//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,q28,q29,q30,q31),dp;

ideal Invariants = 
q28*q30-q29*q31,
q22*q30-q24*q31,
q16*q30-q19*q31,
q9*q30-q12*q31,
q3*q30-q5*q31,
q27*q29-q26*q30,
q23*q29-q24*q30,
q19*q29-q17*q30,
q12*q29-q10*q30,
q4*q29-q5*q30,
q27*q28-q26*q31,
q27*q28-q25*q30,
q26*q28-q25*q29,
q24*q28-q22*q29,
q23*q28-q24*q31,
q19*q28-q17*q31,
q19*q28-q18*q30,
q19*q28-q16*q29,
q17*q28-q18*q29,
q16*q28-q18*q31,
q12*q28-q10*q31,
q12*q28-q11*q30,
q12*q28-q9*q29,
q10*q28-q11*q29,
q9*q28-q11*q31,
q5*q28-q3*q29,
q4*q28-q5*q31,
q24*q27-q21*q30,
q19*q27-q14*q30,
q16*q27-q14*q31,
q12*q27-q7*q30,
q9*q27-q7*q31,
q5*q27-q2*q30,
q24*q26-q21*q29,
q23*q26-q24*q27,
q19*q26-q15*q30,
q19*q26-q14*q29,
q19*q26-q17*q27,
q18*q26-q13*q29,
q18*q26-q15*q28,
q17*q26-q15*q29,
q14*q26-q15*q27,
q12*q26-q8*q30,
q12*q26-q7*q29,
q12*q26-q10*q27,
q11*q26-q6*q29,
q11*q26-q8*q28,
q10*q26-q8*q29,
q7*q26-q8*q27,
q5*q26-q2*q29,
q4*q26-q5*q27,
q24*q25-q21*q28,
q24*q25-q22*q26,
q23*q25-q21*q31,
q23*q25-q20*q29,
q23*q25-q22*q27,
q19*q25-q15*q31,
q19*q25-q13*q30,
q19*q25-q14*q28,
q19*q25-q18*q27,
q19*q25-q16*q26,
q18*q25-q13*q28,
q17*q25-q18*q26,
q16*q25-q13*q31,
q15*q25-q13*q26,
q14*q25-q13*q27,
q12*q25-q8*q31,
q12*q25-q6*q30,
q12*q25-q7*q28,
q12*q25-q11*q27,
q12*q25-q9*q26,
q11*q25-q6*q28,
q10*q25-q11*q26,
q9*q25-q6*q31,
q8*q25-q6*q26,
q7*q25-q6*q27,
q5*q25-q2*q28,
q5*q25-q3*q26,
q4*q25-q2*q31,
q4*q25-q1*q29,
q4*q25-q3*q27,
q17*q23-q19*q24,
q10*q23-q12*q24,
q5*q23-q4*q24,
q19*q22-q18*q23,
q19*q22-q16*q24,
q17*q22-q18*q24,
q12*q22-q11*q23,
q12*q22-q9*q24,
q10*q22-q11*q24,
q5*q22-q3*q24,
q4*q22-q3*q23,
q19*q21-q15*q23,
q19*q21-q14*q24,
q18*q21-q15*q22,
q18*q21-q13*q24,
q17*q21-q15*q24,
q12*q21-q8*q23,
q12*q21-q7*q24,
q11*q21-q8*q22,
q11*q21-q6*q24,
q10*q21-q8*q24,
q5*q21-q2*q24,
q4*q21-q2*q23,
q3*q21-q2*q22,
q17*q20-q16*q21,
q17*q20-q14*q22,
q17*q20-q13*q23,
q10*q20-q9*q21,
q10*q20-q7*q22,
q10*q20-q6*q23,
q5*q20-q1*q24,
q4*q20-q1*q23,
q3*q20-q1*q22,
q2*q20-q1*q21,
q12*q18-q11*q19,
q16*q17-q18*q19,
q14*q17-q15*q19,
q13*q17-q15*q18,
q12*q17-q10*q19,
q11*q17-q10*q18,
q9*q17-q12*q18,
q4*q17-q5*q19,
q3*q17-q5*q18,
q15*q16-q14*q18,
q15*q16-q13*q19,
q12*q16-q9*q19,
q11*q16-q9*q18,
q10*q16-q12*q18,
q5*q16-q4*q18,
q5*q16-q3*q19,
q12*q15-q8*q19,
q12*q15-q7*q17,
q11*q15-q8*q18,
q11*q15-q6*q17,
q10*q15-q8*q17,
q5*q15-q2*q17,
q12*q14-q7*q19,
q10*q14-q12*q15,
q9*q14-q7*q16,
q8*q14-q7*q15,
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,
q11*q13-q6*q18,
q10*q13-q11*q15,
q9*q13-q6*q16,
q8*q13-q6*q15,
q7*q13-q6*q14,
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,
q20*q30^2-q23*q27*q31,
q20*q28^2-q22*q25*q31,
q16*q23*q27-q19*q20*q30,
q9*q23*q27-q12*q20*q30,
q4*q16*q27-q1*q19*q30,
q4*q9*q27-q1*q12*q30,
q20*q26^2-q21*q25*q27,
q16*q22*q25-q18*q20*q28,
q9*q22*q25-q11*q20*q28,
