\"*/)QM:-%'MATRIXG6#7'7'\"\"!F.F.F.\"\"\"7'F.%#n3G%#n5G%#n7G%#n6G7'
F.F2%#n9G%$n11G%$n10G7'F/F4F8%$n12G%$n13G7'F.F3F7%$n14GF:%'MatrixG" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "The matrices must commute, giving
 a set of diophantine equations." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 189 "comrels:=`minus`(`union`(convert(evalm(M1&*M2-M2&*M1
),set),convert(evalm(M1&*M3-M3&*M1),set),convert(evalm(M2&*M3-M3&*M2),
set),convert(evalm(M3&*Transpose(M3)-Transpose(M3)&*M3),set)),\{0\});
" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(comrelsG<@,&%$n12G\"\"\"%$n13G!
\"\",&F)F(F'F*,.*&%#n3GF(%#n6GF(F(*&%#n5GF(%$n10GF(F(*&%#n7GF(F)F(F(*&
F.F(F4F(F**&F1F(%$n11GF(F**&F4F(%$n14GF(F*,.F5F(F6F(F8F(F-F*F0F*F3F*,.
*&F/F(F1F(F(*&%#n9GF(F2F(F(*&F7F(F)F(F(*&F4F(F1F(F**&F>F(F7F(F**&F7F(F
9F(F*,.F@F(FAF(FBF(F<F*F=F*F?F*,0F(F(*$)F/\"\"#F(F(*$)F2FGF(F(*$)F)FGF
(F(*$)F4FGF(F**$)F7FGF(F**$)F9FGF(F*,0F(F(*&%#n1GF(%#n4GF(F(*&%#n2GF(%
#n8GF(F(*(FGF(F.F(F>F(F(*$)FWFGF(F**$)FUFGF(F**&FGF()F1FGF(F*,0FZF(Ffn
F(*&FGF(FinF(F(F(F*FSF*FVF**(FGF(F.F(F>F(F*,0*$)F.FGF(F(*$FinF(F(*(FGF
(F4F(F/F(F(*&FTF(F4F(F**&FWF(F7F(F**&F.F(F'F(F**&F.F(F9F(F*,0F`oF(*$)F
>FGF(F(*(FGF(F7F(F2F(F(*&FUF(F4F(F**&FXF(F7F(F**&F>F(F'F(F**&F>F(F9F(F
*,0FLF(FNF(FPF(F(F*FEF*FHF*FJF*,0FboF(FcoF(FdoF(FeoF(F^oF*F`oF**(FGF(F
4F(F/F(F*,0FjoF(F[pF(F\\pF(F]pF(F`oF*FgoF**(FGF(F7F(F2F(F*,2*&FTF(F1F(
F(*&FWF(F>F(F(*&F.F(F2F(F(*&F.F(F7F(F(*&F.F(FWF(F**&F1F(FUF(F*F<F*F@F*
,2*&FWF(F1F(F(*&FUF(F>F(F(F0F(F6F(*&F.F(FUF(F**&F1F(FXF(F**&F/F(F>F(F*
*&F4F(F>F(F*,2*&FWF(F/F(F(*&FUF(F2F(F(*&F1F(F)F(F(*&F1F(F'F(F(*&F.F(F1
F(F**&F1F(F>F(F**&F/F(F2F(F**&F4F(F7F(F*,2*&FWF(F4F(F(*&FUF(F7F(F(FeqF
(*&F1F(F9F(F(FfqF*FgqF**&F/F(F7F(F**&F4F(F2F(F*,2FhpF(FipF(F<F(F@F(Fdp
F*FepF*FfpF*FgpF*,2F]qF(F^qF(F_qF(F`qF(F[qF*F\\qF*F0F*F6F*,2FfqF(FgqF(
FhqF(FiqF(FbqF*FcqF*FdqF*FeqF*,2FfqF(FgqF(F^rF(F_rF(F[rF*F\\rF*FeqF*F]
rF*,2F-F(F0F(*&F/F(F)F(F(*&F4F(F'F(F(F5F*F6F**&F/F(F'F(F*F8F*,2F5F(F6F
(FgrF(F8F(F-F*F0F*FerF*FfrF*,2F<F(F=F(*&F2F(F)F(F(*&F7F(F'F(F(F@F*FAF*
*&F2F(F'F(F*FBF*,2F@F(FAF(F\\sF(FBF(F<F*F=F*FjrF*F[sF*,4F(F(*&FTF(F/F(
F(*&FWF(F2F(F(*&F.F(F)F(F(FdoF(F^oF*F`oF*FEF*FLF*,4F(F(*&FUF(F/F(F(*&F
XF(F2F(F(*&F>F(F)F(F(F\\pF(F`oF*FgoF*FHF*FNF*,4F^oF(F`oF(FEF(FLF(F(F*F
_sF*F`sF*FasF*FdoF*,4F`oF(FgoF(FHF(FNF(F(F*FcsF*FdsF*FesF*F\\pF*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "indets(comrels);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#<0%#n1G%$n10G%$n11G%$n12G%$n13G%$n14G%#n2G%#
n3G%#n4G%#n5G%#n6G%#n7G%#n8G%#n9G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
59 "The commutation relations alone leave 6 degrees of freedom." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "HilbertDimension(comrels,tdeg(n1, n
10, n11, n12, n13, n14, n2, n3, n4, n5, n6, n7, n8, n9));" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#\"\"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "
The Characteristic polynomials of M1,M2 and M3 are computed in two way
s. First from the knowledge of the eigenvalues." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 182 "ch1:=collect(expand(product('(X-S[2,i]/S[1,i])',i=1.
.5)),X); ch2:=collect(expand(product('(X-S[3,i]/S[1,i])',i=1..5)),X); \+
ch3:=collect(expand(product('(X-S[4,i]/S[1,i])',i=1..5)),X);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%$ch1G,.\"\"\"!\"\"*$)%\"XG\"\"&F&F&*&,(*&F
&F&%\"dGF'F'F&F&F/F'F&)F*\"\"%F&F&*&,&F.F'F/F'F&)F*\"\"$F&F&*&,&F.F&F/
F&F&)F*\"\"#F&F&*&,(F.F&F&F'F/F&F&F*F&F&" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%$ch2G,**$)%\"XG\"\"&\"\"\"F**&,**&F*F*%\"fG!\"\"F**&F.F*%\"dGF
/F/*&F.F/F1F*F*F.F/F*)F(\"\"%F*F**&,*\"\"#F/F1F/*&F.F7F1F/F**&F*F*F1F/
F/F*)F(\"\"$F*F**&,&F.F*F0F*F*)F(F7F*F*" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%$ch3G,,*$)%\"XG\"\"&\"\"\"F**&,**&%\"gG!\"\"%\"dGF*F/*&F.F*F0F
/F**&F*F*F.F/F*F.F/F*)F(\"\"%F*F**&,2*&%#h2G\"\"#F.!\"#F**&F.F9F0F/F/F
9F/*(F4F/F.F:F0F9F**&F*F*F0F/F*F0F**&F*F**&F4F*)F.F9F*F/F**(F9F/F.F:F0
F*F/F*)F(\"\"$F*F**&,2**FCF*F4F/F.F/F0F*F*F.F**&F8F9F.F/F/*(FCF*F4F/F.
F/F/*&F*F**(F4F*F0F*F.F*F/F**(F0F/F8F9F.F/F**(F4F/F0F9F.F/F/F1F/F*)F(F
9F*F**&,*#F*F9F**&F*F**&F4F*F0F*F/F/*&F4F/F0F*F/*&F0F/F8F9F/F*F(F*F*" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 34 "set m=(d+1)/f which in an integer." }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 97 "simplify(\{subs(\{m=(d+1)/f\},coeff(ch2,X,4)+coeff(
ch2,X,2)-m),subs(\{m=(d+1)/f\},subs(\{X=-m\},ch2))\});" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#<#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "Th
e form of these polynomials imply further relations." }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 134 "simplify(\{coeff(ch1,X,0)+1,coeff(ch2,X,0),coef
f(ch2,X,1),coeff(ch3,X,0),coeff(ch1,X,4)+coeff(ch1,X,1),coeff(ch1,X,3)
+coeff(ch1,X,2)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#\"\"!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "factor(ch1); factor(ch2); fa
ctor(ch3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*,,&%\"XG\"\"\"F&!\"\"F&
,&F%F&F&F&\"\"#,&*&%\"dGF&F%F&F&F&F'F&,&F%F&F,F'F&F,F'" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#*.%\"XG\"\"#,&F$\"\"\"%\"fG!\"\"F',(%\"dGF'F'F'*&F
(F'F$F'F'F',&*&F+F'F$F'F'F(F)F'F+F)F(F)" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,$*0\"\"%!\"\"%\"XG\"\"\",&F'F(%\"gGF&F(,&*&%\"dGF(F'F(F(F*F(F(,
0*(F%F()F'\"\"#F()F*F1F(F(**F%F(F*F(F'F(F-F(F&*(F%F(F*F(F'F(F(*&F1F(F-
F(F&F(F(*$)F-F1F(F(*&F%F()%#h2GF1F(F(F(F-F&F*!\"#F(" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 107 "p1:=CharacteristicPolynomial(M1,X): p2:=
CharacteristicPolynomial(M2,X): p3:=CharacteristicPolynomial(M3,X):" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "factor(p1); factor(p2);" }
}{PARA 12 "" 1 "" {XPPMATH 20 "6#*&,(%#n6G!\"\"%#n7G\"\"\"%\"XGF(F(,X*
&\"\"#F()%#n5GF,F(F(*&%#n4GF(F'F(F&*&F0F(F%F(F&*&)F)\"\"$F(F0F(F&*&F3F
(F%F(F&*(F,F()F)F,F(F-F(F&*(F7F(F%F(F0F(F(*$F7F(F&*$)F)\"\"%F(F(*&F)F(
F0F(F(*(F7F(%#n1GF(F0F(F(*(F7F(F?F(F%F(F(*(F7F(F0F(F'F(F(*(F7F(F?F(F'F
(F(**F,F(F)F(F?F(F-F(F(**F,F(F)F()%#n3GF,F(F0F(F(*(F)F(F%F()%#n2GF,F(F
(*(F)F(FHF(F'F(F(*&F)F(F%F(F(*&F)F(F'F(F(*,F<F(F)F(FIF(FFF(F.F(F&**F)F
(F%F(F?F(F0F(F&**F)F(F?F(F0F(F'F(F&*&F3F(F'F(F&*&F3F(F?F(F&*(F,F(F7F(F
EF(F&*&F7F(FHF(F&F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*&,(%\"XG\"\"\"
%$n10G!\"\"%$n11GF&F&,X*&\"\"#F&)%#n5GF,F&F&*&%#n2GF&F)F&F(*&F0F&F'F&F
(*(F,F&)F%F,F&F-F&F(*&)F%\"\"$F&F'F&F(*(F,F&F3F&)%#n9GF,F&F(*&F5F&%#n8
GF&F(*&F%F&F'F&F&*(F3F&F'F&F;F&F&*$F3F&F(*$)F%\"\"%F&F&*(F3F&F;F&F)F&F
&*(F3F&F0F&F'F&F&*(F3F&F0F&F;F&F&*(F3F&F0F&F)F&F&**F,F&F%F&F0F&F8F&F&*
(F%F&F'F&)%#n4GF,F&F&*(F%F&FHF&F)F&F&*&F%F&F)F&F&*,FAF&F%F&FIF&F.F&F9F
&F(**F%F&F'F&F0F&F;F&F(**F%F&F0F&F;F&F)F&F(*&F5F&F)F&F(*&F5F&F0F&F(*&F
3F&FHF&F(*&F%F&F0F&F&**F,F&F%F&F-F&F;F&F&F&" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 135 "Observe that the only linear terms of p1 are (X-1) and
 (X+1), while the linear terms of p2 are X and (X+m) (provided d is no
t integral)" }}{PARA 0 "" 0 "" {TEXT -1 52 "we may use this later, but
 for now we avoid using m." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "solve
(X+n11-n10=(X+m)); solve(X+n11-n10=X);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#<&/%\"XGF%/%\"mG,&%$n10G!\"\"%$n11G\"\"\"/F)F)/F+F+" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#<%/%\"XGF%/%$n10G%$n11G/F(F(" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 34 "mlinrels:=\{(n10-n11)*(n10-n11+m)\};" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)mlinrelsG<#*&,&%$n10G\"\"\"%$n11G!
