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\f0\fs24 \cf0 #The following code checks that the intersection matrix of the roots\
#in the Coxeter diagram obtained by Sterk for cusp 3 has rank 10.\
\
#The following vector records the self intersections of the roots.\
SelfIntersections=[-4,-4,-4,-4,-2,-4,-4,-2,-4,-4,-4,-4]\
\
#The following matrix records the miltiplicities of the edges of the Coxeter diagram.\
var('infinity')\
EdgesCoxeterDiagram=[\
    [0,1,0,0,0,0,0,0,0,0,1,1],\
    [1,0,1,0,0,0,0,0,0,0,0,0],\
    [0,1,0,1,0,0,0,0,0,0,0,0],\
    [0,0,1,0,2,0,0,0,0,0,0,0],\
    [0,0,0,2,0,2,0,0,0,0,0,0],\
    [0,0,0,0,2,0,infinity,0,0,0,0,0],\
    [0,0,0,0,0,infinity,0,2,0,0,0,0],\
    [0,0,0,0,0,0,2,0,2,0,0,0],\
    [0,0,0,0,0,0,0,2,0,1,0,0],\
    [0,0,0,0,0,0,0,0,1,0,1,0],\
    [1,0,0,0,0,0,0,0,0,1,0,0],\
    [1,0,0,0,0,0,0,0,0,0,0,0]\
]\
\
#We now are ready to form the intersection matrix of the roots of the Coxeter diagram\
RootsIntersectionMatrix=[]\
for i in range(0,12):\
    row=[]\
    for j in range(0,12):\
        if i==j:\
            number=SelfIntersections[i]\
        if i!=j:\
            edgemult=EdgesCoxeterDiagram[i][j]\
            visquared=SelfIntersections[i]\
            vjsquared=SelfIntersections[j]\
            if edgemult!=infinity:\
                number=sqrt(visquared*vjsquared)*cos(pi/(edgemult+2))\
            if edgemult==infinity:\
                number=sqrt(visquared*vjsquared)\
        row.append(number)\
    RootsIntersectionMatrix.append(row)\
\
#We now compute the rank.\
matrix(RootsIntersectionMatrix).rank()}