q14*q21*q25-q15*q20*q26,
q7*q21*q25-q8*q20*q26,
q3*q16*q25-q1*q18*q28,
q2*q14*q25-q1*q15*q26,
q3*q9*q25-q1*q11*q28,
q2*q7*q25-q1*q8*q26,
q12*q19*q24-q5*q30^2,
q12*q18*q24-q5*q29*q31,
q11*q18*q24-q5*q28^2,
q12*q17*q24-q5*q29*q30,
q11*q17*q24-q5*q28*q29,
q10*q17*q24-q5*q29^2,
q21*q22*q23-q20*q24^2,
q12*q19*q23-q4*q30^2,
q12*q16*q23-q4*q30*q31,
q9*q16*q23-q4*q31^2,
q12*q14*q23-q4*q27*q30,
q7*q14*q23-q4*q27^2,
q4*q21*q22-q5*q20*q24,
q12*q19*q22-q5*q30*q31,
q12*q18*q22-q5*q28*q31,
q11*q18*q22-q3*q28^2,
q12*q16*q22-q5*q31^2,
q11*q16*q22-q3*q28*q31,
q9*q16*q22-q3*q31^2,
q12*q13*q22-q5*q25*q31,
q11*q13*q22-q3*q25*q28,
q6*q13*q22-q3*q25^2,
q12*q19*q21-q5*q27*q30,
q12*q18*q21-q5*q27*q28,
q11*q18*q21-q5*q25*q28,
q12*q17*q21-q5*q27*q29,
q11*q17*q21-q5*q26*q28,
q10*q17*q21-q5*q26*q29,
q12*q15*q21-q5*q26*q27,
q11*q15*q21-q5*q25*q26,
q10*q15*q21-q5*q26^2,
q8*q15*q21-q2*q26^2,
q12*q14*q21-q5*q27^2,
q8*q14*q21-q2*q26*q27,
q7*q14*q21-q2*q27^2,
q12*q13*q21-q5*q25*q27,
q11*q13*q21-q5*q25^2,
q8*q13*q21-q2*q25*q26,
q6*q13*q21-q2*q25^2,
q19^2*q20-q14*q16*q23,
q12*q19*q20-q9*q14*q23,
q12*q19*q20-q1*q30^2,
q12*q19*q20-q4*q27*q31,
q18^2*q20-q13*q16*q22,
q12*q18*q20-q4*q25*q31,
q11*q18*q20-q9*q13*q22,
q11*q18*q20-q1*q28^2,
q11*q18*q20-q3*q25*q31,
q12*q17*q20-q5*q27*q31,
q12*q16*q20-q1*q30*q31,
q11*q16*q20-q1*q28*q31,
q9*q16*q20-q1*q31^2,
q15^2*q20-q13*q14*q21,
q12*q15*q20-q4*q25*q27,
q11*q15*q20-q1*q26*q28,
q8*q15*q20-q7*q13*q21,
q8*q15*q20-q1*q26^2,
q8*q15*q20-q2*q25*q27,
q12*q14*q20-q1*q27*q30,
q9*q14*q20-q1*q27*q31,
q8*q14*q20-q1*q26*q27,
q7*q14*q20-q1*q27^2,
q12*q13*q20-q1*q27*q28,
q11*q13*q20-q1*q25*q28,
q9*q13*q20-q1*q25*q31,
q8*q13*q20-q1*q25*q26,
q7*q13*q20-q1*q25*q27,
q6*q13*q20-q1*q25^2,
q12^2*q20-q7*q9*q23,
q11^2*q20-q6*q9*q22,
q8^2*q20-q6*q7*q21,
q5^2*q20-q3*q4*q21,
q4*q14*q16-q1*q19^2,
q3*q13*q16-q1*q18^2,
q2*q13*q14-q1*q15^2,
q4*q9*q14-q1*q12*q19,
q3*q9*q13-q1*q11*q18,
q2*q7*q13-q1*q8*q15,
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][31] =
1/256,3/256,3/256,3/256,3/128,3/256,3/256,3/128,3/256,21/256,3/128,3/128,3/256,3/256,3/128,3/256,21/256,3/128,3/128,3/256,9/256,9/256,9/256,9/128,3/128,3/64,3/128,3/64,21/128,3/64,3/128,
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,-9/256,-9/256,27/256,-9/128,-3/128,-3/64,9/128,-3/64,-21/128,9/64,9/128,
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,-9/256,27/256,-9/256,-9/128,9/128,-3/64,-3/128,9/64,-21/128,-3/64,9/128,
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,-3/128,9/256,27/256,-9/256,-9/256,-9/128,-3/128,-3/64,-3/128,-3/64,15/128,-3/64,9/128,
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,-9/128,-9/128,-9/128,9/64,-3/64,3/32,-3/64,-3/32,3/64,-3/32,9/64,