\"\"F),(%\"mGF)F(F)F*F+F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
23 "linrels:=\{(n6-n7)^2-1\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(li
nrelsG<#,&*$),&%#n6G\"\"\"%#n7G!\"\"\"\"#F+F+F+F-" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 76 "The relations implied by the coefficients of the C
haracteristic polynomials." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "chre
ls:=factor(\{coeff(p1,X,0)+1,coeff(p2,X,0),coeff(p2,X,1),coeff(p3,X,0)
,coeff(p1,X,4)+coeff(p1,X,1),coeff(p1,X,3)+coeff(p1,X,2)\});" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#>%'chrelsG<(*&,&%$n10G\"\"\"%$n11G!\"\"F),(*
&\"\"#F))%#n5GF.F)F+*&%#n2GF)F*F)F)*&F2F)F(F)F)F),,*&%#n4GF))%#n6GF.F)
F)*&F6F))%#n7GF.F)F+*(F.F)F/F)F8F)F+*(F.F)F/F)F;F)F)F)F),,*&F:F)%#n9GF
)F)**F.F)F*F)F0F)F;F)F+*&%$n14GF)F/F)F)*(FCF)%#n3GF)F@F)F+*&)F*F.F)FEF
)F),>*$)F(F.F)F+*,\"\"%F)F0F)F6F)F@F)F*F)F+*,FLF)F(F)F6F)F0F)F@F)F)*&F
GF))F6F.F)F)*(FJF)F2F)%#n8GF)F)*&F.F)F/F)F)**F.F)F/F)FQF)F(F)F+*&FJF)F
OF)F+**F.F)F*F)F/F)FQF)F)*(F.F)F2F)F(F)F+**F.F))F@F.F)F2F)F(F)F+*(FGF)
F2F)FQF)F+**F.F)FXF)F*F)F2F)F)*$FGF)F),D*&F.F)F8F)F+F6F+%#n1GF+*,FLF)F
2F)FEF)F0F)F;F)F+*,FLF)F2F)FEF)F0F)F8F)F)*$F7F)F+*$F:F)F)**F.F)FhnF)F/
F)F8F)F+*(F.F)F6F)F8F)F+*&F:F))F2F.F)F)*&F7F)F`oF)F+*(FhnF)F6F)F7F)F)*
*F.F)FhnF)F/F)F;F)F)*(FhnF)F6F)F:F)F+*&F.F)F/F)F)**F.F))FEF.F)F;F)F6F)
F)**F.F)FgoF)F6F)F8F)F+,R*&FhnF)F6F)F)*&F.F)FgoF)F+*(F.F)FhnF)F8F)F)*(
F.F)F6F)F8F)F)*$F`oF)F+*&F.F)F/F)F+F\\oF+F[oF)F)F+*(F.F)F8F)F`oF)F)F9F
)*(F.F)FgoF)F8F)F)*(F.F)FgoF)F6F)F)**FLF)F2F)FEF)F0F)F+**F.F)F8F)FhnF)
F6F)F+F5F+F6F)*&F.F)F8F)F)*&FhnF)F7F)F+*(F.F)FgoF)F;F)F+*&FhnF)F:F)F)*
(F.F)FhnF)F/F)F)*(F.F)F/F)F;F)F+*(F.F)F/F)F8F)F)" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 13 "nops(chrels);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#\"\"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "these involve the \+
integer m=z/f=(d+1)/f." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "intchar:=
Vector([1,1,-m,0,0]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "intcharre
ls:=`union`(convert(evalm(M1&*intchar-intchar),set),convert(evalm(M2&*
intchar+m*intchar),set),convert(evalm(M3&*intchar),set),convert(evalm(
Transpose(M3)&*intchar),set));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,i
ntcharrelsG<*\"\"!,&%#n1G\"\"\"*&%#n2GF)%\"mGF)!\"\",&%#n3GF)*&%#n5GF)
F,F)F-,&F1F)*&%#n9GF)F,F)F-,&%#n7GF)*&%$n11GF)F,F)F-,(F)F)%#n6GF)*&%$n
10GF)F,F)F-,(F+F)*&%#n4GF)F,F)F-F,F),*F)F)F?F)*&%#n8GF)F,F)F-*$)F,\"\"
#F)F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "mrels:=`union`(\{c
oeff(p2,X,4)+coeff(p2,X,2)-m,subs(\{X=-m\},p2)\},mlinrels,intcharrels)
;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&mrelsG<-\"\"!*&,&%$n10G\"\"\"%
$n11G!\"\"F*,(%\"mGF*F)F*F+F,F*,&%#n1GF**&%#n2GF*F.F*F,,&%#n3GF**&%#n5
GF*F.F*F,,&F6F**&%#n9GF*F.F*F,,&%#n7GF**&F+F*F.F*F,,(F*F*%#n6GF**&F)F*
F.F*F,,(F2F**&%#n4GF*F.F*F,F.F*,*F*F*FBF**&%#n8GF*F.F*F,*$)F.\"\"#F*F,
,4*$)F.\"\"&F*F,*&,(*&FHF*F)F*F*FEF*F2F*F*)F.\"\"%F*F,*&,4*&FHF*)F6FHF
*F**&FHF*)F9FHF*F**$)F+FHF*F**$)F)FHF*F,*(FHF*FEF*F)F*F,*(FHF*F2F*F)F*
F,F*F**$)FBFHF*F**&F2F*FEF*F,F*)F.\"\"$F*F**&,@F2F,*&FHF*F)F*F,*(FHF*F
+F*FUF*F**(FHF*FUF*FEF*F,**FQF*FBF*F6F*F9F*F**(FHF*F2F*FWF*F,**FHF*F)F
*F2F*FEF*F**(FHF*F)F*FinF*F,*(FHF*FWF*F+F*F**(FHF*F)F*FUF*F,*(FHF*F)F*
FWF*F,*&FYF*FEF*F,*&FYF*F2F*F,*&FEF*FenF*F**&F2F*FenF*F*F*FGF*F,*&,>*&
FHF*FUF*F,*(FHF*F2F*F)F*F*FZF*FXF,**FHF*F+F*FUF*FEF*F,*,FQF*F6F*FBF*F9
F*F+F*F**,FQF*F)F*FBF*F6F*F9F*F,**FHF*FWF*F+F*F2F*F,**FHF*FUF*FEF*F)F*
F***FHF*FWF*F2F*F)F*F**(FYF*F2F*FEF*F**(FenF*F2F*FEF*F,*&FYF*FinF*F,*&
FenF*FinF*F*F*F.F*F**(FHF*F+F*FUF*F**(FHF*F)F*FUF*F,FjoF,F\\pF*,@FEF,*
(FHF*FUF*FEF*F***FQF*FBF*F6F*F9F*F,FioF**(FHF*FWF*F+F*F,*(FHF*F)F*FUF*
F*F\\pF,*(FHF*F)F*FinF*F*FjoF**(FHF*F2F*FWF*F***FHF*F)F*F2F*FEF*F,*(FH
F*F)F*FWF*F*F[pF,*(FHF*F+F*FUF*F,F.F," }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 60 "We take all of the relations together and process them. .
   " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "allNrels:=subs(\{n8=u,n11=v,
n13=t,n14=w\},`union`(linrels,comrels,chrels,mrels)):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "factor(Basis(allNrels,lexdeg([w, v,
 t, u,n1,n6, n9,n10,n12, n2, n3, n4, n5,n7],[m])));" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#7jn,&%#n3G\"\"\"*&%#n5GF&%\"mGF&!\"\",(%#n2GF&*&%#n4G
F&F)F&F*F)F&,(F&F*F.F**(\"\"#F&%$n10GF&F)F&F&,&F(F**&%#n9GF&F)F&F&,(F.
F*F&F&*&F1F&%#n6GF&F&,(%#n1GF&*&F.F&)F)F1F&F**$F<F&F&,*F&F*F.F**&%\"uG
F&F)F&F&F=F&,&%\"tGF&%$n12GF*,&%#n7GF**&%\"vGF&F)F&F&*(,&F1F&F=F&F&,&F
EF&F&F&F&,(F&F&*&F1F&FEF&F*F.F&F&*&FEF&,(*&F1F&F2F&F*F)F&F@F&F&,.F&F*F
.F**&F1F&)F(F1F&F&*&F1F&)FEF1F&F&*&\"\"$F&FEF&F&*(FVF&F.F&FEF&F*,2F(F&
*&F(F&F.F&F&*(F1F&FEF&F(F&F***F1F&FEF&%\"wGF&F)F&F**(F.F&F<F&F(F&F&*&F
<F&F(F&F***F1F&FEF&F)F&FCF&F&**F1F&FEF&F<F&F(F&F*,2F&F*F.F**&FVF&FEF&F
&*&F1F&FTF&F&*(FVF&F.F&FEF&F**&F1F&F<F&F&*(F1F&F.F&F<F&F***\"\"%F&F(F&
FCF&F)F&F&,0*(F1F&F(F&F2F&F&*(F1F&FEF&F5F&F*F'F**(F1F&FEF&FCF&F&*(F1F&
FEF&FfnF&F***F1F&FEF&F(F&F)F&F**(F)F&F(F&F.F&F&,*F)F**&F(F&F5F&F&*(F1F
&F(F&FCF&F*F-F&,.*(F1F&FEF&F5F&F**(F1F&FEF&FCF&F&*(F1F&FEF&FfnF&F**&F(
F&F@F&F&**F1F&FEF&F(F&F)F&F*FioF&,&*&FEF&F5F&F**&F(F&FGF&F&,0*&FVF&F)F
&F**(F.F&FEF&F)F&F**(F1F&F(F&FCF&F*F-F&*(F1F&F(F&FfnF&F&*&FEF&F)F&F&*(
F1F&FTF&F)F&F&*&FKF&,(F.F&FVF**&F1F&FEF&F*F&,8F(F&**F1F&F)F&FCF&F.F&F&
*(F1F&FfnF&F)F&F***F1F&FEF&FfnF&F)F&F*FYF**(FVF&F<F&F(F&F***F1F&FEF&F)
F&FCF&F&*(FEF&F<F&F(F&F**&FfnF&)F)FVF&F**&)F)FboF&F(F&F**(FCF&F.F&FhqF
&F&,2F-F&F)F&F2F&FhpF&F[qF**&F.F&F2F&F&*(F1F&FEF&FGF&F**(F1F&FTF&F)F&F
*,2F5F&*&F.F&F5F&F&*(F1F&FEF&F5F&F*F'F**(F1F&FEF&FCF&F&*(F1F&FEF&FfnF&
F***F1F&FEF&F(F&F)F&F*FioF&,2*(FVF&F.F&F)F&F&*&FVF&F)F&F&F@F&**F1F&F.F
&FEF&F)F&F&*(F1F&FEF&F)F&F**(FboF&FEF&FGF&F**(FboF&FTF&F)F&F**&F.F&F@F
&F&,(FGF&*(F1F&FEF&F2F&F**&F.F&FGF&F&,4FCF*FfnF&*(FVF&F(F&F)F&F&*(F1F&
FEF&FCF&F**(F1F&FEF&FfnF&F&**F1F&FEF&F(F&F)F&F&FioF**&FfnF&F.F&F&*&FCF
&F.F&F*,.FVF*F.F&*(FboF&F5F&FCF&F**&F1F&)F5F1F&F&*&FboF&)F2F1F&F&*(F1F
&F@F&F2F&F*,4\"\"&F*F.F*FEF&*&F1F&FTF&F&*&F.F&FEF&F**&\"\")F&F]tF&F&*(
F1F&F5F&FfnF&F&*(FboF&F@F&F2F&F**(F1F&F5F&FCF&F*,0F&F&F.F&*&F1F&FEF&F&
*(FboF&FGF&F2F&F**(F1F&F@F&FGF&F&*&FboF&F]tF&F**(F1F&F@F&F2F&F&,0F&F*F
.F**&F1F&FTF&F&*&F1F&F]tF&F*FEF&FbtF**&F1F&)FGF1F&F&,0F(F&*(F1F&FGF&Ff
nF&F&*(F1F&FGF&FCF&F*FYF**(F1F&FEF&F(F&F&*(F1F&F5F&F2F&F**(F1F&F5F&FGF
&F&,0F.F&FEF&*&F1F&)FfnF1F&F&*&F1F&FTF&F&FbtF**&F1F&)FCF1F&F*FVF*,:F(F
&*(F1F&FEF&F(F&F&*&F(F&FTF&F***F1F&F)F&FCF&F.F&F&*(FVF&FfnF&F)F&F**(FE
F&FfnF&F)F&F**(F1F&F(F&F.F&F**(FVF&F<F&F(F&F**(FEF&F)F&FCF&F&FfqF**&F)
F&FCF&F***F1F&FCF&FTF&F)F&F&,<*&F1F&FCF&F&*(F1F&FEF&F5F&F**(F1F&FEF&FC
F&F&*(F1F&FTF&F5F&F&*(F1F&F(F&F)F&F**(F1F&FEF&FfnF&F**(FboF&FCF&FTF&F*
*(FVF&FfnF&F<F&F**(FEF&F(F&F)F&F&*(FVF&FhqF&F(F&F*FioF***FVF&FCF&F.F&F
<F&F&*(F1F&FCF&F.F&F&,$*(FEF&FJF&,,*(F1F&FEF&F)F&F**&FboF&F2F&F&F-F&F)
F&*&FboF&FGF&F*F&F*,B*(FboF&FEF&FCF&F**(FEF&FhqF&F(F&F&*(FEF&F<F&FfnF&
F&*(F`tF&F(F&F)F&F&*(FboF&FEF&FfnF&F&*(FboF&FCF&FTF&F**(FVF&FfnF&F<F&F
***\"\"'F&FEF&F(F&F)F&F&*&FhqF&F(F&F***F1F&FCF&F.F&F<F&F&*(FTF&F(F&F)F
&F**(FboF&FfnF&FTF&F&*(FCF&FEF&F<F&F***F1F&FTF&F<F&FfnF&F&**F1F&F)F&F(
F&F.F&F**&F<F&FCF&F*,<*&F1F&F(F&F&FYF**&FEF&F(F&F&*(FVF&F(F&FTF&F**(F.
