//This is the ideal Fourier invariants.

ring rQ = 0,(q1,q2,q3,q4,q5,q6,q7,q8,q9,q10,q11,q12,q13,q14,q15,q16,q17,q18,q19,q20,q21,q22,q23,q24,q25,q26,q27),dp;

ideal Invariants =  
q23*q25-q22*q26,
q23*q25-q21*q27,
q19*q25-q15*q27,
q19*q25-q18*q26,
q12*q25-q8*q27,
q12*q25-q11*q26,
q20*q24-q23*q25,
q22*q23-q24*q27,
q21*q23-q24*q26,
q20*q23-q26*q27,
q17*q23-q19*q24,
q16*q23-q19*q27,
q14*q23-q19*q26,
q13*q23-q19*q25,
q10*q23-q12*q24,
q9*q23-q12*q27,
q7*q23-q12*q26,
q6*q23-q12*q25,
q21*q22-q24*q25,
q20*q22-q25*q27,
q19*q22-q17*q27,
q19*q22-q16*q24,
q19*q22-q18*q23,
q17*q22-q18*q24,
q16*q22-q18*q27,
q14*q22-q19*q25,
q13*q22-q18*q25,
q12*q22-q10*q27,
q12*q22-q9*q24,
q12*q22-q11*q23,
q10*q22-q11*q24,
q9*q22-q11*q27,
q7*q22-q12*q25,
q6*q22-q11*q25,
q20*q21-q25*q26,
q19*q21-q17*q26,
q19*q21-q14*q24,
q19*q21-q15*q23,
q18*q21-q17*q25,
q18*q21-q13*q24,
q18*q21-q15*q22,
q17*q21-q15*q24,
q16*q21-q19*q25,
q14*q21-q15*q26,
q13*q21-q15*q25,
q12*q21-q10*q26,
q12*q21-q7*q24,
q12*q21-q8*q23,
q11*q21-q10*q25,
q11*q21-q6*q24,
q11*q21-q8*q22,
q10*q21-q8*q24,
q9*q21-q12*q25,
q7*q21-q8*q26,
q6*q21-q8*q25,
q19*q20-q14*q27,
q19*q20-q16*q26,
q18*q20-q13*q27,
q18*q20-q16*q25,
q17*q20-q19*q25,
q15*q20-q13*q26,
q15*q20-q14*q25,
q12*q20-q7*q27,
q12*q20-q9*q26,
q11*q20-q6*q27,
q11*q20-q9*q25,
q10*q20-q12*q25,
q8*q20-q6*q26,
q8*q20-q7*q25,
q12*q19-q4*q23,
q12*q18-q11*q19,
q12*q18-q5*q27,
q12*q18-q3*q23,
q12*q18-q4*q22,
q11*q18-q3*q22,
q16*q17-q18*q19,
q14*q17-q15*q19,
q13*q17-q15*q18,
q12*q17-q10*q19,
q12*q17-q4*q24,
q12*q17-q5*q23,
q11*q17-q10*q18,
q11*q17-q3*q24,
q11*q17-q5*q22,
q10*q17-q5*q24,
q9*q17-q12*q18,
q4*q17-q5*q19,
q3*q17-q5*q18,
q15*q16-q14*q18,
q15*q16-q13*q19,
q12*q16-q9*q19,
q12*q16-q4*q27,
q11*q16-q9*q18,
q11*q16-q3*q27,
q10*q16-q12*q18,
q5*q16-q4*q18,
q5*q16-q3*q19,
q12*q15-q8*q19,
q12*q15-q7*q17,
q12*q15-q5*q26,
q12*q15-q2*q23,
q12*q15-q4*q21,
q11*q15-q8*q18,
q11*q15-q6*q17,
q11*q15-q5*q25,
q11*q15-q2*q22,
q11*q15-q3*q21,
q10*q15-q8*q17,
q10*q15-q2*q24,
q10*q15-q5*q21,
q8*q15-q2*q21,
q5*q15-q2*q17,
q12*q14-q7*q19,
q12*q14-q4*q26,
q10*q14-q12*q15,
q9*q14-q7*q16,
q9*q14-q1*q23,
q9*q14-q4*q20,
q8*q14-q7*q15,
q8*q14-q2*q26,
q5*q14-q4*q15,
q5*q14-q2*q19,
q12*q13-q11*q14,
q12*q13-q9*q15,
q12*q13-q8*q16,
q12*q13-q7*q18,
q12*q13-q6*q19,
q12*q13-q2*q27,
q12*q13-q3*q26,
q12*q13-q4*q25,
q12*q13-q1*q24,
q12*q13-q5*q20,
q11*q13-q6*q18,
q11*q13-q3*q25,
q10*q13-q11*q15,
q9*q13-q6*q16,
q9*q13-q1*q22,
q9*q13-q3*q20,
q8*q13-q6*q15,
q8*q13-q2*q25,
q7*q13-q6*q14,
q7*q13-q1*q21,
q7*q13-q2*q20,
q5*q13-q3*q15,
q5*q13-q2*q18,
q4*q13-q3*q14,
q4*q13-q2*q16,
q4*q13-q1*q17,
q9*q10-q11*q12,
q7*q10-q8*q12,
q6*q10-q8*q11,
q4*q10-q5*q12,
q3*q10-q5*q11,
q8*q9-q7*q11,
q8*q9-q6*q12,
q5*q9-q4*q11,
q5*q9-q3*q12,
q5*q8-q2*q10,
q5*q7-q4*q8,
q5*q7-q2*q12,
q5*q6-q3*q8,
q5*q6-q2*q11,
q4*q6-q3*q7,
q4*q6-q2*q9,
q4*q6-q1*q10,
q9*q16*q20-q1*q27^2,
q7*q14*q20-q1*q26^2,
q6*q13*q20-q1*q25^2,
q4*q14*q16-q1*q19^2,
q3*q13*q16-q1*q18^2,
q2*q13*q14-q1*q15^2,
q4*q7*q9-q1*q12^2,
q3*q6*q9-q1*q11^2,
q2*q6*q7-q1*q8^2,
q2*q3*q4-q1*q5^2;