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,-3/128,9/256,27/256,-9/256,-9/256,-9/128,9/128,9/64,9/128,-3/64,-21/128,-3/64,-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,-9/256,27/256,-9/256,-9/128,-3/128,-3/64,9/128,-3/64,15/128,-3/64,-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,-9/128,-9/128,-9/128,9/64,-3/64,-3/32,9/64,3/32,3/64,-3/32,-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,-9/256,-9/256,27/256,-9/128,9/128,-3/64,-3/128,-3/64,15/128,-3/64,-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,-3/128,9/256,27/256,27/256,27/256,27/128,-3/128,-3/64,-3/128,-3/64,-21/128,-3/64,-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,-9/128,-9/128,27/128,-9/64,-3/64,3/32,-3/64,3/32,3/64,-3/32,-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,-9/128,-9/128,-9/128,9/64,9/64,-3/32,-3/64,-3/32,3/64,3/32,-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,-9/128,27/128,-9/128,-9/64,-3/64,3/32,-3/64,-3/32,3/64,3/32,-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,-3/128,3/128,3/64,3/64,9/128,27/128,-9/128,-9/128,-9/64,-3/64,-3/32,-3/64,3/32,3/64,3/32,-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,-9/128,-9/128,-9/128,9/64,-3/64,3/32,-3/64,3/32,-9/64,3/32,-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,-9/256,-9/256,-9/256,-9/128,-3/128,-3/64,-3/128,-3/64,-21/128,-3/64,-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,-3/256,3/256,3/256,-9/256,3/128,-3/128,3/64,-3/128,3/64,3/128,-3/64,-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,-9/128,3/64,3/64,0,-3/64,0,9/64,-3/32,-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,-3/256,3/256,-9/256,3/256,3/128,-3/128,3/64,-3/128,-3/64,3/128,3/64,-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,-9/256,3/256,3/256,3/128,-3/128,-3/64,-3/128,3/64,3/128,3/64,-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,-3/128,3/128,3/128,3/128,-3/64,3/64,0,-3/64,0,-3/64,3/32,-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,-9/128,3/128,3/64,-3/64,0,3/64,-3/32,9/64,0,-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,-3/128,3/128,3/128,3/128,-3/64,-3/64,0,3/64,3/32,-3/64,0,-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,-9/128,3/128,3/128,3/64,3/64,3/32,3/64,0,-9/64,0,-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,-3/128,3/128,3/128,3/128,-3/64,3/64,-3/32,3/64,0,3/64,0,-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,-9/256,3/256,3/256,3/128,-3/128,-3/64,-3/128,3/64,3/128,3/64,-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,-3/256,3/256,-9/256,3/256,3/128,-3/128,3/64,-3/128,-3/64,3/128,3/64,-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,-3/128,3/128,3/128,3/128,-3/64,3/64,0,-3/64,0,-3/64,3/32,-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,-3/256,3/256,3/256,-9/256,3/128,-3/128,3/64,-3/128,3/64,3/128,-3/64,-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,-9/256,-9/256,-9/256,-9/128,-3/128,-3/64,-3/128