F&F(F&FEF&F&**F1F&F)F&FCF&F.F&F&*(FVF&FfnF&F)F&F*FdvF&FhnF*FgvF*FfqF&F
hvF***F1F&FTF&F)F&FfnF&F&*&FEF&,2*(FVF&F.F&F)F&F**(FdtF&FEF&F2F&F**&Fd
tF&FGF&F&*&F`tF&F)F&F&FhpF&F[qF**(F1F&FTF&F)F&F**(FdtF&F(F&FCF&F&F&,<*
(FEF&FCF&F.F&F&FCF&*(F1F&FEF&FCF&F*F'F&*&FEF&FfnF&F&*(FVF&FCF&FTF&F**&
FfnF&F<F&F*FgxF**(FCF&F.F&F<F&F&*&FfnF&FTF&F&FgsF&FioF***F1F&FEF&F(F&F
)F&F&,8**FboF&FCF&FEF&FfnF&F&*&F1F&F.F&F&*&F`tF&FEF&F*F1F&*&FVF&FTF&F*
*(FVF&F.F&FEF&F&*&F1F&)FEFVF&F&*&F.F&FTF&F**&F1F&F<F&F**(FboF&F^vF&FEF
&F**(F1F&F.F&F<F&F&,P**F1F&FTF&F)F&FfnF&F*FgvF&*&FVF&F(F&F**(F1F&F(F&F
.F&F**(FfxF&FEF&F(F&F&*(FdtF&F5F&F2F&F&*(F1F&F2F&FCF&F*FavF&**FdtF&F2F
&FEF&F5F&F&**FfxF&F)F&FCF&F.F&F&**FdtF&FGF&FCF&FEF&F**(\"#7F&F5F&FGF&F
**(FdtF&FGF&FCF&F&*(FVF&FCF&F@F&F&*(FboF&FfnF&F)F&F*FdvF**(F1F&F2F&Ffn
F&F&*(F1F&F@F&F5F&F**&F@F&FfnF&F&*(\"\"(F&F<F&F(F&F**(FboF&F)F&FCF&F&*
*FdtF&F2F&FCF&FEF&F*FfqF*,6F&F*F.F*FEF***F1F&FEF&F2F&FGF&F&*&F1F&FTF&F
&FbtF&**FboF&F5F&FCF&FEF&F**&F1F&F]tF&F**(F1F&FGF&F2F&F**(F1F&FEF&F]tF
&F&,L**F1F&FTF&F)F&FfnF&F&FgvF**&FVF&F(F&F&*(F1F&F(F&F.F&F**(FfxF&FEF&
F(F&F&*(FdtF&F5F&F2F&F**(FfxF&F2F&FCF&F&FavF***F1F&F)F&FCF&F.F&F&*(Fbo
F&F5F&FGF&F&*(FVF&FCF&F@F&F**(FboF&FfnF&F)F&F*FdvF&*(F1F&F2F&FfnF&F&*(
F1F&F@F&F5F&F&Fe\\lF*FhnF**(FboF&F)F&FCF&F***FboF&F2F&FCF&FEF&F***FboF
&FfnF&FEF&F2F&F&FfqF&,P**F1F&FTF&F)F&FfnF&F&FgvF**&FVF&F(F&F&*(FfxF&F(
F&F.F&F**(\"#5F&FEF&F(F&F&*(FdtF&F5F&F2F&F**(Ff^lF&F2F&FCF&F&FavF***Ff
^lF&F)F&FCF&F.F&F&**FdtF&FGF&FCF&FEF&F**(FboF&F5F&FGF&F**(FdtF&FGF&FCF
&F&*(FVF&FCF&F@F&F**(F_\\lF&FfnF&F)F&F*FdvF&*(FfxF&F2F&FfnF&F&*(F1F&F@
F&F5F&F&Fe\\lF**(\"\"*F&F<F&F(F&F**(FboF&F)F&FCF&F***FdtF&F2F&FCF&FEF&
F***FdtF&FGF&FEF&F5F&F&FfqF&,8*&F1F&F)F&F***FVF&F.F&FEF&F)F&F**(FdtF&F
2F&FTF&F**(F1F&F.F&F)F&F**(FdtF&FEF&FGF&F&*(FboF&F(F&FCF&F&*(Fb_lF&FEF
&F)F&F&*&FTF&F)F&F&**FVF&F.F&FTF&F)F&F&**FboF&FCF&F.F&F(F&F&*(FfxF&F_[
lF&F)F&F*,N**FboF&FTF&F)F&FfnF&F***F1F&FEF&F)F&FCF&F&*(\"#;F&F^vF&F(F&
F&*(FboF&F(F&F.F&F**(FboF&FEF&F(F&F&*(FdtF&F5F&F2F&F&*(F1F&F2F&FCF&F&*
(F1F&F(F&FTF&F&**FboF&F)F&FCF&F.F&F***FdtF&FGF&FCF&FEF&F**(FboF&F5F&FG
F&F**(FdtF&FGF&FCF&F&*&FCF&F@F&F&*(FVF&FfnF&F)F&F***F1F&FEF&FfnF&F)F&F
**(FfxF&F2F&FfnF&F&*(F1F&F@F&F5F&F*Fe\\lF**(FfxF&F<F&F(F&F**(\"#6F&F)F
&FCF&F&**FdtF&F2F&FCF&FEF&F***F1F&FEF&F<F&F(F&F*,N*(FdtF&FCF&FfnF&F*\"
#8F&*&F\\blF&F.F&F&*&F1F&FEF&F**&F\\blF&FTF&F**(FdtF&FTF&F<F&F**(FboF&
F.F&FEF&F**(FdtF&F5F&FCF&F&*&Ff`lF&F]tF&F**&FfxF&F_[lF&F&*(F`tF&F.F&FT
F&F**(F1F&FjqF&F.F&F**&FfxF&F<F&F&*(Ff`lF&FGF&F2F&F&*(FdtF&F@F&F2F&F&*
(Ff`lF&FEF&F]tF&F**&F1F&FjqF&F&*(FdtF&F^vF&F.F&F&*(F1F&F.F&F<F&F&**Fbo
F&F^vF&F.F&F<F&F&**FboF&FCF&FfnF&F<F&F***FdtF&F5F&FCF&FEF&F&,:*&Ff`lF&
F2F&F**&FboF&F@F&F&*&FboF&F)F&F***FVF&F.F&FEF&F)F&F**(FdtF&F(F&FCF&F**
*FdtF&F5F&FGF&FCF&F&*(FdtF&FEF&FGF&F&**FdtF&FCF&F5F&F2F&F**(F`tF&FEF&F
)F&F&*(F.F&FTF&F)F&F&*(Fg\\lF&FTF&F)F&F&*(F1F&F_[lF&F)F&F*,D**FboF&Ffn
F&F2F&FCF&F**&Ff`lF&F2F&F**&FboF&F@F&F&*&FVF&F)F&F***F1F&FfnF&F)F&FCF&
F&**F1F&F@F&FfnF&FCF&F&**FVF&F.F&FEF&F)F&F**(F_\\lF&F(F&FCF&F**(FfxF&F
)F&F^vF&F&*(F_\\lF&F2F&F^vF&F*F-F&*(FdtF&FEF&FGF&F&**FdtF&FCF&F5F&F2F&
F&**FboF&FCF&F@F&F5F&F**(FVF&FEF&F)F&F&*(FfxF&FTF&F)F&F&*(FfxF&F^vF&F@
F&F&,<*&FboF&F2F&F**&FdtF&FGF&F**&F1F&F@F&F&*&Fg\\lF&F)F&F***F`tF&F.F&
FEF&F)F&F**(FdtF&FEF&F2F&F&*(FdtF&F(F&FCF&F*F-F**(FdtF&FEF&FGF&F&*(Fdt
F&F@F&F]tF&F&*(F`tF&FEF&F)F&F&*(Ff^lF&FTF&F)F&F&*&Ff`lF&)F2FVF&F*,>**F
boF&F2F&FCF&F@F&F**&F1F&F5F&F&*&F1F&FCF&F**&F1F&FfnF&F**(F1F&FEF&F5F&F
&*(F1F&F(F&F)F&F*FdzF**(FdtF&F]tF&F5F&F***FboF&F@F&F2F&F5F&F&FcwF&FgxF
*FioF*FezF&*(FdtF&F]tF&FCF&F&,6*(F1F&FEF&F5F&F**(F1F&FEF&FCF&F&**F1F&F
5F&FGF&F2F&F&F'F**(F1F&FEF&FfnF&F**(F1F&F]tF&FfnF&F**(F1F&F]tF&F5F&F**
*F1F&FEF&F(F&F)F&F*FioF&*(F1F&F]tF&FCF&F&,F*&\"#WF&F2F&F**&\"#CF&FGF&F
**&\"#9F&F@F&F&*(FboF&F2F&)F@F1F&F&*&\"#DF&F)F&F***\"#<F&F.F&FEF&F)F&F
**(FdglF&FEF&F2F&F&*(FbglF&F(F&FCF&F**(FdtF&F)F&F^vF&F&*(Ff`lF&F2F&F^v
F&F*F-F&*(\"#KF&FEF&FGF&F&**FdtF&FCF&F5F&F2F&F&**FboF&FCF&F@F&F5F&F**(
F\\hlF&FEF&F)F&F&*(\"#MF&FTF&F)F&F&*(FdtF&F^vF&F@F&F&*&Ff`lF&F\\flF&F*
,@**F1F&F2F&FCF&F@F&F&*&F1F&F5F&F**&F1F&FCF&F&*&F1F&FfnF&F&*(F1F&FEF&F
5F&F&*(F1F&FEF&FCF&F*F'F&*(F1F&FEF&FfnF&F&FdzF&*(FboF&F]tF&FfnF&F*FcwF
&FgxF&FezF**(FboF&F]tF&FCF&F***F1F&F@F&FfnF&F2F&F&,F**FboF&F2F&FCF&F@F
&F**(F1F&F<F&FCF&F&*&FfxF&F5F&F&*&FfxF&FCF&F**&FfxF&FfnF&F**(F1F&FEF&F
5F&F&*(F1F&FhglF&F5F&F&*(FboF&FEF&FCF&F&*(Ff^lF&F(F&F)F&F**(FboF&FEF&F
fnF&F**(FVF&FfnF&F<F&F**(FdtF&F]tF&F5F&F*FcwF**(FVF&FhqF&F(F&F**(F1F&F
CF&FhglF&F*FioF***FVF&FCF&F.F&F<F&F&*(Ff`lF&F]tF&FCF&F&,F**FboF&F2F&FC
F&F@F&F&F^yF**&FfxF&F5F&F**&FfxF&FCF&F&*&FfxF&FfnF&F&*(FfxF&FEF&F5F&F&
*(FdtF&FEF&FCF&F**(FfxF&F(F&F)F&F&*(FdtF&FEF&FfnF&F&*(F1F&FfnF&F<F&F&*
(FboF&F]tF&FfnF&F***F`tF&FEF&F(F&F)F&F&*(FVF&FhqF&F(F&F&*&FCF&FhglF&F&
FioF***FVF&FCF&F.F&F<F&F**(F_\\lF&F]tF&FCF&F**&FhglF&FfnF&F&,P*(FdglF&
FCF&FfnF&F&FfglF**&FfglF&F.F&F**&\"#OF&FEF&F**&F`tF&FTF&F**(FdtF&FTF&F
<F&F&*(\"#SF&F.F&FEF&F&*(FdglF&F5F&FCF&F**&F_\\mF&F]tF&F&*&FdtF&F^vF&F
&*&F`tF&F_[lF&F**&F1F&)FEFboF&F**&F.F&F_[lF&F&*(FgalF&F.F&FTF&F&*(FdtF
&FTF&F]tF&F&*&F_\\lF&F<F&F**(\"#cF&FGF&F2F&F**(FdglF&F@F&F2F&F**(F[]mF
&FEF&F]tF&F&*(Ff`lF&F^vF&F.F&F**(FboF&F.F&F<F&F&*(Ff`lF&F^vF&FTF&F***
\"#[F&F5F&FCF&FEF&F*,X**F`tF&F_[lF&F)F&F.F&F***FdglF&FfnF&F2F&FCF&F&*&
\"#?F&FhqF&F&*&F_\\mF&F2F&F**&Ff`lF&FGF&F**&F_\\lF&F@F&F&*&\"#QF&F)F&F
***F_\\mF&FfnF&F)F&FCF&F***\"#]F&F.F&FEF&F)F&F**(Ff`lF&FEF&F2F&F&*(Fdg
lF&F2F&FTF&F&*(FbhlF&FEF&F\\flF&F&**Fb]mF&FGF&F^vF&FEF&F**(FdglF&F2F&F
^vF&F&*(\"#=F&F.F&F)F&F**(FdtF&FEF&FGF&F***FbhlF&FCF&F5F&F2F&F**(\"#mF
&FEF&F)F&F&**F_\\mF&F)F&F^vF&F.F&F&**FVF&F.F&FTF&F)F&F&*(\"#bF&FTF&F)F
&F&*(Fg]mF&F.F&FhqF&F***Ff`lF&F2F&F^vF&FEF&F**(Ff^lF&Fe\\mF&F)F&F&*(Fg
alF&F_[lF&F)F&F**(FbhlF&FGF&F]tF&F**(Ff`lF&F^vF&FGF&F&,FFboF&*&FboF&F.