// This is the inverse of the Fourier transform.

matrix ptoq[51][27] =
1/256,3/256,3/256,3/256,3/128,3/256,3/256,3/128,3/256,21/256,3/128,3/128,3/256,3/256,3/128,3/256,21/256,3/128,3/128,3/256,21/256,21/256,21/256,15/64,3/128,3/128,3/128,
3/256,-3/256,-3/256,9/256,-3/128,-3/256,9/256,-3/128,9/256,-21/256,-3/128,9/128,-3/256,9/256,-3/128,9/256,-21/256,-3/128,9/128,9/256,-21/256,-21/256,63/256,-15/64,-3/128,9/128,9/128,
3/256,-3/256,9/256,-3/256,-3/128,9/256,-3/256,-3/128,9/256,-21/256,9/128,-3/128,9/256,-3/256,-3/128,9/256,-21/256,9/128,-3/128,9/256,-21/256,63/256,-21/256,-15/64,9/128,-3/128,9/128,
3/256,9/256,-3/256,-3/256,-3/128,-3/256,-3/256,-3/128,9/256,15/256,-3/128,-3/128,-3/256,-3/256,-3/128,9/256,15/256,-3/128,-3/128,9/256,15/256,-21/256,-21/256,3/64,-3/128,-3/128,9/128,
3/128,-3/128,-3/128,-3/128,3/64,-3/128,-3/128,3/64,9/128,3/128,-3/64,-3/64,-3/128,-3/128,3/64,9/128,3/128,-3/64,-3/64,9/128,3/128,-21/128,-21/128,3/16,-3/64,-3/64,9/64,
3/256,9/256,-3/256,-3/256,-3/128,9/256,9/256,9/128,-3/256,-21/256,-3/128,-3/128,9/256,9/256,9/128,-3/256,-21/256,-3/128,-3/128,9/256,63/256,-21/256,-21/256,-15/64,9/128,9/128,-3/128,
3/256,-3/256,9/256,-3/256,-3/128,-3/256,9/256,-3/128,-3/256,15/256,-3/128,-3/128,-3/256,9/256,-3/128,-3/256,15/256,-3/128,-3/128,9/256,-21/256,15/256,-21/256,3/64,-3/128,9/128,-3/128,
3/128,-3/128,-3/128,-3/128,3/64,-3/128,9/128,-3/64,-3/128,3/128,3/64,-3/64,-3/128,9/128,-3/64,-3/128,3/128,3/64,-3/64,9/128,-21/128,3/128,-21/128,3/16,-3/64,9/64,-3/64,
3/256,-3/256,-3/256,9/256,-3/128,9/256,-3/256,-3/128,-3/256,15/256,-3/128,-3/128,9/256,-3/256,-3/128,-3/256,15/256,-3/128,-3/128,9/256,-21/256,-21/256,15/256,3/64,9/128,-3/128,-3/128,
3/256,9/256,9/256,9/256,9/128,-3/256,-3/256,-3/128,-3/256,-21/256,-3/128,-3/128,-3/256,-3/256,-3/128,-3/256,-21/256,-3/128,-3/128,9/256,15/256,15/256,15/256,3/64,-3/128,-3/128,-3/128,
3/128,-3/128,-3/128,9/128,-3/64,-3/128,-3/128,3/64,-3/128,3/128,3/64,-3/64,-3/128,-3/128,3/64,-3/128,3/128,3/64,-3/64,9/128,3/128,3/128,15/128,-3/32,-3/64,-3/64,-3/64,
3/128,-3/128,-3/128,-3/128,3/64,9/128,-3/128,-3/64,-3/128,3/128,-3/64,3/64,9/128,-3/128,-3/64,-3/128,3/128,-3/64,3/64,9/128,-21/128,-21/128,3/128,3/16,9/64,-3/64,-3/64,
3/128,-3/128,9/128,-3/128,-3/64,-3/128,-3/128,3/64,-3/128,3/128,-3/64,3/64,-3/128,-3/128,3/64,-3/128,3/128,-3/64,3/64,9/128,3/128,15/128,3/128,-3/32,-3/64,-3/64,-3/64,
3/128,9/128,-3/128,-3/128,-3/64,-3/128,-3/128,-3/64,-3/128,3/128,3/64,3/64,-3/128,-3/128,-3/64,-3/128,3/128,3/64,3/64,9/128,15/128,3/128,3/128,-3/32,-3/64,-3/64,-3/64,
3/128,-3/128,-3/128,-3/128,3/64,-3/128,-3/128,3/64,-3/128,-9/128,3/64,3/64,-3/128,-3/128,3/64,-3/128,-9/128,3/64,3/64,9/128,3/128,3/128,3/128,0,-3/64,-3/64,-3/64,
3/256,9/256,9/256,9/256,9/128,-3/256,-3/256,-3/128,-3/256,-21/256,-3/128,-3/128,9/256,9/256,9/128,9/256,63/256,9/128,9/128,-3/256,-21/256,-21/256,-21/256,-15/64,-3/128,-3/128,-3/128,