,-3/64,-21/128,-3/64,-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,-9/128,3/64,3/64,0,-3/64,0,9/64,-3/32,-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,-3/128,3/128,3/128,3/128,-3/64,-3/64,0,3/64,3/32,-3/64,0,-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,-9/128,3/128,3/64,-3/64,0,3/64,-3/32,9/64,0,-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,-9/128,3/128,3/128,3/64,3/64,3/32,3/64,0,-9/64,0,-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,-3/128,3/128,3/128,3/128,-3/64,3/64,-3/32,3/64,0,3/64,0,-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,-9/128,3/128,3/128,3/64,-3/64,-3/32,-3/64,0,9/64,0,3/64,
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,-3/128,3/128,3/128,3/128,-3/64,-3/64,3/32,-3/64,0,-3/64,0,3/64,
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,-9/128,3/128,3/64,3/64,0,-3/64,3/32,-9/64,0,3/64,
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,-3/128,3/128,3/128,3/128,-3/64,3/64,0,-3/64,-3/32,3/64,0,3/64,
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,-3/128,3/128,3/128,3/128,-3/64,-3/64,3/32,-3/64,0,-3/64,0,3/64,
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,-9/128,3/128,3/128,3/64,-3/64,-3/32,-3/64,0,9/64,0,3/64,
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,-9/128,3/128,3/64,3/64,0,-3/64,3/32,-9/64,0,3/64,
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,-3/128,3/128,3/128,3/128,-3/64,3/64,0,-3/64,-3/32,3/64,0,3/64,
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,-9/128,3/64,-3/64,0,3/64,0,-9/64,3/32,3/64,
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,-9/128,3/64,-3/64,0,3/64,0,-9/64,3/32,3/64,
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,-9/128,-9/128,-9/128,-9/64,3/64,3/32,3/64,3/32,21/64,3/32,3/64,
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,-3/128,3/128,3/128,-9/128,3/64,3/64,-3/32,3/64,-3/32,-3/64,3/32,3/64,
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,-3/128,3/128,3/128,3/128,-3/64,-3/64,0,3/64,0,3/64,-3/32,3/64,
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,-3/128,3/128,3/128,3/128,-3/64,-3/64,0,3/64,0,3/64,-3/32,3/64,
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,-3/128,3/128,-9/128,3/128,3/64,3/64,-3/32,3/64,3/32,-3/64,-3/32,3/64,
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,-9/128,3/128,3/128,3/64,3/64,3/32,3/64,-3/32,-3/64,-3/32,3/64;

// 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[31][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,-1/3,1,-1/3,1,-1/3,-1/3,-1/3,1,-1/3,-1/3,-1/3,1,-1/3,-1/3,1/9,1/9,1/9,-1/3,1/9,1/9,1/9,-1/3,1/9,-1/3,1/9,1/9,1/9,-1/3,1/9,1/9,1/9,-1/3,1/9,-1/3,1/9,1/9,1/9,1/9,-1/3,1/9,1/9,1/9,1/9,-1/3,1/9,1/9,1/9,1/9,-1/3,
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,-1/3,1/9,1/9,-1/3,1/9,1/9,-1/3,1/9,1/9,1/9,1/9,-1/3,1/9,1/9,-1/3,1/9,1/9,-1/3,1/9,1/9,1/9,1/9,-1/3,1/9,1/9,1/9,-1/3,1/9,1/9,1/9,-1/3,1/9,1/9,1/9,-1/3,1/9,