F&F&*&FVF&FEF&F&*$FTF&F&*(F`tF&F.F&FEF&F**&FdglF&F]tF&F**&FboF&F^vF&F&
*&F1F&F_[lF&F&*(Ff`lF&F^vF&F]tF&F*F`[lF**&F1F&F<F&F&*(FdtF&FGF&F2F&F&*
(Ff`lF&F@F&F2F&F&*(FdtF&FEF&F]tF&F**&F1F&FhglF&F**(FboF&F^vF&F.F&F&**F
dtF&F^vF&F@F&F2F&F&**FdtF&F5F&FCF&FEF&F&,BF1F&*&F1F&F.F&F&*&F1F&FEF&F&
*&FVF&FTF&F&*(FfxF&F.F&FEF&F**&FVF&F_[lF&F&*&F1F&Fe\\mF&F**(FdtF&F^vF&
F]tF&F*Ff\\mF&F`[lF**(FdtF&FGF&F2F&F&*&FdtF&F]tF&F**(FboF&F@F&F2F&F&**
FdtF&FCF&F]tF&FfnF&F&*(FdtF&FEF&F]tF&F***Ff`lF&F5F&FCF&FEF&F&,jn**F[]m
F&FfnF&F2F&FCF&F**&FdtF&FhqF&F**&\"$w*F&F2F&F&*&\"$C#F&FGF&F&*&\"$s#F&
F@F&F**&\"$D$F&F)F&F&*&FdtF&)F@FVF&F***\"$Q#F&F.F&FEF&F)F&F&*(FjamF&FE
F&F2F&F**(\"$O(F&F(F&FCF&F&*(\"$G\"F&F2F&)FCFboF&F&**Ff`lF&FGF&F^vF&FE
F&F**(\"$7$F&F)F&F^vF&F**(\"$k'F&F2F&F^vF&F&*(\"#fF&F.F&F)F&F**(\"$G&F
&FEF&FGF&F***F\\bmF&FCF&F5F&F2F&F***\"$O\"F&FCF&F@F&F5F&F&*(\"#kF&F)F&
FhbmF&F**(\"$_#F&FEF&F)F&F***Ff`lF&F)F&F^vF&F.F&F&**Fg\\lF&F.F&FTF&F)F
&F**(\"$$[F&FTF&F)F&F**(\"$'HF&F^vF&F@F&F***FbhlF&)FCFVF&F@F&F5F&F&**F
b]mF&F2F&F^vF&FEF&F&*(FfglF&F_[lF&F)F&F&**FfcmF&F`dmF&F5F&F2F&F**&Ffcm
F&F\\flF&F&*(Fb]mF&F^vF&FGF&F**(FfcmF&FhbmF&F@F&F*,ho*(FdglF&FCF&FfnF&
F&FdtF**&FdtF&F.F&F**&\"#oF&FEF&F&*&\"$E\"F&FTF&F&*(\"$c\"F&F.F&FEF&F*
*(FdglF&F5F&FCF&F**&FbhlF&F]tF&F**&\"#!)F&F^vF&F&*&FbhlF&)F2FboF&F**&F
1F&)F@FboF&F&*&\"#()F&F_[lF&F&*&F`tF&Fe\\mF&F**(FgbmF&F^vF&F]tF&F**(FV
F&F.F&F_[lF&F&*(\"#UF&F.F&FTF&F**(FdtF&FhbmF&F<F&F**&F[\\mF&F<F&F**(F[
]mF&FGF&F2F&F&*(Ff`lF&F@F&F2F&F***FbhlF&F5F&F`dmF&FEF&F&**FbhlF&F]tF&F
^vF&FEF&F**(F[]mF&FEF&F]tF&F**&F1F&FjqF&F**&FbhlF&FhglF&F&*(F[]mF&F^vF
&F.F&F&*(FdtF&FhbmF&FhglF&F&*&F1F&)FEF`tF&F**(Ff`lF&FhbmF&F.F&F&*(Fg]m
F&F.F&F<F&F&*(FbhlF&F<F&F^vF&F**(FbhlF&F^vF&FhglF&F&*(FbhlF&F^vF&FTF&F
**&Ff`lF&FhbmF&F&*(FbhlF&FhbmF&F]tF&F**&Fe\\mF&F.F&F&**\"$W\"F&F5F&FCF
&FEF&F&**FbhlF&F^vF&FGF&F2F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 
"m>0 an integer implies:." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 155 "factor(Basis(`union`(allNrels,\{1-2*n7+n4,m^2*n
5+m*n12+w*m-2*n7*m*n12-n5+n7*n5,n5*m-n7*n12+n7*w\}),plex(n1,n6, n9,n12
, n2, n3, n4,w, v,n10, t, u, n5,n7,m)));" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#7<,(*$)%#n5G\"\"#\"\"\"F)*&F(F)%#n7GF)F)*&F(F))F+F(F)!\"\",(*&F(
F)F+F)F.*$)%\"mGF(F)F)*&%\"uGF)F3F)F),$*&,&F+F)F)F.F),(*(\"\"%F)F+F)%
\"tGF)F.*(\"\"$F)F'F)F3F)F)*&F'F)F5F)F)F)F.,,F+F)*(F'F)F3F)F<F)F)*$F-F
)F.F1F)*&F+F)F2F)F.,$*&F+F),,*&F>F)F3F)F.F5F.*&F5F)F+F)F)*(F(F)F'F)F<F
)F.*(F>F)F+F)F3F)F)F)F.,0*&\"\"'F)F+F)F)**F(F)F5F)F'F)F<F)F)*&)F5F(F)F
+F)F.*&F>F)F2F)F.*$FPF)F)*&FMF)F-F)F.*(F>F)F+F)F2F)F),&F+F.*&%$n10GF)F
3F)F),$*&F+F),(*&F(F)FWF)F.F3F)F5F)F)F.,$*&F'F)FZF)F.,**&F(F)F+F)F.F1F
)*(F(F)F5F)FWF)F)FRF.,**&F;F)F+F)F.*&F;F))FWF(F)F)F1F)FRF.,0*(FMF)F+F)
F3F)F.*&F>F)F3F)F)*&F(F)FWF)F)*&F;F)%\"vGF)F)F5F.*(F;F)F'F)F<F)F)*(F(F
)F5F)F+F)F.,.F'F.*&F<F)F3F)F)*&%\"wGF)F3F)F)*&F+F)F'F)F)*&F2F)F'F)F)**
F(F)F+F)F3F)F<F)F.,(*&F'F)F3F)F)*&F+F)F<F)F.*&F+F)FjoF)F),**(F(F)F+F)F
3F)F)*&F(F)F3F)F.*&F'F)F<F)F.*&F'F)FjoF)F),2*(F>F)F'F)F3F)F.*(F(F)F'F)
F5F)F)*(F(F)F<F)F2F)F)*(F+F)F'F)F3F)F)*&FPF)F<F)F.*&)F3F>F)F'F)F)**F(F
)F+F)F2F)F<F)F.*&FPF)FjoF)F),D*&\"\")F)F3F)F)*&FcqF)F5F)F.*&\"#KF)FWF)
F)*(\"#>F)F+F)F3F)F.*(\"\"(F)F5F)F+F)F.*(\"#AF)F'F)F<F)F)*&F(F)F^qF)F)
*(\"\"&F)F-F)F3F)F.*(F(F))F<F(F)F3F)F)*&F-F)F5F)F.*(F;F)F5F)FarF)F.*(
\"#7F)FWF)FarF)F)*(F(F)F+F)F^qF)F.**FcqF)F+F)F3F)FarF)F.*(FcqF))F<F>F)
F'F)F)**F;F)FarF)F5F)F+F)F.**F(F)F5F)FjoF)F<F)F),6F'F)*(F_rF)F<F)F3F)F
)F[pF)*(F(F)FWF)F<F)F)F\\pF)**FcqF)F+F)F3F)F<F)F.*(F;F)F'F)FarF)F)**F(
F)F5F)F+F)F<F)F.*&F5F)FjoF)F.*(F;F)FWF)FjoF)F),**&F(F)F+F)F)F(F.*$)Fjo
F(F)F)*$FarF)F.,(F)F)*&F(F)F+F)F.%#n4GF),&%#n3GF)F_pF.,(%#n2GF)*(F(F)F
+F)F3F)F.*&F(F)F3F)F),&%$n12GF)F<F.,.*&F(F)F<F)F.*&F(F)FjoF)F.*(F>F)F'
F)F3F)F.*(F;F)F+F)F<F)F)F?F.*&F(F)%#n9GF)F),(F+F.F)F)%#n6GF),(*&F(F)F2
F)F)*(F(F)F+F)F2F)F.%#n1GF)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 " a
ssuming n7=0 gives m=0." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "factor(
Basis(`union`(allNrels,\{1-2*n7+n4,m^2*n5+m*n12+w*m-2*n7*m*n12-n5+n7*n
5,n5*m-n7*n12+n7*w,n7\}),plex(n1,n6, n9,n12, n2, n3, n4,w, v,n10, t, u
, n5,n7,m)));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7:*$)%\"mG\"\"#\"\"\"
%#n7G*&%#n5GF(F&F(*$)F+F'F(*&%\"uGF(F&F(*&F+F(F/F(*$)F/F'F(*&%$n10GF(F
&F(*&F+F(F4F(*&F/F(F4F(*$)F4F'F(,,*(\"\"%F(F+F(%\"tGF(F(F/!\"\"*&F;F(%
\"vGF(F(*&F'F(F4F(F(*&\"\"$F(F&F(F(,(*&%\"wGF(F&F(F(F+F=*&F<F(F&F(F(,(
*&F'F(F&F(F=*&F+F(F<F(F=*&F+F(FEF(F(,4*&F;F(F&F(F(*&F;F(F/F(F=*&\"#;F(
F4F(F(*(\"#6F(F+F(F<F(F(*&)F<F'F(F&F(F(*(F'F(F/F(FSF(F=*(\"\"'F(F4F(FS
F(F(*(F;F()F<FBF(F+F(F(*(F/F(FEF(F<F(F(,.F+F(*(\"\"&F(F<F(F&F(F(*(F;F(
F+F(FSF(F(*(F'F(F4F(F<F(F(*&F/F(FEF(F=*(F;F(F4F(FEF(F(,(*$)FEF'F(F(F'F
=*$FSF(F=,&%#n4GF(F(F(%#n3G,&%#n2GF(*&F'F(F&F(F(,&%$n12GF(F<F=,(%#n9GF
(FEF=F<F=,&%#n6GF(F(F(%#n1G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Th
us we may assume n7 not zero since then we have m=0 ." }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 192 "factor(Basis(`union`(allNrels,\{1-2*n7+n4,m^2*n
5+m*n12+w*m-2*n7*m*n12-n5+n7*n5,n5*m-n7*n12+n7*w,-3*m-u-2*n5*t+u*n7+3*
n7*m,-2*n10+u+m\}),plex(n1,n6, n9,n12, n2, n3, n4,w, v,n10, t, u, n5,n
7,m)));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#74,(*$)%#n5G\"\"#\"\"\"F)*&
F(F)%#n7GF)F)*&F(F))F+F(F)!\"\",(*&F(F)F+F)F.*$)%\"mGF(F)F)*&%\"uGF)F3
F)F),$*&,&F+F)F)F.F),(*(\"\"%F)F+F)%\"tGF)F.*(\"\"$F)F'F)F3F)F)*&F'F)F
5F)F)F)F.,,*&F>F)F3F)F)F5F)*&F5F)F+F)F.*(F(F)F'F)F<F)F)*(F>F)F+F)F3F)F
.,(*&F(F)%$n10GF)F)F3F.F5F.,(F3F.*&F(F)%\"vGF)F)F5F.,.F'F.*&F<F)F3F)F)
*&%\"wGF)F3F)F)*&F+F)F'F)F)*&F2F)F'F)F)**F(F)F+F)F3F)F<F)F.,(*&F'F)F3F
)F)*&F+F)F<F)F.*&F+F)FNF)F),,F3F.F5F)*&F+F)F3F)F)FBF.*(F(F)F'F)FNF)F),
0*&F>F)F'F)F)*(F(F)F<F)F3F)F.FOF.FPF.**F(F)F+F)F3F)F<F)F)*&F5F)FNF)F)*
&F5F)F<F)F.,**&F(F)F+F)F)F(F.*$)FNF(F)F)*$)F<F(F)F.,(F)F)*&F(F)F+F)F.%
#n4GF),&%#n3GF)FSF.,(%#n2GF)*(F(F)F+F)F3F)F.*&F(F)F3F)F),&%$n12GF)F<F.