3/256,-3/256,-3/256,9/256,-3/128,9/256,-3/256,-3/128,-3/256,15/256,-3/128,-3/128,-3/256,9/256,-3/128,9/256,-21/256,-3/128,9/128,-3/256,15/256,15/256,-21/256,3/64,-3/128,-3/128,-3/128,
3/128,-3/128,-3/128,9/128,-3/64,-3/128,-3/128,3/64,-3/128,3/128,3/64,-3/64,-3/128,9/128,-3/64,9/128,-21/128,-3/64,9/64,-3/128,3/128,3/128,-21/128,3/16,3/64,-3/64,-3/64,
3/256,-3/256,9/256,-3/256,-3/128,-3/256,9/256,-3/128,-3/256,15/256,-3/128,-3/128,9/256,-3/256,-3/128,9/256,-21/256,9/128,-3/128,-3/256,15/256,-21/256,15/256,3/64,-3/128,-3/128,-3/128,
3/256,9/256,-3/256,-3/256,-3/128,9/256,9/256,9/128,-3/256,-21/256,-3/128,-3/128,-3/256,-3/256,-3/128,9/256,15/256,-3/128,-3/128,-3/256,-21/256,15/256,15/256,3/64,-3/128,-3/128,-3/128,
3/128,-3/128,-3/128,-3/128,3/64,-3/128,9/128,-3/64,-3/128,3/128,3/64,-3/64,-3/128,-3/128,3/64,9/128,3/128,-3/64,-3/64,-3/128,3/128,3/128,15/128,-3/32,3/64,-3/64,-3/64,
3/128,-3/128,9/128,-3/128,-3/64,-3/128,-3/128,3/64,-3/128,3/128,-3/64,3/64,9/128,-3/128,-3/64,9/128,-21/128,9/64,-3/64,-3/128,3/128,-21/128,3/128,3/16,-3/64,3/64,-3/64,
3/128,-3/128,-3/128,-3/128,3/64,9/128,-3/128,-3/64,-3/128,3/128,-3/64,3/64,-3/128,-3/128,3/64,9/128,3/128,-3/64,-3/64,-3/128,3/128,15/128,3/128,-3/32,-3/64,3/64,-3/64,
3/128,9/128,-3/128,-3/128,-3/64,-3/128,-3/128,-3/64,-3/128,3/128,3/64,3/64,-3/128,-3/128,-3/64,9/128,15/128,-3/64,-3/64,-3/128,3/128,3/128,3/128,-3/32,3/64,3/64,-3/64,
3/128,-3/128,-3/128,-3/128,3/64,-3/128,-3/128,3/64,-3/128,-9/128,3/64,3/64,-3/128,-3/128,3/64,9/128,3/128,-3/64,-3/64,-3/128,-9/128,3/128,3/128,0,3/64,3/64,-3/64,
3/256,9/256,-3/256,-3/256,-3/128,-3/256,-3/256,-3/128,9/256,15/256,-3/128,-3/128,9/256,9/256,9/128,-3/256,-21/256,-3/128,-3/128,-3/256,-21/256,15/256,15/256,3/64,-3/128,-3/128,-3/128,
3/256,-3/256,9/256,-3/256,-3/128,9/256,-3/256,-3/128,9/256,-21/256,9/128,-3/128,-3/256,9/256,-3/128,-3/256,15/256,-3/128,-3/128,-3/256,15/256,-21/256,15/256,3/64,-3/128,-3/128,-3/128,
3/128,-3/128,-3/128,-3/128,3/64,-3/128,-3/128,3/64,9/128,3/128,-3/64,-3/64,-3/128,9/128,-3/64,-3/128,3/128,3/64,-3/64,-3/128,3/128,3/128,15/128,-3/32,3/64,-3/64,-3/64,
3/256,-3/256,-3/256,9/256,-3/128,-3/256,9/256,-3/128,9/256,-21/256,-3/128,9/128,9/256,-3/256,-3/128,-3/256,15/256,-3/128,-3/128,-3/256,15/256,15/256,-21/256,3/64,-3/128,-3/128,-3/128,
3/256,9/256,9/256,9/256,9/128,9/256,9/256,9/128,9/256,63/256,9/128,9/128,-3/256,-3/256,-3/128,-3/256,-21/256,-3/128,-3/128,-3/256,-21/256,-21/256,-21/256,-15/64,-3/128,-3/128,-3/128,
3/128,-3/128,-3/128,9/128,-3/64,-3/128,9/128,-3/64,9/128,-21/128,-3/64,9/64,-3/128,-3/128,3/64,-3/128,3/128,3/64,-3/64,-3/128,3/128,3/128,-21/128,3/16,3/64,-3/64,-3/64,
3/128,-3/128,-3/128,-3/128,3/64,-3/128,-3/128,3/64,9/128,3/128,-3/64,-3/64,9/128,-3/128,-3/64,-3/128,3/128,-3/64,3/64,-3/128,3/128,15/128,3/128,-3/32,-3/64,3/64,-3/64,
3/128,-3/128,9/128,-3/128,-3/64,9/128,-3/128,-3/64,9/128,-21/128,9/64,-3/64,-3/128,-3/128,3/64,-3/128,3/128,-3/64,3/64,-3/128,3/128,-21/128,3/128,3/16,-3/64,3/64,-3/64,