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,-1/3,-1/3,-1/3,1/9,1/9,1/9,1/9,1/9,1/9,1/9,1/9,1/9,1/9,-1/3,-1/3,-1/3,1/9,1/9,1/9,1/9,1/9,1/9,1/9,1/9,1/9,1/9,1/9,1/9,-1/3,-1/3,-1/3,-1/3,1/9,1/9,1/9,1/9,
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,-1/3,1/9,1/9,1/9,1/9,-1/9,1/9,-1/9,1/9,-1/9,1/9,1/9,-1/9,1/9,-1/3,1/9,-1/9,1/9,1/9,-1/9,1/9,-1/9,1/9,-1/9,-1/9,1/9,1/9,-1/9,1/9,1/9,-1/3,1/9,-1/9,-1/9,1/9,1/9,
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,-1/3,-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,-1/3,1/3,0,1/3,-1/3,0,0,0,1/3,-1/3,-1/3,1/3,0,1/3,-1/3,0,0,0,1/3,-1/3,-1/3,1/3,0,0,1/3,-1/3,0,0,0,0,1/3,-1/3,0,0,-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,-1/3,1,-1/3,-1/3,-1/3,-1/3,1/3,-1/3,-1/3,1/3,-1/3,-1/3,1/3,1/3,-1/3,1/3,0,-1/3,1/3,0,-1/3,1/3,0,0,1/3,-1/3,0,1/3,-1/3,0,1/3,-1/3,0,0,0,0,1/3,-1/3,0,0,1/3,-1/3,0,0,1/3,-1/3,0,0,1/3,-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,-1/3,1/21,1/7,1/21,1/21,-1/21,1/7,-1/21,-1/7,1/21,1/21,1/21,-1/21,1/21,-1/3,1/7,-1/21,1/7,-1/7,1/21,1/7,-1/21,-1/7,1/21,-1/21,1/7,-1/7,1/21,-1/7,-1/7,1/3,-1/21,1/21,1/21,-1/21,-1/21,
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,-1/3,1/3,1/3,1/3,0,0,0,0,1/3,1/3,1/3,-1/3,-1/3,-1/3,0,0,0,0,0,0,0,0,0,0,0,0,1/3,1/3,1/3,1/3,-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,f0,f1),dp;

ideal P = 
a0*b0*c0*d0*e0*f0+3*a0*b1*c1*d1*e0*f1+3*a1*b0*c1*d0*e1*f0+3*a1*b1*c0*d1*e1*f1+6*a1*b1*c1*d1*e1*f1,
3*a0*b0*c1*d1*e0*f1+3*a0*b1*c0*d0*e0*f0+6*a0*b1*c1*d1*e0*f1+3*a1*b0*c0*d1*e1*f1+6*a1*b0*c1*d1*e1*f1+6*a1*b1*c0*d1*e1*f1+9*a1*b1*c1*d0*e1*f0+12*a1*b1*c1*d1*e1*f1,
3*a0*b0*c0*d1*e0*f0+3*a0*b1*c1*d0*e0*f1+6*a0*b1*c1*d1*e0*f1+9*a1*b0*c1*d1*e1*f0+3*a1*b1*c0*d0*e1*f1+6*a1*b1*c0*d1*e1*f1+6*a1*b1*c1*d0*e1*f1+12*a1*b1*c1*d1*e1*f1,
3*a0*b0*c1*d0*e0*f1+3*a0*b1*c0*d1*e0*f0+6*a0*b1*c1*d1*e0*f1+3*a1*b0*c0*d0*e1*f1+6*a1*b0*c1*d0*e1*f1+6*a1*b1*c0*d1*e1*f1+9*a1*b1*c1*d1*e1*f0+12*a1*b1*c1*d1*e1*f1,
6*a0*b0*c1*d1*e0*f1+6*a0*b1*c0*d1*e0*f0+6*a0*b1*c1*d0*e0*f1+6*a0*b1*c1*d1*e0*f1+6*a1*b0*c0*d1*e1*f1+12*a1*b0*c1*d1*e1*f1+6*a1*b1*c0*d0*e1*f1+6*a1*b1*c0*d1*e1*f1+12*a1*b1*c1*d0*e1*f1+18*a1*b1*c1*d1*e1*f0+12*a1*b1*c1*d1*e1*f1,
3*a0*b0*c0*d0*e0*f1+3*a0*b1*c1*d1*e0*f0+6*a0*b1*c1*d1*e0*f1+9*a1*b0*c1*d0*e1*f1+3*a1*b1*c0*d1*e1*f0+6*a1*b1*c0*d1*e1*f1+6*a1*b1*c1*d1*e1*f0+12*a1*b1*c1*d1*e1*f1,
3*a0*b0*c1*d1*e0*f0+3*a0*b1*c0*d0*e0*f1+6*a0*b1*c1*d1*e0*f1+3*a1*b0*c0*d1*e1*f0+6*a1*b0*c1*d1*e1*f0+6*a1*b1*c0*d1*e1*f1+9*a1*b1*c1*d0*e1*f1+12*a1*b1*c1*d1*e1*f1,
6*a0*b0*c1*d1*e0*f1+6*a0*b1*c0*d0*e0*f1+6*a0*b1*c1*d1*e0*f0+6*a0*b1*c1*d1*e0*f1+6*a1*b0*c0*d1*e1*f1+12*a1*b0*c1*d1*e1*f1+6*a1*b1*c0*d1*e1*f0+6*a1*b1*c0*d1*e1*f1+18*a1*b1*c1*d0*e1*f1+12*a1*b1*c1*d1*e1*f0+12*a1*b1*c1*d1*e1*f1,