,.*&F(F)F<F)F.*&F(F)FNF)F.*(F>F)F'F)F3F)F.*(F;F)F+F)F<F)F)F?F.*&F(F)%#
n9GF)F),(F+F.F)F)%#n6GF),(*&F(F)F2F)F)*(F(F)F+F)F2F)F.%#n1GF)" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "finally, we solve for all but w,v,
t,u." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 206 "intrels:=factor(Basis(`uni
on`(allNrels,\{1-2*n7+n4,m^2*n5+m*n12+w*m-2*n7*m*n12-n5+n7*n5,n5*m-n7*
n12+n7*w,-3*m-u-2*n5*t+u*n7+3*n7*m,-2*n10+u+m\}),lexdeg([m,n1,n6, n9,n
10,n12, n2, n3, n4,n5,n7],[u, v, t, w])));" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%(intrelsG70,,\"\"#\"\"\"*$)%\"tGF'F(F(*$)%\"wGF'F(!\"
\"*(F'F(%\"uGF(%\"vGF(F(*&\"\"%F()F2F'F(F/*&,&F.F/F+F(F(,**(\"\"$F(F2F
(F.F(F/*&F2F(F+F(F/*&F1F(F+F(F(*&F1F(F.F(F(F(,$*&F7F(,2*$)F+F:F(F(*&F*
F(F.F(F(*&F'F(F+F(F(*(F'F(F+F(F5F(F/*&F+F(F-F(F/*(F'F(F.F(F5F(F(*&F'F(
F.F(F(*$)F.F:F(F/F(F/,**&F'F(%#n7GF(F(F'F/F,F(F)F/,(*&F2F(F.F(F(F;F/%#
n5GF(,*F(F/%#n4GF(F,F(F)F/,0*&F'F(%#n3GF(F(*&F'F(F.F(F(FCF(FIF/*&F'F(F
+F(F/FAF/FFF(,***F'F(F.F(F+F(F2F(F(%#n2GF(*&F*F(F2F(F/*&F2F(F-F(F/,&%$
n12GF(F+F/,&%$n10GF(F2F/,0FAF(FCF(FFF/FIF/*&F'F(%#n9GF(F(*(F'F(F.F(F5F
(F(*(F'F(F+F(F5F(F/,(F,F(F)F/*&F'F(%#n6GF(F(,2*&F'F(%#n1GF(F(*(F4F(F+F
(F.F(F(*(F'F(F.F(FBF(F(*(F'F(F+F(FJF(F/*&F'F(F-F(F/*$)F.F4F(F(*&F'F(F*
F(F/*$)F+F4F(F/,(%\"mGF(*&F'F(F2F(F/F1F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 75 "rules1:=`union`(\{n8=u,n11=v,n13=t,n14=w\},solve(\{se
q(intrels[i],i=4..14)\}));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'rules
1G<5/%\"mG,&%\"uG!\"\"*&\"\"#\"\"\"%\"vGF-F-/%#n1G,0*$)%\"tGF,F-F-*(F,
F-F4F-%\"wGF-F**$)F6F,F-F-*&#F-F,F-*$)F4\"\"%F-F-F-*&F6F-)F4\"\"$F-F**
&F4F-)F6F@F-F-*&#F-F,F-*$)F6F=F-F-F*/%$n10GF./%$n11GF./%$n12GF4/%$n13G
F4/%$n14GF6/%#n2G,(*&F.F-F8F-F-**F,F-F6F-F4F-F.F-F**&F3F-F.F-F-/%#n3G,
.F4F-F6F**&F:F-*$F?F-F-F-*&#F-F,F-*&F3F-F6F-F-F**&#F-F,F-*&F4F-F8F-F-F
**&F:F-*$FBF-F-F-/%#n4G,(F-F-F2F-F7F*/%#n5G,&*&F.F-F6F-F**&F.F-F4F-F-/
%#n6G,&*&F,F*F4F,F-*&F,F*F6F,F*/%#n7G,(F-F-*&F,F*F4F,F-*&F,F*F6F,F*/%#
n8GF)/%#n9G,.*&F:F-F]oF-F-*&F:F-F[oF-F-*&#F-F,F-FhnF-F**&F6F-)F.F,F-F*
*&#F-F,F-FenF-F**&F4F-FjpF-F-/F4F4/F)F)/F.F./F6F6" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 142 "indets(subs(rules1,`union`(allNrels,\{1-2*n
7+n4,m^2*n5+m*n12+w*m-2*n7*m*n12-n5+n7*n5,n5*m-n7*n12+n7*w,-3*m-u-2*n5
*t+u*n7+3*n7*m,-2*n10+u+m\})));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&%
\"tG%\"uG%\"vG%\"wG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "agb
:=factor(Basis(subs(rules1,`union`(allNrels,\{1-2*n7+n4,m^2*n5+m*n12+w
*m-2*n7*m*n12-n5+n7*n5,n5*m-n7*n12+n7*w,-3*m-u-2*n5*t+u*n7+3*n7*m,-2*n
10+u+m\})),plex( u, v, t,w )));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$
agbG7%,$*&,&%\"wG!\"\"%\"tG\"\"\"F,,2*$)F+\"\"$F,F,*&)F+\"\"#F,F)F,F,*
&F3F,F+F,F,*(F3F,F+F,)%\"vGF3F,F**&F+F,)F)F3F,F**(F3F,F)F,F6F,F,*&F3F,
F)F,F,*$)F)F0F,F*F,F**&F(F,,**(F0F,F7F,F)F,F**&F7F,F+F,F**&%\"uGF,F+F,
F,*&FCF,F)F,F,F,,,F3F,*$F2F,F,*$F9F,F**(F3F,FCF,F7F,F,*&\"\"%F,F6F,F*
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "M1s:=subs(rules1,M1); M
2s:=subs(rules1,M2); M3s:=subs(rules1,M3);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$M1sG-%'RTABLEG6%\"*WAEq\"-%'MATRIXG6#7'7'\"\"!\"\"\"
F.F.F.7'F/,0*$)%\"tG\"\"#F/F/*(F5F/F4F/%\"wGF/!\"\"*$)F7F5F/F/*&#F/F5F
/*$)F4\"\"%F/F/F/*&F7F/)F4\"\"$F/F8*&F4F/)F7FBF/F/*&#F/F5F/*$)F7F?F/F/
F8,(*&%\"vGF/F:F/F/**F5F/F7F/F4F/FKF/F8*&F3F/FKF/F/,.F4F/F7F8*&F<F/*$F
AF/F/F/*&#F/F5F/*&F3F/F7F/F/F8*&#F/F5F/*&F4F/F:F/F/F8*&F<F/*$FDF/F/F/F
N7'F.FI,(F/F/F2F/F9F8,&*&FKF/F7F/F8*&FKF/F4F/F/Fen7'F.FNFen,&*&F5F8F4F
5F/*&F5F8F7F5F8,(F/F/*&F5F8F4F5F/*&F5F8F7F5F87'F.FNFenF\\oFin%'MatrixG
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$M2sG-%'RTABLEG6%\"*g=Eq\"-%'MAT
RIXG6#7'7'\"\"!F.\"\"\"F.F.7'F.,(*&%\"vGF/)%\"wG\"\"#F/F/**F6F/F5F/%\"
tGF/F3F/!\"\"*&)F8F6F/F3F/F/,(F/F/*$F;F/F/*$F4F/F9,&*&F3F/F5F/F9*&F3F/
F8F/F/F?7'F/F<%\"uG,.*&#F/F6F/*$)F5\"\"$F/F/F/*&FFF/*&F8F/F4F/F/F/*&#F
/F6F/*&F;F/F5F/F/F9*&F5F/)F3F6F/F9*&#F/F6F/*$)F8FIF/F/F9*&F8F/FPF/F/FD
7'F.F?FDF3F3FV%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$M3sG-%'R
TABLEG6%\"*K4Eq\"-%'MATRIXG6#7'7'\"\"!F.F.F.\"\"\"7'F.,.%\"tGF/%\"wG!