3/128,9/128,-3/128,-3/128,-3/64,-3/128,-3/128,-3/64,9/128,15/128,-3/64,-3/64,-3/128,-3/128,-3/64,-3/128,3/128,3/64,3/64,-3/128,3/128,3/128,3/128,-3/32,3/64,3/64,-3/64,
3/128,-3/128,-3/128,-3/128,3/64,-3/128,-3/128,3/64,9/128,3/128,-3/64,-3/64,-3/128,-3/128,3/64,-3/128,-9/128,3/64,3/64,-3/128,-9/128,3/128,3/128,0,3/64,3/64,-3/64,
3/128,9/128,-3/128,-3/128,-3/64,-3/128,-3/128,-3/64,-3/128,3/128,3/64,3/64,9/128,9/128,9/64,-3/128,-21/128,-3/64,-3/64,-3/128,-21/128,3/128,3/128,3/16,-3/64,-3/64,3/64,
3/128,-3/128,-3/128,-3/128,3/64,9/128,-3/128,-3/64,-3/128,3/128,-3/64,3/64,-3/128,9/128,-3/64,-3/128,3/128,3/64,-3/64,-3/128,15/128,3/128,3/128,-3/32,-3/64,-3/64,3/64,
3/128,-3/128,9/128,-3/128,-3/64,-3/128,-3/128,3/64,-3/128,3/128,-3/64,3/64,-3/128,9/128,-3/64,-3/128,15/128,-3/64,-3/64,-3/128,3/128,3/128,3/128,-3/32,3/64,-3/64,3/64,
3/128,-3/128,-3/128,-3/128,3/64,-3/128,-3/128,3/64,-3/128,-9/128,3/64,3/64,-3/128,9/128,-3/64,-3/128,3/128,3/64,-3/64,-3/128,3/128,-9/128,3/128,0,3/64,-3/64,3/64,
3/128,-3/128,-3/128,-3/128,3/64,-3/128,9/128,-3/64,-3/128,3/128,3/64,-3/64,9/128,-3/128,-3/64,-3/128,3/128,-3/64,3/64,-3/128,15/128,3/128,3/128,-3/32,-3/64,-3/64,3/64,
3/128,9/128,-3/128,-3/128,-3/64,9/128,9/128,9/64,-3/128,-21/128,-3/64,-3/64,-3/128,-3/128,-3/64,-3/128,3/128,3/64,3/64,-3/128,-21/128,3/128,3/128,3/16,-3/64,-3/64,3/64,
3/128,-3/128,9/128,-3/128,-3/64,-3/128,9/128,-3/64,-3/128,15/128,-3/64,-3/64,-3/128,-3/128,3/64,-3/128,3/128,-3/64,3/64,-3/128,3/128,3/128,3/128,-3/32,3/64,-3/64,3/64,
3/128,-3/128,-3/128,-3/128,3/64,-3/128,9/128,-3/64,-3/128,3/128,3/64,-3/64,-3/128,-3/128,3/64,-3/128,-9/128,3/64,3/64,-3/128,3/128,-9/128,3/128,0,3/64,-3/64,3/64,
3/128,-3/128,-3/128,9/128,-3/64,-3/128,-3/128,3/64,-3/128,3/128,3/64,-3/64,9/128,-3/128,-3/64,-3/128,15/128,-3/64,-3/64,-3/128,3/128,3/128,3/128,-3/32,-3/64,3/64,3/64,
3/128,-3/128,-3/128,9/128,-3/64,9/128,-3/128,-3/64,-3/128,15/128,-3/64,-3/64,-3/128,-3/128,3/64,-3/128,3/128,3/64,-3/64,-3/128,3/128,3/128,3/128,-3/32,-3/64,3/64,3/64,
3/128,9/128,9/128,9/128,9/64,-3/128,-3/128,-3/64,-3/128,-21/128,-3/64,-3/64,-3/128,-3/128,-3/64,-3/128,-21/128,-3/64,-3/64,-3/128,3/128,3/128,3/128,3/16,3/64,3/64,3/64,
3/128,-3/128,-3/128,9/128,-3/64,-3/128,-3/128,3/64,-3/128,3/128,3/64,-3/64,-3/128,-3/128,3/64,-3/128,3/128,3/64,-3/64,-3/128,-9/128,-9/128,3/128,0,3/64,3/64,3/64,
3/128,-3/128,-3/128,-3/128,3/64,-3/128,-3/128,3/64,-3/128,-9/128,3/64,3/64,9/128,-3/128,-3/64,-3/128,3/128,-3/64,3/64,-3/128,3/128,3/128,-9/128,0,-3/64,3/64,3/64,
3/128,-3/128,-3/128,-3/128,3/64,9/128,-3/128,-3/64,-3/128,3/128,-3/64,3/64,-3/128,-3/128,3/64,-3/128,-9/128,3/64,3/64,-3/128,3/128,3/128,-9/128,0,-3/64,3/64,3/64,
3/128,-3/128,9/128,-3/128,-3/64,-3/128,-3/128,3/64,-3/128,3/128,-3/64,3/64,-3/128,-3/128,3/64,-3/128,3/128,-3/64,3/64,-3/128,-9/128,3/128,-9/128,0,3/64,3/64,3/64,
3/128,9/128,-3/128,-3/128,-3/64,-3/128,-3/128,-3/64,-3/128,3/128,3/64,3/64,-3/128,-3/128,-3/64,-3/128,3/128,3/64,3/64,-3/128,3/128,-9/128,-9/128,0,3/64,3/64,3/64;