3*a0*b0*c0*d1*e0*f1+3*a0*b1*c1*d0*e0*f0+6*a0*b1*c1*d1*e0*f1+9*a1*b0*c1*d1*e1*f1+3*a1*b1*c0*d0*e1*f0+6*a1*b1*c0*d1*e1*f1+6*a1*b1*c1*d0*e1*f0+12*a1*b1*c1*d1*e1*f1,
3*a0*b0*c1*d0*e0*f0+3*a0*b1*c0*d1*e0*f1+6*a0*b1*c1*d1*e0*f1+3*a1*b0*c0*d0*e1*f0+6*a1*b0*c1*d0*e1*f0+6*a1*b1*c0*d1*e1*f1+21*a1*b1*c1*d1*e1*f1,
6*a0*b0*c1*d1*e0*f1+6*a0*b1*c0*d1*e0*f1+6*a0*b1*c1*d0*e0*f0+6*a0*b1*c1*d1*e0*f1+6*a1*b0*c0*d1*e1*f1+12*a1*b0*c1*d1*e1*f1+6*a1*b1*c0*d0*e1*f0+6*a1*b1*c0*d1*e1*f1+12*a1*b1*c1*d0*e1*f0+30*a1*b1*c1*d1*e1*f1,
6*a0*b0*c0*d1*e0*f1+6*a0*b1*c1*d0*e0*f1+6*a0*b1*c1*d1*e0*f0+6*a0*b1*c1*d1*e0*f1+18*a1*b0*c1*d1*e1*f1+6*a1*b1*c0*d0*e1*f1+6*a1*b1*c0*d1*e1*f0+6*a1*b1*c0*d1*e1*f1+12*a1*b1*c1*d0*e1*f1+12*a1*b1*c1*d1*e1*f0+12*a1*b1*c1*d1*e1*f1,
6*a0*b0*c1*d1*e0*f0+6*a0*b1*c0*d1*e0*f1+6*a0*b1*c1*d0*e0*f1+6*a0*b1*c1*d1*e0*f1+6*a1*b0*c0*d1*e1*f0+12*a1*b0*c1*d1*e1*f0+6*a1*b1*c0*d0*e1*f1+6*a1*b1*c0*d1*e1*f1+12*a1*b1*c1*d0*e1*f1+30*a1*b1*c1*d1*e1*f1,
6*a0*b0*c1*d0*e0*f1+6*a0*b1*c0*d1*e0*f1+6*a0*b1*c1*d1*e0*f0+6*a0*b1*c1*d1*e0*f1+6*a1*b0*c0*d0*e1*f1+12*a1*b0*c1*d0*e1*f1+6*a1*b1*c0*d1*e1*f0+6*a1*b1*c0*d1*e1*f1+12*a1*b1*c1*d1*e1*f0+30*a1*b1*c1*d1*e1*f1,
6*a0*b0*c1*d1*e0*f1+6*a0*b1*c0*d1*e0*f1+6*a0*b1*c1*d0*e0*f1+6*a0*b1*c1*d1*e0*f0+6*a1*b0*c0*d1*e1*f1+12*a1*b0*c1*d1*e1*f1+6*a1*b1*c0*d0*e1*f1+6*a1*b1*c0*d1*e1*f0+12*a1*b1*c1*d0*e1*f1+12*a1*b1*c1*d1*e1*f0+18*a1*b1*c1*d1*e1*f1,
3*a0*b0*c0*d0*e1*f0+9*a0*b1*c1*d1*e1*f1+3*a1*b0*c1*d0*e0*f0+6*a1*b0*c1*d0*e1*f0+3*a1*b1*c0*d1*e0*f1+6*a1*b1*c0*d1*e1*f1+6*a1*b1*c1*d1*e0*f1+12*a1*b1*c1*d1*e1*f1,
3*a0*b0*c1*d1*e1*f1+3*a0*b1*c0*d0*e1*f0+6*a0*b1*c1*d1*e1*f1+3*a1*b0*c0*d1*e0*f1+6*a1*b0*c1*d1*e1*f1+6*a1*b1*c0*d1*e1*f1+3*a1*b1*c1*d0*e0*f0+6*a1*b1*c1*d0*e1*f0+6*a1*b1*c1*d1*e0*f1+6*a1*b1*c1*d1*e1*f1,
6*a0*b0*c1*d1*e1*f1+6*a0*b1*c0*d0*e1*f0+12*a0*b1*c1*d1*e1*f1+6*a1*b0*c0*d1*e1*f1+6*a1*b0*c1*d1*e0*f1+6*a1*b0*c1*d1*e1*f1+6*a1*b1*c0*d1*e0*f1+6*a1*b1*c0*d1*e1*f1+6*a1*b1*c1*d0*e0*f0+12*a1*b1*c1*d0*e1*f0+6*a1*b1*c1*d1*e0*f1+18*a1*b1*c1*d1*e1*f1,
3*a0*b0*c0*d1*e1*f0+3*a0*b1*c1*d0*e1*f1+6*a0*b1*c1*d1*e1*f1+3*a1*b0*c1*d1*e0*f0+6*a1*b0*c1*d1*e1*f0+3*a1*b1*c0*d0*e0*f1+6*a1*b1*c0*d1*e1*f1+6*a1*b1*c1*d0*e1*f1+6*a1*b1*c1*d1*e0*f1+6*a1*b1*c1*d1*e1*f1,
3*a0*b0*c1*d0*e1*f1+3*a0*b1*c0*d1*e1*f0+6*a0*b1*c1*d1*e1*f1+3*a1*b0*c0*d0*e0*f1+6*a1*b0*c1*d0*e1*f1+6*a1*b1*c0*d1*e1*f1+3*a1*b1*c1*d1*e0*f0+6*a1*b1*c1*d1*e0*f1+6*a1*b1*c1*d1*e1*f0+6*a1*b1*c1*d1*e1*f1,
6*a0*b0*c1*d1*e1*f1+6*a0*b1*c0*d1*e1*f0+6*a0*b1*c1*d0*e1*f1+6*a0*b1*c1*d1*e1*f1+6*a1*b0*c0*d1*e1*f1+6*a1*b0*c1*d1*e0*f1+6*a1*b0*c1*d1*e1*f1+6*a1*b1*c0*d0*e0*f1+6*a1*b1*c0*d1*e1*f1+12*a1*b1*c1*d0*e1*f1+6*a1*b1*c1*d1*e0*f0+6*a1*b1*c1*d1*e0*f1+12*a1*b1*c1*d1*e1*f0+6*a1*b1*c1*d1*e1*f1,
6*a0*b0*c0*d1*e1*f0+6*a0*b1*c1*d0*e1*f1+12*a0*b1*c1*d1*e1*f1+6*a1*b0*c1*d1*e0*f0+12*a1*b0*c1*d1*e1*f0+6*a1*b1*c0*d0*e1*f1+6*a1*b1*c0*d1*e0*f1+6*a1*b1*c0*d1*e1*f1+6*a1*b1*c1*d0*e0*f1+6*a1*b1*c1*d0*e1*f1+6*a1*b1*c1*d1*e0*f1+18*a1*b1*c1*d1*e1*f1,