\"\"*&#F/\"\"#F/*$)F2\"\"$F/F/F/*&#F/F7F/*&)F2F7F/F3F/F/F4*&#F/F7F/*&F
2F/)F3F7F/F/F4*&F6F/*$)F3F:F/F/F/,&*&%\"vGF/F3F/F4*&FHF/F2F/F/,(F/F/*&
F7F4F2F7F/*&F7F4F3F7F4,&*&F7F4F2F7F/*&F7F4F3F7F47'F.FF,.*&F6F/FDF/F/*&
F6F/FAF/F/*&#F/F7F/F=F/F4*&F3F/)FHF7F/F4*&#F/F7F/F8F/F4*&F2F/FWF/F/FHF
H7'F/FMFHF2F27'F.FJFHF3F2%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "The unique po
sitive character is psi0." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "psi0:=
Vector([1,d,f,g,g]): psi2:=subs(\{m=(d+1)/f\},intchar):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 296 "psirels:=numer(`minus`(`union`(con
vert(evalm(M1s&*psi0-psi0[2]*psi0),set),convert(evalm(M2s&*psi0-psi0[3
]*psi0),set),convert(evalm(M3s&*psi0-psi0[4]*psi0),set),convert(evalm(
M1s&*psi2-psi2[2]*psi2),set),convert(evalm(M2s&*psi2-psi2[3]*psi2),set
),convert(evalm(M3s&*psi2-psi2[4]*psi2),set)),\{0\}));" }}{PARA 12 "" 
1 "" {XPPMATH 20 "6#>%(psirelsG</,,*&\"\"#\"\"\"%\"fGF)F)*&)%\"tGF(F)F
*F)F)*&)%\"wGF(F)F*F)!\"\"*(F(F)%\"vGF)%\"dGF)F1*&F(F)F3F)F1,.F(F)*&F4
F)F,F)F)*&F4F)F/F)F1*(F(F)F3F)F*F)F)*(\"\"%F)F-F)%\"gGF)F)*&F(F))F<F(F
)F1,0*&F(F)F4F)F)F7F)F8F1*(F(F)F3F)F*F)F)*(F(F)F0F)F<F)F)*(F(F)F-F)F<F
)F)*&F(F)F>F)F1,0*(F*F)F3F)F/F)F)*,F(F)F*F)F0F)F-F)F3F)F1*(F*F)F,F)F3F
)F)F7F1*$F,F)F1F8F)*$F/F)F),2*&F(F))F*F(F)F)*&F,F)FMF)F)*&F/F)FMF)F1*(
%\"uGF)F*F)F4F)F1*&FQF)F*F)F1*$)F4F(F)F1*&F(F)F4F)F1F)F1,4*(F4F)F3F)F/
F)F)*,F(F)F4F)F0F)F-F)F3F)F1*(F4F)F,F)F3F)F)F*F)F+F)F.F1**F(F)F<F)F3F)
F0F)F1**F(F)F<F)F3F)F-F)F)*&F4F)F*F)F1,6*(F(F)F-F)F*F)F)*(F(F)F0F)F*F)
F1*&F*F))F-\"\"$F)F)*(F*F)F,F)F0F)F1*(F*F)F-F)F/F)F1*&F*F))F0F\\oF)F)*
*F(F)F4F)F3F)F0F)F)*(F(F)F3F)F0F)F)**F(F)F4F)F3F)F-F)F1*(F(F)F3F)F-F)F
1,6**F(F)F4F)F3F)F0F)F1**F(F)F4F)F3F)F-F)F)F_oF)F^oF)F]oF1**F(F)F*F)F0
F))F3F(F)F1FjnF1**F(F)F*F)F-F)FioF)F)*(F;F)F3F)F<F)F)*(F(F)F*F)F<F)F1,
:F)F)F4F)F7F)F8F1FRF)*&F<F)F`oF)F)*(F<F)F-F)F/F)F)*(F<F)F,F)F0F)F1**F(
F)F<F)F0F)FioF)F1*&F<F)F[oF)F1**F(F)F<F)F-F)FioF)F)*$FMF)F1,:*(F(F)F4F
)F-F)F)*(F(F)F4F)F0F)F1*&F[oF)F4F)F)*(F,F)F0F)F4F)F1*(F-F)F/F)F4F)F1*&
F`oF)F4F)F)**F(F)F3F)F0F)F*F)F1**F(F)F3F)F-F)F*F)F)*(F(F)F<F)F,F)F)*(F
(F)F<F)F/F)F1*&F(F)F<F)F)*(F(F)F4F)F<F)F1,<*(F(F)F,F)F*F)F)**F;F)F-F)F
0F)F*F)F1*(F(F)F/F)F*F)F)*&)F-F;F)F*F)F)**F(F)F0F)F[oF)F*F)F1**F(F)F-F
)F`oF)F*F)F)*&)F0F;F)F*F)F1**F(F)F4F)F,F)F3F)F1*(F(F)F,F)F3F)F1*,F;F)F
4F)F0F)F-F)F3F)F)**F;F)F0F)F-F)F3F)F)**F(F)F4F)F3F)F/F)F1*(F(F)F3F)F/F
)F1,>**F(F)F3F)F0F)F*F)F1**F(F)F3F)F-F)F*F)F)FhpF)*$F[oF)F)FipF)*&F,F)
F0F)F)FjpF1*&F-F)F/F)F1F[qF1*$F`oF)F1**F(F)F0F)FioF)F4F)F)*(F(F)F0F)Fi
oF)F)**F(F)F-F)FioF)F4F)F1*(F(F)F-F)FioF)F1,FF(F)*(F(F)F4F)F,F)F)**F;F
)F4F)F-F)F0F)F1*(F(F)F4F)F/F)F)*&F4F)FgqF)F)**F(F)F4F)F0F)F[oF)F1**F(F
)F4F)F-F)F`oF)F)*&F4F)F[rF)F1**F(F)F*F)F3F)F/F)F)*,F;F)F*F)F0F)F-F)F3F
)F1**F(F)F*F)F,F)F3F)F)*(F;F)F-F)F<F)F)*(F;F)F0F)F<F)F1*(F(F)F<F)F[oF)
F)**F(F)F<F)F,F)F0F)F1**F(F)F<F)F-F)F/F)F1*(F(F)F<F)F`oF)F)*&F(F)FTF)F
1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "later we will consider the c
ase d=1, f=2 and g=sqrt(3)." }{MPLTEXT 1 0 317 "psirels2:=subs(\{d=1,f
=2,g=sqrt(3)\},numer(`minus`(`union`(convert(evalm(M1&*psi0-psi0[2]*ps
i0),set),convert(evalm(M2&*psi0-psi0[3]*psi0),set),convert(evalm(M3&*p
si0-psi0[4]*psi0),set),convert(evalm(M1&*psi2-psi2[2]*psi2),set),conve
rt(evalm(M2&*psi2-psi2[3]*psi2),set),convert(evalm(M3&*psi2-psi2[4]*ps
i2),set)),\{0\})));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)psirels2G<0,
&*&\"\"#\"\"\"%#n1GF)F)*&F(F)%#n2GF)!\"\",&*&F(F)%#n3GF)F)*&F(F)%#n5GF
)F-,&*&\"\"%F)%#n4GF)F)*&F5F)%#n8GF)F-,&*&F(F)F2F)F)*&F(F)%#n9GF)F-,&*
&F(F)%#n7GF)F)*&F(F)%$n11GF)F-,(F(F)*&F(F)%#n6GF)F)*&F(F)%$n10GF)F-,(F
*F)*&F(F)F,F)F)*(F(F)F0F)\"\"$#F)F(F),(*&F(F)F,F)F)*&F(F)F6F)F-F(F),*F
JF-F6F)*&F(F)F8F)F)*(F(F)F<F)FJFKF),*F,F)*&F(F)F6F)F)*(F(F)F2F)FJFKF)F
(F-,,F(F-FDF)*&F(F)FFF)F)*&%$n12GF)FJFKF)*&%$n13GF)FJFKF),,F0F)*&F(F)F
2F)F)*&FDF)FJFKF)*&F?F)FJFKF)*$FJFKF-,,F2F)*&F(F)F<F)F)*&FFF)FJFKF)*&F
AF)FJFKF)*&F(F)FJFKF-,,F?F)*&F(F)FAF)F)*&%$n14GF)FJFKF)FWF)FJF-" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "indets(psirels2);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#<0%#n1G%$n10G%$n11G%$n12G%$n13G%$n14G%#n2G%#
n3G%#n4G%#n5G%#n6G%#n7G%#n8G%#n9G" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 99 "d1basis:=factor(Basis(psirels2,tdeg(n1, n10, n11, n12
, n13, n14, n2, n3, n4, n5, n6, n7, n8, n9)));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%(d1basisG7-,(\"\"$!\"\"*(\"\"#\"\"\"%#n9GF+F'#F+F*F+*
&F'F+%#n8GF+F+,*F+F(*&F,F+F'F-F+%#n7GF+%#n6GF+,&%#n5GF+F,F(,(F'F(*(F*F
+F,F+F'F-F+*&F'F+%#n4GF+F+,&F,F(%#n3GF+,&*(F*F+F,F+F'F-F+*&F'F+%#n2GF+
F+,,*&F*F+F'F-F+*&F'F+F,F+F(*(F*F+F2F+F'F-F(%$n14GF(%$n13GF+,*FDF+%$n1
2GF+*&F2F+F'F-F+*$F'F-F(,&F2F(%$n11GF+,*F*F(F1F+F2F+%$n10GF+,&*(F*F+F,
F+F'F-F+*&F'F+%#n1GF+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "
solve(convert(d1basis,set));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<0/%#n
1G,$**\"\"#\"\"\"\"\"$!\"\"%#n9GF)F*#F)F(F+/%$n10G,(F(F)*&F,F)F*F-F+%#
n7GF+/%$n11GF2/%$n12G,(%$n14GF+*&F2F)F*F-F+*$F*F-F)/%$n13G,**&F(F)F*F-
F+*&F*F)F,F)F)*(F(F)F2F)F*F-F)F8F)/F8F8/%#n2GF&/%#n3GF,/%#n4G,&F)F)**F
(F)F*F+F,F)F*F-F+/%#n5GF,/%#n6G,(F)F)F1F+F2F+/F2F2/%#n8GFH/F,F," }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "since these are integers, we see t
hat if d=1 we get the following rules:" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 86 "d1rules:=\{n1=0,n9=0,n8=1,n7=1,n10=1,n12=0,n14=0,n13=0,n2=0,n4
=1,n5=0,n6=0,n3=0,n11=1\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(d1rul
esG<0/%#n1G\"\"!/%$n10G\"\"\"/%$n11GF+/%$n12GF(/%$n13GF(/%$n14GF(/%#n2
GF(/%#n3GF(/%#n4GF+/%#n5GF(/%#n6GF(/%#n7GF+/%#n8GF+/%#n9GF(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "subs(d1rules,[M1,M2,M3]);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%-%'RTABLEG6%\"*G9y!=-%'MATRIXG6#7'7
'\"\"!\"\"\"F-F-F-7'F.F-F-F-F-7'F-F-F.F-F-7'F-F-F-F-F.7'F-F-F-F.F-%'Ma
trixG-F%6%\"*'\\')4=-F)6#7'F0F07'F.F.F.F-F-7'F-F-F-F.F.F;F3-F%6%\"*?SW
\"=-F)6#7'F1F2F;7'F.F-F.F-F-7'F-F.F.F-F-F3" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 19 "map(Eigenvalues,%);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#7%-%'RTABLEG6%\"*Sgg\"=-%'MATRIXG6#7'7#!\"\"F,7#\"\"\"F.F.&%'Vec
torG6#%'columnG-F%6%\"*?Nv#=-F)6#7'F,7#\"\"!F:7#\"\"#F<F0-F%6%\"*'pFG=
-F)6#7'F:7#*$\"\"$#F/F=7#,$FEF-7#^#F/7#^#F-F0" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 53 "We combine the nontrivial relations obtained so far. \+
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "SandNrels:=`union`(psirels,conv
ert(agb,set),orthrel);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%*SandNrels
G<3*&,&%\"wG!\"\"%\"tG\"\"\"F+,**(\"\"$F+%\"vGF+F(F+F)*&F/F+F*F+F)*&%
\"uGF+F*F+F+*&F2F+F(F+F+F+,$*&F'F+,2*$)F*F.F+F+*&)F*\"\"#F+F(F+F+*&F;F
+F*F+F+*(F;F+F*F+)F/F;F+F)*&F*F+)F(F;F+F)*(F;F+F(F+F>F+F+*&F;F+F(F+F+*
$)F(F.F+F)F+F),(*&F;F+%\"dGF+F+*$)%\"fGF;F+F+*&F;F+)%\"gGF;F+F),,F;F+*