// This is the ring of probability distributions. 

ring rP = 0,(p1,p2,p3,p4,p5,p6,p7,p8,p9,p10,p11,p12,p13,p14,p15,p16,p17,p18,p19,p20,p21,p22,p23,p24,p25,p26,p27,p28,p29,p30,p31,p32,p33,p34,p35,p36,p37,p38,p39,p40,p41,p42,p43,p44,p45,p46,p47,p48,p49,p50,p51),dp;

//This is the Fourier transform.

matrix qtop[27][51] =
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,-1/3,-1/3,1,-1/3,1,-1/3,-1/3,-1/3,1,-1/3,-1/3,-1/3,1,-1/3,1,-1/3,-1/3,-1/3,1,-1/3,-1/3,-1/3,1,-1/3,1,-1/3,-1/3,-1/3,1,-1/3,-1/3,-1/3,1,-1/3,1,-1/3,-1/3,-1/3,-1/3,1,-1/3,-1/3,-1/3,-1/3,1,-1/3,-1/3,-1/3,-1/3,1,
1,-1/3,1,-1/3,-1/3,-1/3,1,-1/3,-1/3,1,-1/3,-1/3,1,-1/3,-1/3,1,-1/3,-1/3,1,-1/3,-1/3,1,-1/3,-1/3,-1/3,-1/3,1,-1/3,-1/3,1,-1/3,-1/3,1,-1/3,-1/3,-1/3,-1/3,1,-1/3,-1/3,-1/3,1,-1/3,-1/3,-1/3,1,-1/3,-1/3,-1/3,1,-1/3,
1,1,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,1,1,1,-1/3,-1/3,-1/3,-1/3,1,1,1,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,1,1,1,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,1,1,1,1,-1/3,-1/3,-1/3,-1/3,
1,-1/3,-1/3,-1/3,1/3,-1/3,-1/3,1/3,-1/3,1,-1/3,1/3,-1/3,-1/3,1/3,1,-1/3,-1/3,-1/3,-1/3,1/3,-1/3,1/3,-1/3,1/3,-1/3,-1/3,1/3,-1/3,1,-1/3,1/3,-1/3,-1/3,1/3,-1/3,1/3,-1/3,1/3,1/3,-1/3,-1/3,1/3,-1/3,-1/3,1,-1/3,1/3,1/3,-1/3,-1/3,
1,-1/3,1,-1/3,-1/3,1,-1/3,-1/3,1,-1/3,-1/3,1,-1/3,-1/3,-1/3,-1/3,1,-1/3,-1/3,1,-1/3,-1/3,1,-1/3,-1/3,-1/3,1,-1/3,-1/3,1,-1/3,-1/3,1,-1/3,-1/3,-1/3,1,-1/3,-1/3,-1/3,1,-1/3,-1/3,-1/3,1,-1/3,-1/3,-1/3,1,-1/3,-1/3,
1,1,-1/3,-1/3,-1/3,1,1,1,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,1,1,1,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,1,1,1,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,1,1,1,1,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,
1,-1/3,-1/3,-1/3,1/3,1,-1/3,-1/3,-1/3,-1/3,1/3,-1/3,1/3,-1/3,1/3,-1/3,-1/3,1/3,-1/3,1,-1/3,1/3,-1/3,-1/3,1/3,-1/3,-1/3,1/3,-1/3,1,-1/3,1/3,-1/3,-1/3,1/3,-1/3,-1/3,1/3,1/3,-1/3,1,-1/3,-1/3,1/3,-1/3,-1/3,1/3,1/3,-1/3,1/3,-1/3,
1,1,1,1,1,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,1,1,1,1,1,1,1,1,1,1,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,
1,-1/3,-1/3,5/21,1/21,-1/3,5/21,1/21,5/21,-1/3,1/21,1/21,1/21,1/21,-1/7,-1/3,5/21,1/21,5/21,-1/3,1/21,1/21,1/21,1/21,-1/7,5/21,-1/3,1/21,-1/3,1,-1/3,1/21,-1/3,5/21,1/21,1/21,1/21,1/21,-1/7,1/21,-1/3,5/21,1/21,1/21,5/21,-1/3,1/21,-1/7,1/21,1/21,1/21,
1,-1/3,1,-1/3,-1/3,-1/3,-1/3,1/3,-1/3,-1/3,1/3,-1/3,-1/3,1/3,1/3,-1/3,-1/3,1/3,-1/3,-1/3,1/3,-1/3,-1/3,1/3,1/3,-1/3,1,-1/3,-1/3,1,-1/3,-1/3,1,-1/3,-1/3,1/3,-1/3,-1/3,1/3,1/3,-1/3,-1/3,1/3,1/3,-1/3,-1/3,1/3,1/3,-1/3,-1/3,1/3,
1,1,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,1/3,1/3,1/3,1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,1/3,1/3,1/3,1/3,-1/3,-1/3,-1/3,1,1,1,-1/3,-1/3,-1/3,-1/3,1/3,1/3,1/3,1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,1/3,1/3,1/3,1/3,
1,-1/3,1,-1/3,-1/3,1,-1/3,-1/3,1,-1/3,-1/3,1,-1/3,-1/3,-1/3,1,-1/3,-1/3,1,-1/3,-1/3,1,-1/3,-1/3,-1/3,1,-1/3,-1/3,1,-1/3,-1/3,1,-1/3,-1/3,-1/3,1,-1/3,-1/3,-1/3,1,-1/3,-1/3,-1/3,1,-1/3,-1/3,-1/3,1,-1/3,-1/3,-1/3,