6*a0*b0*c1*d1*e1*f1+6*a0*b1*c0*d1*e1*f0+6*a0*b1*c1*d0*e1*f1+6*a0*b1*c1*d1*e1*f1+6*a1*b0*c0*d1*e0*f1+12*a1*b0*c1*d1*e1*f1+6*a1*b1*c0*d0*e1*f1+6*a1*b1*c0*d1*e1*f1+6*a1*b1*c1*d0*e0*f1+6*a1*b1*c1*d0*e1*f1+6*a1*b1*c1*d1*e0*f0+6*a1*b1*c1*d1*e0*f1+12*a1*b1*c1*d1*e1*f0+6*a1*b1*c1*d1*e1*f1,
6*a0*b0*c1*d0*e1*f1+6*a0*b1*c0*d1*e1*f0+12*a0*b1*c1*d1*e1*f1+6*a1*b0*c0*d0*e1*f1+6*a1*b0*c1*d0*e0*f1+6*a1*b0*c1*d0*e1*f1+6*a1*b1*c0*d1*e0*f1+6*a1*b1*c0*d1*e1*f1+6*a1*b1*c1*d1*e0*f0+6*a1*b1*c1*d1*e0*f1+12*a1*b1*c1*d1*e1*f0+18*a1*b1*c1*d1*e1*f1,
6*a0*b0*c1*d1*e1*f1+6*a0*b1*c0*d1*e1*f0+6*a0*b1*c1*d0*e1*f1+6*a0*b1*c1*d1*e1*f1+6*a1*b0*c0*d1*e1*f1+6*a1*b0*c1*d1*e0*f1+6*a1*b0*c1*d1*e1*f1+6*a1*b1*c0*d0*e1*f1+6*a1*b1*c0*d1*e0*f1+6*a1*b1*c1*d0*e0*f1+6*a1*b1*c1*d0*e1*f1+6*a1*b1*c1*d1*e0*f0+12*a1*b1*c1*d1*e1*f0+12*a1*b1*c1*d1*e1*f1,
3*a0*b0*c0*d0*e1*f1+3*a0*b1*c1*d1*e1*f0+6*a0*b1*c1*d1*e1*f1+3*a1*b0*c1*d0*e0*f1+6*a1*b0*c1*d0*e1*f1+3*a1*b1*c0*d1*e0*f0+6*a1*b1*c0*d1*e1*f1+6*a1*b1*c1*d1*e0*f1+6*a1*b1*c1*d1*e1*f0+6*a1*b1*c1*d1*e1*f1,
3*a0*b0*c1*d1*e1*f0+3*a0*b1*c0*d0*e1*f1+6*a0*b1*c1*d1*e1*f1+3*a1*b0*c0*d1*e0*f0+6*a1*b0*c1*d1*e1*f0+6*a1*b1*c0*d1*e1*f1+3*a1*b1*c1*d0*e0*f1+6*a1*b1*c1*d0*e1*f1+6*a1*b1*c1*d1*e0*f1+6*a1*b1*c1*d1*e1*f1,
6*a0*b0*c1*d1*e1*f1+6*a0*b1*c0*d0*e1*f1+6*a0*b1*c1*d1*e1*f0+6*a0*b1*c1*d1*e1*f1+6*a1*b0*c0*d1*e1*f1+6*a1*b0*c1*d1*e0*f1+6*a1*b0*c1*d1*e1*f1+6*a1*b1*c0*d1*e0*f0+6*a1*b1*c0*d1*e1*f1+6*a1*b1*c1*d0*e0*f1+12*a1*b1*c1*d0*e1*f1+6*a1*b1*c1*d1*e0*f1+12*a1*b1*c1*d1*e1*f0+6*a1*b1*c1*d1*e1*f1,
3*a0*b0*c0*d1*e1*f1+3*a0*b1*c1*d0*e1*f0+6*a0*b1*c1*d1*e1*f1+3*a1*b0*c1*d1*e0*f1+6*a1*b0*c1*d1*e1*f1+3*a1*b1*c0*d0*e0*f0+6*a1*b1*c0*d1*e1*f1+6*a1*b1*c1*d0*e1*f0+6*a1*b1*c1*d1*e0*f1+6*a1*b1*c1*d1*e1*f1,
3*a0*b0*c1*d0*e1*f0+3*a0*b1*c0*d1*e1*f1+6*a0*b1*c1*d1*e1*f1+3*a1*b0*c0*d0*e0*f0+6*a1*b0*c1*d0*e1*f0+6*a1*b1*c0*d1*e1*f1+9*a1*b1*c1*d1*e0*f1+12*a1*b1*c1*d1*e1*f1,
6*a0*b0*c1*d1*e1*f1+6*a0*b1*c0*d1*e1*f1+6*a0*b1*c1*d0*e1*f0+6*a0*b1*c1*d1*e1*f1+6*a1*b0*c0*d1*e1*f1+6*a1*b0*c1*d1*e0*f1+6*a1*b0*c1*d1*e1*f1+6*a1*b1*c0*d0*e0*f0+6*a1*b1*c0*d1*e1*f1+12*a1*b1*c1*d0*e1*f0+12*a1*b1*c1*d1*e0*f1+18*a1*b1*c1*d1*e1*f1,
6*a0*b0*c0*d1*e1*f1+6*a0*b1*c1*d0*e1*f1+6*a0*b1*c1*d1*e1*f0+6*a0*b1*c1*d1*e1*f1+6*a1*b0*c1*d1*e0*f1+12*a1*b0*c1*d1*e1*f1+6*a1*b1*c0*d0*e1*f1+6*a1*b1*c0*d1*e0*f0+6*a1*b1*c0*d1*e1*f1+6*a1*b1*c1*d0*e0*f1+6*a1*b1*c1*d0*e1*f1+6*a1*b1*c1*d1*e0*f1+12*a1*b1*c1*d1*e1*f0+6*a1*b1*c1*d1*e1*f1,
6*a0*b0*c1*d1*e1*f0+6*a0*b1*c0*d1*e1*f1+6*a0*b1*c1*d0*e1*f1+6*a0*b1*c1*d1*e1*f1+6*a1*b0*c0*d1*e0*f0+12*a1*b0*c1*d1*e1*f0+6*a1*b1*c0*d0*e1*f1+6*a1*b1*c0*d1*e1*f1+6*a1*b1*c1*d0*e0*f1+6*a1*b1*c1*d0*e1*f1+12*a1*b1*c1*d1*e0*f1+18*a1*b1*c1*d1*e1*f1,