$F:F+F+*$F@F+F)*(F;F+F2F+F/F+F+*&\"\"%F+F>F+F),,*&F;F+FJF+F+*&F:F+FJF+
F+*&F@F+FJF+F)*(F;F+F/F+FGF+F)*&F;F+F/F+F),.F;F+*&FGF+F:F+F+*&FGF+F@F+
F)*(F;F+F/F+FJF+F+*(FSF+F*F+FMF+F+*&F;F+FLF+F),0*&F;F+FGF+F+FenF+FfnF)
*(F;F+F/F+FJF+F+*(F;F+F(F+FMF+F+*(F;F+F*F+FMF+F+*&F;F+FLF+F),0*(FJF+F/
F+F@F+F+*,F;F+FJF+F(F+F*F+F/F+F)*(FJF+F:F+F/F+F+FenF)FOF)FfnF+FPF+,2*&
F;F+FIF+F+*&F:F+FIF+F+*&F@F+FIF+F)*(F2F+FJF+FGF+F)*&F2F+FJF+F)*$)FGF;F
+F)*&F;F+FGF+F)F+F),4*(FGF+F/F+F@F+F+*,F;F+FGF+F(F+F*F+F/F+F)*(FGF+F:F
+F/F+F+FJF+FVF+FWF)**F;F+FMF+F/F+F(F+F)**F;F+FMF+F/F+F*F+F+*&FGF+FJF+F
),6*(F;F+F*F+FJF+F+*(F;F+F(F+FJF+F)*&FJF+F8F+F+*(FJF+F:F+F(F+F)*(FJF+F
*F+F@F+F)*&FJF+FDF+F+**F;F+FGF+F/F+F(F+F+*(F;F+F/F+F(F+F+**F;F+FGF+F/F
+F*F+F)*(F;F+F/F+F*F+F),6**F;F+FGF+F/F+F(F+F)**F;F+FGF+F/F+F*F+F+FjpF+
FipF+FhpF)**F;F+FJF+F(F+F>F+F)FgpF)**F;F+FJF+F*F+F>F+F+*(FSF+F/F+FMF+F
+*(F;F+FJF+FMF+F),:F+F+FGF+FenF+FfnF)FioF+*&FMF+FDF+F+*(FMF+F*F+F@F+F+
*(FMF+F:F+F(F+F)**F;F+FMF+F(F+F>F+F)*&FMF+F8F+F)**F;F+FMF+F*F+F>F+F+FH
F),:*(F;F+FGF+F*F+F+*(F;F+FGF+F(F+F)*&F8F+FGF+F+*(F:F+F(F+FGF+F)*(F*F+
F@F+FGF+F)*&FDF+FGF+F+**F;F+F/F+F(F+FJF+F)**F;F+F/F+F*F+FJF+F+*(F;F+FM
F+F:F+F+*(F;F+FMF+F@F+F)*&F;F+FMF+F+*(F;F+FGF+FMF+F),<*(F;F+F:F+FJF+F+
**FSF+F*F+F(F+FJF+F)*(F;F+F@F+FJF+F+*&)F*FSF+FJF+F+**F;F+F(F+F8F+FJF+F
)**F;F+F*F+FDF+FJF+F+*&)F(FSF+FJF+F)**F;F+FGF+F:F+F/F+F)*(F;F+F:F+F/F+
F)*,FSF+FGF+F(F+F*F+F/F+F+**FSF+F(F+F*F+F/F+F+**F;F+FGF+F/F+F@F+F)*(F;
F+F/F+F@F+F),>**F;F+F/F+F(F+FJF+F)**F;F+F/F+F*F+FJF+F+F`rF+F7F+FarF+F9
F+FbrF)F?F)FcrF)FCF)**F;F+F(F+F>F+FGF+F+*(F;F+F(F+F>F+F+**F;F+F*F+F>F+
FGF+F)*(F;F+F*F+F>F+F),FF;F+*(F;F+FGF+F:F+F+**FSF+FGF+F*F+F(F+F)*(F;F+
FGF+F@F+F+*&FGF+F_sF+F+**F;F+FGF+F(F+F8F+F)**F;F+FGF+F*F+FDF+F+*&FGF+F
csF+F)**F;F+FJF+F/F+F@F+F+*,FSF+FJF+F(F+F*F+F/F+F)**F;F+FJF+F:F+F/F+F+
*(FSF+F*F+FMF+F+*(FSF+F(F+FMF+F)*(F;F+FMF+F8F+F+**F;F+FMF+F:F+F(F+F)**
F;F+FMF+F*F+F@F+F)*(F;F+FMF+FDF+F+*&F;F+F[pF+F)" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 18 "indets(SandNrels);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#<)%\"dG%\"fG%\"gG%\"tG%\"uG%\"vG%\"wG" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 62 "SandNbasis:=factor(Basis(SandNrels,plex(d
, f, g, u, v,t, w)));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+SandNbasis
G7.,$*&,&%\"wG!\"\"%\"tG\"\"\"F,,2*$)F+\"\"$F,F,*&)F+\"\"#F,F)F,F,*&F3
F,F+F,F,*(F3F,F+F,)%\"vGF3F,F**&F+F,)F)F3F,F**(F3F,F)F,F6F,F,*&F3F,F)F
,F,*$)F)F0F,F*F,F*,$*&F-F,,,*&F3F,F6F,F,F,F,*$F2F,F,*(F3F,F+F,F)F,F,*&
F0F,F9F,F*F,F*,6*(F3F,F+F,)F7F0F,F,*&F/F,F7F,F**(F3F,F)F,FGF,F**(F2F,F
)F,F7F,F**(F9F,F+F,F7F,F,*&F=F,F7F,F,*&%\"uGF,F+F,F,*(F0F,F7F,F+F,F**&
FNF,F)F,F,*(\"\"&F,F7F,F)F,F*,,F3F,FBF,*$F9F,F**(F3F,FNF,F7F,F,*&\"\"%
F,F6F,F*,$*&%\"gGF,F-F,F*,DF3F*FTF*FBF,*&F3F,)FZF3F,F,*&FZF,F=F,F**&FZ
F,F/F,F**(\"\"'F,F+F,FZF,F**(F3F,F)F,FZF,F**(F3F,F)F,F/F,F**(F3F,F+F,F
=F,F,*&F3F,)F)FWF,F,*(FWF,F6F,F9F,F***FWF,F)F,F+F,F6F,F,*&FWF,F6F,F**(
F3F,F2F,F9F,F**(FZF,F+F,F9F,F,*(FZF,F2F,F)F,F,,.*&F3F,%\"fGF,F,*&FWF,F
7F,F**&F9F,FioF,F**&F2F,FioF,F,**F3F,FZF,F7F,F+F,F***F3F,FZF,F7F,F)F,F
,,:*&F+F,FZF,F**&F)F,FZF,F**&F)F,F/F,F**&F+F,F=F,F,FTF**$F`oF,F,FBF,*(
F3F,F6F,F9F,F***F3F,F)F,F+F,F6F,F,*&F3F,F6F,F**&F2F,F9F,F**&F7F,FioF,F
,,<F3F,*&FNF,FioF,F,F`pF**(F0F,F)F,FZF,F**(F3F,F)F,F/F,F**(F3F,F+F,F=F
,F,*&F3F,F9F,F**&F3F,F`oF,F,*&F3F,F2F,F,*(FWF,F6F,F9F,F***FWF,F)F,F+F,
F6F,F,*&FWF,F6F,F**(F3F,F2F,F9F,F*,@*(F0F,F7F,F)F,F,*&F7F,F+F,F,FLF*FK
F*FJF,FHF,*(F3F,F)F,FGF,F,*(F3F,F+F,FGF,F**(F3F,F7F,FZF,F**(F7F,FZF,F9
F,F**,F3F,FZF,F7F,F+F,F)F,F,*(F2F,FZF,F7F,F**&F)F,FioF,F**&F+F,FioF,F*
*&FioF,FZF,F,,BFTF*FBF,*&FWF,F6F,F**&F3F,F`oF,F,*(F3F,F+F,F=F,F,*(F3F,
F2F,F9F,F**(FWF,F6F,F9F,F**(F3F,F)F,F/F,F**(FWF,F)F,FZF,F***FWF,F)F,F+
F,F6F,F,*(FWF,F+F,FZF,F*FhnF*FeoF,FfoF,FinF**$)FioF3F,F,,*F`pF*F,F*Fap
F,%\"dGF," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 179 "The next step is to introduce the relati
ons involving the roots of unity theta_i.  The first goal is to show t
hat th2 satisfies a nontrivial degree 3 (or less) polynomial in Q[d].
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "psith0:=Vector([1,d*th1
,f*th2,g*th3,g*th3]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 339 "T
hetarels:=simplify(subs(\{\},[th1^2*S[2,2]-(sum('M1[k,2]*psith0[k]',k=
1..5)),th1*th2*S[2,3]-(sum('M1[k,3]*psith0[k]',k=1..5)),th1*th3*S[2,4]
-(sum('M1[k,4]*psith0[k]',k=1..5)),th3*th2*S[4,3]-(sum('M2[k,4]*psith0
[k]',k=1..5)),th2^2*S[3,3]-(sum('M2[k,3]*psith0[k]',k=1..5)),th3^2*(S[
4,4]+S[4,5])-(sum('(M3[k,4]+M3[k,5])*psith0[k]',k=1..5))]));" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#>%*ThetarelsG7(,,*$)%$th1G\"\"#\"\"\"F+F+!\"
\"*(%#n1GF+%\"dGF+F)F+F,*(%#n2GF+%\"fGF+%$th2GF+F,**F*F+%#n3GF+%\"gGF+
%$th3GF+F,,**(F)F+F3F+F2F+F+*(F1F+F/F+F)F+F,*(%#n4GF+F2F+F3F+F,**F*F+%
#n5GF+F6F+F7F+F,,,*(F)F+F7F+F6F+F,*(F5F+F/F+F)F+F,*(F>F+F2F+F3F+F,*(%#
n6GF+F6F+F7F+F,*(%#n7GF+F6F+F7F+F,,**(F>F+F/F+F)F+F,*(%#n9GF+F2F+F3F+F
,*(%$n10GF+F6F+F7F+F,*(%$n11GF+F6F+F7F+F,,.*&)F3F*F+F/F+F,*$FQF+F,F+F,
*(F<F+F/F+F)F+F,*(%#n8GF+F2F+F3F+F,**F*F+FJF+F6F+F7F+F,,6*&)F7F*F+F/F+
F+*$FYF+F,F+F,*(F/F+F)F+FFF+F,*(F/F+F)F+FDF+F,*(F2F+F3F+FNF+F,*(F2F+F3
F+FLF+F,**F*F+F6F+F7F+%$n12GF+F,*(F6F+F7F+%$n13GF+F,*(F6F+F7F+%$n14GF+
F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 31 "we immediately find that th1=1." }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 107 "factor(Basis(`union`(SandNrels,convert(subs(rules
1,Thetarels),set)),tdeg(th1,u, v,t, w,d,th2,th3,f,g)))[7];" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#*(%\"gG\"\"\"%$th3GF%,&%$th1GF%F%!\"\"F%" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "Thetarels1:=factor(numer(su
bs(\{th1=1\},Thetarels)));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+Theta
rels1G7(,(*&%#n1G\"\"\"%\"dGF)!\"\"*(%#n2GF)%\"fGF)%$th2GF)F+**\"\"#F)
%#n3GF)%\"gGF)%$th3GF)F+,**&F.F)F/F)F)*&F-F)F*F)F+*(%#n4GF)F.F)F/F)F+*
*F1F)%#n5GF)F3F)F4F)F+,,*&F3F)F4F)F+*&F2F)F*F)F+*(F;F)F.F)F/F)F+*(%#n6
GF)F3F)F4F)F+*(%#n7GF)F3F)F4F)F+,**&F;F)F*F)F+*(%#n9GF)F.F)F/F)F+*(%$n
10GF)F3F)F4F)F+*(%$n11GF)F3F)F4F)F+,.*&)F/F1F)F*F)F+*$FNF)F+F)F+*&F9F)
F*F)F+*(%#n8GF)F.F)F/F)F+**F1F)FGF)F3F)F4F)F+,6*&)F4F1F)F*F)F)*$FVF)F+
F)F+*&F*F)FCF)F+*&F*F)FAF)F+*(F.F)F/F)FKF)F+*(F.F)F/F)FIF)F+**F1F)F3F)
F4F)%$n12GF)F+*(F3F)F4F)%$n13GF)F+*(F3F)F4F)%$n14GF)F+" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "indets(Thetarels1);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#<5%\"dG%\"fG%\"gG%#n1G%$n10G%$n11G%$n12G%$n13G%$n1
4G%#n2G%#n3G%#n4G%#n5G%#n6G%#n7G%#n8G%#n9G%$th2G%$th3G" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "map(degree,Thetarels1,th3);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"\"F$F$F$F$\"\"#" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "solve(Thetarels1[2],th3);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,$**\"\"#!\"\",(*&%\"fG\"\"\"%$th2GF*F&*&%#n