1,1,-1/3,-1/3,-1/3,1,1,1,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,1,1,1,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,1,1,1,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,1,1,1,1,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,
1,-1/3,-1/3,-1/3,1/3,1,-1/3,-1/3,-1/3,-1/3,1/3,-1/3,1/3,-1/3,1/3,1,-1/3,-1/3,-1/3,-1/3,1/3,-1/3,1/3,-1/3,1/3,1,-1/3,-1/3,-1/3,-1/3,1/3,-1/3,1/3,-1/3,1/3,1,-1/3,-1/3,-1/3,-1/3,-1/3,1/3,1/3,-1/3,1/3,-1/3,1/3,-1/3,1/3,1/3,-1/3,
1,1,1,1,1,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,1,1,1,1,1,1,1,1,1,1,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,
1,-1/3,-1/3,5/21,1/21,-1/3,5/21,1/21,5/21,-1/3,1/21,1/21,1/21,1/21,-1/7,1,-1/3,-1/3,-1/3,5/21,1/21,-1/3,1/21,5/21,1/21,-1/3,5/21,1/21,5/21,-1/3,1/21,1/21,1/21,1/21,-1/7,-1/3,1/21,5/21,1/21,1/21,1/21,1/21,-1/7,5/21,1/21,-1/3,1/21,1/21,-1/7,1/21,1/21,
1,-1/3,1,-1/3,-1/3,-1/3,-1/3,1/3,-1/3,-1/3,1/3,-1/3,-1/3,1/3,1/3,1,-1/3,-1/3,1,-1/3,-1/3,1,-1/3,-1/3,-1/3,-1/3,-1/3,1/3,-1/3,-1/3,1/3,-1/3,-1/3,1/3,1/3,-1/3,1/3,-1/3,1/3,-1/3,1/3,-1/3,1/3,-1/3,1/3,-1/3,1/3,-1/3,1/3,-1/3,1/3,
1,1,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,1/3,1/3,1/3,1/3,1,1,1,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,1/3,1/3,1/3,1/3,-1/3,-1/3,-1/3,-1/3,1/3,1/3,1/3,1/3,-1/3,-1/3,-1/3,-1/3,1/3,1/3,1/3,1/3,
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,
1,-1/3,-1/3,5/21,1/21,1,-1/3,-1/3,-1/3,5/21,1/21,-1/3,1/21,5/21,1/21,-1/3,5/21,1/21,5/21,-1/3,1/21,1/21,1/21,1/21,-1/7,-1/3,5/21,1/21,5/21,-1/3,1/21,1/21,1/21,1/21,-1/7,-1/3,5/21,1/21,1/21,5/21,-1/3,1/21,1/21,1/21,1/21,1/21,-1/7,1/21,1/21,-1/7,1/21,
1,-1/3,1,-1/3,-1/3,-1/3,5/21,1/21,-1/3,5/21,1/21,-1/3,5/21,1/21,1/21,-1/3,5/21,1/21,-1/3,5/21,1/21,-1/3,5/21,1/21,1/21,5/21,-1/3,1/21,5/21,-1/3,1/21,5/21,-1/3,1/21,1/21,1/21,1/21,1/21,-1/7,1/21,1/21,1/21,-1/7,1/21,1/21,1/21,-1/7,1/21,1/21,1/21,-1/7,
1,1,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,5/21,5/21,5/21,1/21,1/21,1/21,1/21,-1/3,-1/3,-1/3,5/21,5/21,5/21,1/21,1/21,1/21,1/21,5/21,5/21,5/21,-1/3,-1/3,-1/3,1/21,1/21,1/21,1/21,1/21,1/21,1/21,1/21,1/21,1/21,1/21,1/21,1/21,1/21,1/21,1/21,-1/7,-1/7,-1/7,-1/7,
1,-1/3,-1/3,1/15,2/15,-1/3,1/15,2/15,1/15,1/15,-1/15,2/15,-1/15,-1/15,0,-1/3,1/15,2/15,1/15,1/15,-1/15,2/15,-1/15,-1/15,0,1/15,1/15,-1/15,1/15,-1/3,2/15,-1/15,2/15,-1/15,0,2/15,-1/15,-1/15,0,-1/15,2/15,-1/15,0,-1/15,-1/15,2/15,0,0,0,0,0,
1,-1/3,1,-1/3,-1/3,1,-1/3,-1/3,1,-1/3,-1/3,1,-1/3,-1/3,-1/3,-1/3,-1/3,1/3,-1/3,-1/3,1/3,-1/3,-1/3,1/3,1/3,-1/3,-1/3,1/3,-1/3,-1/3,1/3,-1/3,-1/3,1/3,1/3,-1/3,-1/3,1/3,1/3,-1/3,-1/3,1/3,1/3,-1/3,-1/3,1/3,1/3,-1/3,-1/3,1/3,1/3,
1,1,-1/3,-1/3,-1/3,1,1,1,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,1/3,1/3,1/3,1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,1/3,1/3,1/3,1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,1/3,1/3,1/3,1/3,1/3,1/3,1/3,1/3,
1,1,1,1,1,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,1/3,1/3,1/3,1/3,1/3,1/3,1/3,1/3,1/3,1/3,1/3,1/3,1/3,1/3,1/3,1/3;