2GF*%\"dGF*F**(%#n4GF*F)F*F+F*F*F*%#n5GF&%\"gGF&F&" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 72 "This is valid unless  n5=0, which implies d=1 (wh
ich we consider below)." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "factor(B
asis(subs(rules1,`union`(SandNrels,\{n5\})),plex(u,t,v,w,f,g,d)));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7+*&,&%\"dG\"\"\"F'!\"\"F',&F&F'F'F'F'
,(*&\"\"#F'F&F'F'*$)%\"fGF,F'F'*&F,F')%\"gGF,F'F(,,*&F&F'F1F'F'*(\"\"%
F'%\"wGF'F2F'F'*$F1F'F(F,F(*&F,F'F&F'F',(*&F6F'%\"vGF'F'F/F(*&F&F'F/F'
F(,**&F&F'F2F'F'*&F,F'%\"tGF'F'*(F,F'F&F'F7F'F(F2F(*&%\"uGF'F%F',.*&F,
F'F/F'F'*(F,F'F&F'F/F'F'*&F/F'F1F'F(*(F&F'F/F'F1F'F(*&F,F'FDF'F(*(F,F'
F1F'FDF'F',,*&F,F'F&F'F'F8F(*&FDF'F/F'F'F4F(F,F',$*&F7F',(F=F'F/F'FDF(
F'F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 135 "We substitute back in to
 eliminate th3 and take at the numerators of the resulting rational fu
nctions, which give us our new relations." }{MPLTEXT 1 0 0 "" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 75 "Thetarels2:=factor(numer(subs(\{th3=solve
(Thetarels1[2],th3)\},Thetarels1)));" }}{PARA 12 "" 1 "" {XPPMATH 20 "
6#>%+Thetarels2G7(,,*(%#n1G\"\"\"%\"dGF)%#n5GF)!\"\"**%#n2GF)%\"fGF)%$
th2GF)F+F)F,*(%#n3GF)F/F)F0F)F,*(F2F)F.F)F*F)F)**F2F)%#n4GF)F/F)F0F)F)
\"\"!,8*&F/F)F0F)F,*&F.F)F*F)F)*(F5F)F/F)F0F)F)**\"\"#F)F2F)F*F)F+F)F,
**F<F))F+F<F)F/F)F0F)F,*(%#n6GF)F/F)F0F)F,*(F@F)F.F)F*F)F)**F@F)F5F)F/
F)F0F)F)*(%#n7GF)F/F)F0F)F,*(FDF)F.F)F*F)F)**FDF)F5F)F/F)F0F)F),2*(F<F
)F>F)F*F)F,*,F<F)%#n9GF)F/F)F0F)F+F)F,*(F/F)F0F)%$n10GF)F,*(FLF)F.F)F*
F)F)**FLF)F5F)F/F)F0F)F)*(F/F)F0F)%$n11GF)F,*(FPF)F.F)F*F)F)**FPF)F5F)
F/F)F0F)F),2*()F0F<F)F*F)F+F)F,*&FUF)F+F)F,F+F,*(F5F)F*F)F+F)F,**%#n8G
F)F/F)F0F)F+F)F,*(FJF)F/F)F0F)F,*(FJF)F.F)F*F)F)**FJF)F5F)F/F)F0F)F),V
*0\"\"%F)%$n12GF)F+F))%\"gGF<F)F5F)F/F)F0F)F)*0F<F)%$n13GF)F+F)F[oF)F5
F)F/F)F0F)F)*0F<F)%$n14GF)F+F)F[oF)F5F)F/F)F0F)F)**F<F))F/F<F)FUF)F5F)
F)*()F5F<F)FboF)FUF)F,*(F*F)FboF)FUF)F)*.FinF)F/F)F0F)FLF)F>F)F[oF)F,*
.FinF)F/F)F0F)FPF)F>F)F[oF)F,*.F<F)F.F)F*F)F5F)F/F)F0F)F,*.F<F)F.F))F*
F<F)F5F)F/F)F0F)F)*.FinF)FjnF)F+F)F[oF)F/F)F0F)F,*.FinF)FjnF)F+F)F[oF)
F.F)F*F)F)*.F<F)F^oF)F+F)F[oF)F/F)F0F)F,*.F<F)F^oF)F+F)F[oF)F.F)F*F)F)
*.F<F)F`oF)F+F)F[oF)F/F)F0F)F,*.F<F)F`oF)F+F)F[oF)F.F)F*F)F)*&FboF)FUF
)F,*&)F.F<F)FjoF)F,*(FinF)F>F)F[oF)F,*,FinF)F*F)FDF)F>F)F[oF)F,*,FinF)
F*F)F@F)F>F)F[oF)F,*,F<F)F/F)F0F)F.F)F*F)F)*,F<F)F/F)F0F)F.F)FjoF)F,*,
F<F)F*F)FboF)FUF)F5F)F,**F*F)FdoF)FboF)FUF)F)*&FcpF))F*\"\"$F)F)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "Thetarels2red:=factor(Reduce
(subs(rules1,Thetarels2),SandNbasis,plex(d, f, g, u, v,t, w,th2))):" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 30 "map(degree,Thetarels2red,th2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#7(,$%)infinityG!\"\"F$F$F$\"\"#F'" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 100 "factor([coeff(Thetarels2red[5],th2,2),coeff(Thetarel
s2red[5],th2,1),coeff(Thetarels2red[5],th2,0)]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7%,$**\"\"#\"\"\"%\"vGF',&%\"wG!\"\"%\"tGF'F',(*&F,F'%
\"gGF'F'F&F'*&F*F'F/F'F+F'F+,<*(\"\"%F')F,\"\"$F'F(F'F'*(\"#7F'F(F'F,F
'F'**F3F')F,F&F'F*F'F(F'F'**F&F'F9F'F/F'F(F'F'*(\"\")F'F,F')F(F5F'F+**
F3F')F*F&F'F,F'F(F'F+*(F<F'F*F'F=F'F'*(F3F')F*F5F'F(F'F+*,F3F'F/F'F(F'
F,F'F*F'F+*(F3F'F,F'%\"fGF'F+**F&F'F(F'F/F'F?F'F'*(F3F'F(F'F*F'F'*(F3F
'F*F'FEF'F+F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 192 "provided v(t-w)
 is not zero, th2 satisfies a nonzero degree 2 polynomial (observe tha
t g cannot be rational). The following shows that if v(t-w)=0, then d=
1, which we will consider separately." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 64 "factor(Basis(`union`(SandNrels,\{v*(t-w)\}),plex(f,g,
u,v,t,w,d)));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#75*&,&%\"dG\"\"\"F'!
\"\"F',&F&F'F'F'F'*&,&%\"wGF(%\"tGF'F'F)F',**$)F,\"\"#F'F(*$)F-F1F'F'F
&F(F'F'*&%\"vGF'F%F'*&F5F'F+F'*&F,F',(*&\"\"%F')F5F1F'F'F&F'F'F'F'*&%
\"uGF'F%F'*&F,F',&F=F'*&F:F'F5F'F(F',&*&F=F'F-F'F'*(F:F'F5F'F,F'F(,**(
F1F'F=F'F5F'F'F'F'*&F:F'F;F'F(F&F',**&F&F'%\"gGF'F'*&F1F'F-F'F'*(F1F'F
&F'F,F'F(FIF(*&F,F',(*&F&F'F,F'F'FIF'F-F(F',,F'F'F&F(*&F-F'FIF'F'*&F&F
'F0F'F'*&F-F'F,F'F(,.F'F'*&\"\"$F'F&F'F(*&F1F')FIF1F'F'*&F:F'F;F'F(*(F
:F'F&F'F0F'F'*(F:F'F-F'F,F'F(,(%\"fGF'*&F&F'FfnF'F'*&F:F'F5F'F(*&F5F',
&*&F1F'F5F'F(FfnF'F',**&F=F'FfnF'F'F'F'*&F:F'F;F'F(F&F',,*&FfnF'FIF'F'
*(F1F'F5F'FIF'F(*&F-F'FfnF'F(*&F,F'FfnF'F(*(F:F'F5F'F,F'F',.F'F'F&F(*(
F:F'F-F'F,F'F(*&F:F'F;F'F(*(F:F'F&F'F0F'F'*$)FfnF1F'F'" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "ThetaElim:=subs(rules1,`union`(conv
ert(Thetarels,set),\{th1-1\}));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%*
ThetaElimG<),&%$th1G\"\"\"F(!\"\",(*(,&*&%\"vGF(%\"wGF(F)*&F.F(%\"tGF(
F(F(%\"dGF(F'F(F)*(,.*&#F(\"\"#F(*$)F/\"\"$F(F(F(*&F6F(*&F1F()F/F7F(F(
F(*&#F(F7F(*&)F1F7F(F/F(F(F)*&F/F()F.F7F(F)*&#F(F7F(*$)F1F:F(F(F)*&F1F
(FCF(F(F(%\"fGF(%$th2GF(F)**F7F(F.F(%\"gGF(%$th3GF(F),**(F'F(FJF(FIF(F
(*(,(*&F.F(F=F(F(**F7F(F/F(F1F(F.F(F)*&FAF(F.F(F(F(F2F(F'F(F)*(,(F(F(*
$FAF(F(*$F=F(F)F(FIF(FJF(F)**F7F(F,F(FLF(FMF(F),,*$)F'F7F(F(F(F)*(,0FW
F(*(F7F(F1F(F/F(F)FXF(*&F6F(*$)F1\"\"%F(F(F(*&F/F(FGF(F)*&F1F(F9F(F(*&
#F(F7F(*$)F/F]oF(F(F)F(F2F(F'F(F)*(FQF(FIF(FJF(F)**F7F(,.F1F(F/F)*&F6F
(FFF(F(*&#F(F7F(F@F(F)*&#F(F7F(F<F(F)*&F6F(F8F(F(F(FLF(FMF(F),,*(F'F(F
MF(FLF(F)*(FfoF(F2F(F'F(F)*(F,F(FIF(FJF(F)*(,&*&F7F)F1F7F(*&F7F)F/F7F)
F(FLF(FMF(F)*(,(F(F(*&F7F)F1F7F(*&F7F)F/F7F)F(FLF(FMF(F),.*&)FJF7F(F2F
(F)*$F[qF(F)F(F)*(FVF(F2F(F'F(F)*(%\"uGF(FIF(FJF(F)**F7F(F4F(FLF(FMF(F
),2*&)FMF7F(F2F(F(*$FcqF(F)F(F)*(F2F(F'F(FfpF(F)*(F2F(F'F(FbpF(F)**F7F
(FIF(FJF(F.F(F)**F:F(FLF(FMF(F1F(F)*(FLF(FMF(F/F(F)" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 18 "indets(ThetaElim);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#<,%\"dG%\"fG%\"gG%\"tG%$th1G%$th2G%$th3G%\"uG%\"vG%\"wG
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "indets(allNrels2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#<1%\"mG%#n1G%$n10G%$n12G%#n2G%#n3G%#n4
G%#n5G%#n6G%#n7G%#n9G%\"tG%\"uG%\"vG%\"wG" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 38 "first suppose d=1.  This forces th3=0." }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 99 "factor(Basis(subs(`union`(\{d=1,f=2,g=sqrt(3)\},d1
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" {XPPMATH 20 "6#7%%$th2G%$th3G,&%$th1G\"\"\"F(F(" }}}{EXCHG {PARA 0 "
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{TEXT -1 79 "Since [Q[d,f,g]:Q]=2, th2 satisfies a cyclotomic polynomi
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s,ThetaElim,\{cyclotomic(i,th2)\}),plex(d,f,g,th1,th3,th2,u,v,t,w)))[1
];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'bases1Gf*6#%\"iG6\"6$%)operat
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"" 1 "" {XPPMATH 20 "6#7&*&,,\"&,A#\"\"\"*&\"&3!>F')%\"wG\"\"'F'!\"\"*
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