ideal Fourier = qtop*transpose(maxideal(1));

// This is the list of polynomial invariants.

map F = rQ, Fourier;
ideal PInvariants =  F(Invariants); 

// This is the polynomial parametrization.

ring r = 0,(a0,a1,b0,b1,c0,c1,d0,d1,e0,e1),dp;

ideal P = 
a0*b0*c0*d0*e0+3*a1*b1*c1*d1*e1,
3*a0*b1*c0*d0*e0+3*a1*b0*c1*d1*e1+6*a1*b1*c1*d1*e1,
3*a0*b0*c1*d0*e0+3*a1*b1*c0*d1*e1+6*a1*b1*c1*d1*e1,
3*a0*b1*c1*d0*e0+3*a1*b0*c0*d1*e1+6*a1*b1*c1*d1*e1,
6*a0*b1*c1*d0*e0+6*a1*b0*c1*d1*e1+6*a1*b1*c0*d1*e1+6*a1*b1*c1*d1*e1,
3*a0*b0*c0*d0*e1+3*a1*b1*c1*d1*e0+6*a1*b1*c1*d1*e1,
3*a0*b1*c0*d0*e1+3*a1*b0*c1*d1*e0+6*a1*b1*c1*d1*e1,
6*a0*b1*c0*d0*e1+6*a1*b0*c1*d1*e1+6*a1*b1*c1*d1*e0+6*a1*b1*c1*d1*e1,
3*a0*b0*c1*d0*e1+3*a1*b1*c0*d1*e0+6*a1*b1*c1*d1*e1,
3*a0*b1*c1*d0*e1+3*a1*b0*c0*d1*e0+6*a1*b1*c1*d1*e1,
6*a0*b1*c1*d0*e1+6*a1*b0*c1*d1*e1+6*a1*b1*c0*d1*e0+6*a1*b1*c1*d1*e1,
6*a0*b0*c1*d0*e1+6*a1*b1*c0*d1*e1+6*a1*b1*c1*d1*e0+6*a1*b1*c1*d1*e1,
6*a0*b1*c1*d0*e1+6*a1*b0*c1*d1*e0+6*a1*b1*c0*d1*e1+6*a1*b1*c1*d1*e1,
6*a0*b1*c1*d0*e1+6*a1*b0*c0*d1*e1+6*a1*b1*c1*d1*e0+6*a1*b1*c1*d1*e1,
6*a0*b1*c1*d0*e1+6*a1*b0*c1*d1*e1+6*a1*b1*c0*d1*e1+6*a1*b1*c1*d1*e0,
3*a0*b0*c0*d1*e0+3*a1*b1*c1*d0*e1+6*a1*b1*c1*d1*e1,
3*a0*b1*c0*d1*e0+3*a1*b0*c1*d0*e1+6*a1*b1*c1*d1*e1,
6*a0*b1*c0*d1*e0+6*a1*b0*c1*d1*e1+6*a1*b1*c1*d0*e1+6*a1*b1*c1*d1*e1,
3*a0*b0*c1*d1*e0+3*a1*b1*c0*d0*e1+6*a1*b1*c1*d1*e1,
3*a0*b1*c1*d1*e0+3*a1*b0*c0*d0*e1+6*a1*b1*c1*d1*e1,
6*a0*b1*c1*d1*e0+6*a1*b0*c1*d1*e1+6*a1*b1*c0*d0*e1+6*a1*b1*c1*d1*e1,
6*a0*b0*c1*d1*e0+6*a1*b1*c0*d1*e1+6*a1*b1*c1*d0*e1+6*a1*b1*c1*d1*e1,
6*a0*b1*c1*d1*e0+6*a1*b0*c1*d0*e1+6*a1*b1*c0*d1*e1+6*a1*b1*c1*d1*e1,
6*a0*b1*c1*d1*e0+6*a1*b0*c0*d1*e1+6*a1*b1*c1*d0*e1+6*a1*b1*c1*d1*e1,
6*a0*b1*c1*d1*e0+6*a1*b0*c1*d1*e1+6*a1*b1*c0*d1*e1+6*a1*b1*c1*d0*e1,
3*a0*b0*c0*d1*e1+3*a1*b1*c1*d0*e0+6*a1*b1*c1*d1*e1,
3*a0*b1*c0*d1*e1+3*a1*b0*c1*d0*e0+6*a1*b1*c1*d1*e1,
6*a0*b1*c0*d1*e1+6*a1*b0*c1*d1*e1+6*a1*b1*c1*d0*e0+6*a1*b1*c1*d1*e1,
3*a0*b0*c1*d1*e1+3*a1*b1*c0*d0*e0+6*a1*b1*c1*d1*e1,
3*a0*b1*c1*d1*e1+3*a1*b0*c0*d0*e0+6*a1*b1*c1*d1*e1,
6*a0*b1*c1*d1*e1+6*a1*b0*c1*d1*e1+6*a1*b1*c0*d0*e0+6*a1*b1*c1*d1*e1,
6*a0*b0*c1*d1*e1+6*a1*b1*c0*d1*e1+6*a1*b1*c1*d0*e0+6*a1*b1*c1*d1*e1,
6*a0*b1*c1*d1*e1+6*a1*b0*c1*d0*e0+6*a1*b1*c0*d1*e1+6*a1*b1*c1*d1*e1,
6*a0*b1*c1*d1*e1+6*a1*b0*c0*d1*e1+6*a1*b1*c1*d0*e0+6*a1*b1*c1*d1*e1,
6*a0*b1*c1*d1*e1+6*a1*b0*c1*d1*e1+6*a1*b1*c0*d1*e1+6*a1*b1*c1*d0*e0,
6*a0*b0*c0*d1*e1+6*a1*b1*c1*d0*e1+6*a1*b1*c1*d1*e0+6*a1*b1*c1*d1*e1,
6*a0*b1*c0*d1*e1+6*a1*b0*c1*d0*e1+6*a1*b1*c1*d1*e0+6*a1*b1*c1*d1*e1,
6*a0*b1*c0*d1*e1+6*a1*b0*c1*d1*e0+6*a1*b1*c1*d0*e1+6*a1*b1*c1*d1*e1,
6*a0*b1*c0*d1*e1+6*a1*b0*c1*d1*e1+6*a1*b1*c1*d0*e1+6*a1*b1*c1*d1*e0,
6*a0*b0*c1*d1*e1+6*a1*b1*c0*d0*e1+6*a1*b1*c1*d1*e0+6*a1*b1*c1*d1*e1,
6*a0*b1*c1*d1*e1+6*a1*b0*c0*d0*e1+6*a1*b1*c1*d1*e0+6*a1*b1*c1*d1*e1,
6*a0*b1*c1*d1*e1+6*a1*b0*c1*d1*e0+6*a1*b1*c0*d0*e1+6*a1*b1*c1*d1*e1,
6*a0*b1*c1*d1*e1+6*a1*b0*c1*d1*e1+6*a1*b1*c0*d0*e1+6*a1*b1*c1*d1*e0,
6*a0*b0*c1*d1*e1+6*a1*b1*c0*d1*e0+6*a1*b1*c1*d0*e1+6*a1*b1*c1*d1*e1,
6*a0*b1*c1*d1*e1+6*a1*b0*c1*d0*e1+6*a1*b1*c0*d1*e0+6*a1*b1*c1*d1*e1,
6*a0*b1*c1*d1*e1+6*a1*b0*c0*d1*e0+6*a1*b1*c1*d0*e1+6*a1*b1*c1*d1*e1,
6*a0*b1*c1*d1*e1+6*a1*b0*c1*d1*e1+6*a1*b1*c0*d1*e0+6*a1*b1*c1*d0*e1,
6*a0*b0*c1*d1*e1+6*a1*b1*c0*d1*e1+6*a1*b1*c1*d0*e1+6*a1*b1*c1*d1*e0,
6*a0*b1*c1*d1*e1+6*a1*b0*c1*d0*e1+6*a1*b1*c0*d1*e1+6*a1*b1*c1*d1*e0,
6*a0*b1*c1*d1*e1+6*a1*b0*c1*d1*e0+6*a1*b1*c0*d1*e1+6*a1*b1*c1*d0*e1,
6*a0*b1*c1*d1*e1+6*a1*b0*c0*d1*e1+6*a1*b1*c1*d0*e1+6*a1*b1*c1*d1*e0;

// This checks that the polynomial parametrization
// lies on the probability simplex.
// It requires suma.sing. Most likely, you should 
// change the directory where you saved this file. 

// If you do have this file, you should uncomment
// the following two lines.

// < "/home/lgp/singular/suma.sing";

// Suma(Substitute(1,P));

// This checks that the PInvariants vanish at
// the polynomial parametrization.

map Evaluate = rP, P;

// The following command takes a lot of space and time to
// finish for larger models.

// ideal Z = Evaluate(PInvariants);

setring rP;
ideal Z;
int i;
for (i=1; i<= size(PInvariants); i++)
{
  i;
  Z = PInvariants[i];
  setring r;
  Evaluate(Z);
  setring rP;
}