6*a0*b0*c1*d0*e1*f1+6*a0*b1*c0*d1*e1*f1+6*a0*b1*c1*d1*e1*f0+6*a0*b1*c1*d1*e1*f1+6*a1*b0*c0*d0*e1*f1+6*a1*b0*c1*d0*e0*f1+6*a1*b0*c1*d0*e1*f1+6*a1*b1*c0*d1*e0*f0+6*a1*b1*c0*d1*e1*f1+12*a1*b1*c1*d1*e0*f1+12*a1*b1*c1*d1*e1*f0+18*a1*b1*c1*d1*e1*f1,
6*a0*b0*c1*d1*e1*f1+6*a0*b1*c0*d1*e1*f1+6*a0*b1*c1*d0*e1*f1+6*a0*b1*c1*d1*e1*f0+6*a1*b0*c0*d1*e1*f1+6*a1*b0*c1*d1*e0*f1+6*a1*b0*c1*d1*e1*f1+6*a1*b1*c0*d0*e1*f1+6*a1*b1*c0*d1*e0*f0+6*a1*b1*c1*d0*e0*f1+6*a1*b1*c1*d0*e1*f1+6*a1*b1*c1*d1*e0*f1+12*a1*b1*c1*d1*e1*f0+12*a1*b1*c1*d1*e1*f1,
6*a0*b0*c0*d0*e1*f1+6*a0*b1*c1*d1*e1*f0+12*a0*b1*c1*d1*e1*f1+6*a1*b0*c1*d0*e0*f1+12*a1*b0*c1*d0*e1*f1+6*a1*b1*c0*d1*e0*f1+6*a1*b1*c0*d1*e1*f0+6*a1*b1*c0*d1*e1*f1+6*a1*b1*c1*d1*e0*f0+6*a1*b1*c1*d1*e0*f1+6*a1*b1*c1*d1*e1*f0+18*a1*b1*c1*d1*e1*f1,
6*a0*b0*c1*d1*e1*f1+6*a0*b1*c0*d0*e1*f1+6*a0*b1*c1*d1*e1*f0+6*a0*b1*c1*d1*e1*f1+6*a1*b0*c0*d1*e0*f1+12*a1*b0*c1*d1*e1*f1+6*a1*b1*c0*d1*e1*f0+6*a1*b1*c0*d1*e1*f1+6*a1*b1*c1*d0*e0*f1+12*a1*b1*c1*d0*e1*f1+6*a1*b1*c1*d1*e0*f0+6*a1*b1*c1*d1*e0*f1+6*a1*b1*c1*d1*e1*f0+6*a1*b1*c1*d1*e1*f1,
6*a0*b0*c1*d1*e1*f0+6*a0*b1*c0*d0*e1*f1+12*a0*b1*c1*d1*e1*f1+6*a1*b0*c0*d1*e1*f0+6*a1*b0*c1*d1*e0*f0+6*a1*b0*c1*d1*e1*f0+6*a1*b1*c0*d1*e0*f1+6*a1*b1*c0*d1*e1*f1+6*a1*b1*c1*d0*e0*f1+12*a1*b1*c1*d0*e1*f1+6*a1*b1*c1*d1*e0*f1+18*a1*b1*c1*d1*e1*f1,
6*a0*b0*c1*d1*e1*f1+6*a0*b1*c0*d0*e1*f1+6*a0*b1*c1*d1*e1*f0+6*a0*b1*c1*d1*e1*f1+6*a1*b0*c0*d1*e1*f1+6*a1*b0*c1*d1*e0*f1+6*a1*b0*c1*d1*e1*f1+6*a1*b1*c0*d1*e0*f1+6*a1*b1*c0*d1*e1*f0+6*a1*b1*c1*d0*e0*f1+12*a1*b1*c1*d0*e1*f1+6*a1*b1*c1*d1*e0*f0+6*a1*b1*c1*d1*e1*f0+12*a1*b1*c1*d1*e1*f1,
6*a0*b0*c0*d1*e1*f1+6*a0*b1*c1*d0*e1*f1+6*a0*b1*c1*d1*e1*f0+6*a0*b1*c1*d1*e1*f1+6*a1*b0*c1*d1*e0*f1+12*a1*b0*c1*d1*e1*f1+6*a1*b1*c0*d0*e0*f1+6*a1*b1*c0*d1*e1*f0+6*a1*b1*c0*d1*e1*f1+12*a1*b1*c1*d0*e1*f1+6*a1*b1*c1*d1*e0*f0+6*a1*b1*c1*d1*e0*f1+6*a1*b1*c1*d1*e1*f0+6*a1*b1*c1*d1*e1*f1,
6*a0*b0*c1*d0*e1*f1+6*a0*b1*c0*d1*e1*f1+6*a0*b1*c1*d1*e1*f0+6*a0*b1*c1*d1*e1*f1+6*a1*b0*c0*d0*e0*f1+12*a1*b0*c1*d0*e1*f1+6*a1*b1*c0*d1*e1*f0+6*a1*b1*c0*d1*e1*f1+6*a1*b1*c1*d1*e0*f0+12*a1*b1*c1*d1*e0*f1+6*a1*b1*c1*d1*e1*f0+18*a1*b1*c1*d1*e1*f1,
6*a0*b0*c1*d1*e1*f0+6*a0*b1*c0*d1*e1*f1+6*a0*b1*c1*d0*e1*f1+6*a0*b1*c1*d1*e1*f1+6*a1*b0*c0*d1*e1*f0+6*a1*b0*c1*d1*e0*f0+6*a1*b0*c1*d1*e1*f0+6*a1*b1*c0*d0*e0*f1+6*a1*b1*c0*d1*e1*f1+12*a1*b1*c1*d0*e1*f1+12*a1*b1*c1*d1*e0*f1+18*a1*b1*c1*d1*e1*f1,
6*a0*b0*c1*d1*e1*f1+6*a0*b1*c0*d1*e1*f1+6*a0*b1*c1*d0*e1*f1+6*a0*b1*c1*d1*e1*f0+6*a1*b0*c0*d1*e1*f1+6*a1*b0*c1*d1*e0*f1+6*a1*b0*c1*d1*e1*f1+6*a1*b1*c0*d0*e0*f1+6*a1*b1*c0*d1*e1*f0+12*a1*b1*c1*d0*e1*f1+6*a1*b1*c1*d1*e0*f0+6*a1*b1*c1*d1*e0*f1+6*a1*b1*c1*d1*e1*f0+12*a1*b1*c1*d1*e1*f1,
6*a0*b0*c0*d1*e1*f1+6*a0*b1*c1*d0*e1*f0+12*a0*b1*c1*d1*e1*f1+6*a1*b0*c1*d1*e0*f1+12*a1*b0*c1*d1*e1*f1+6*a1*b1*c0*d0*e1*f0+6*a1*b1*c0*d1*e0*f1+6*a1*b1*c0*d1*e1*f1+6*a1*b1*c1*d0*e0*f0+6*a1*b1*c1*d0*e1*f0+6*a1*b1*c1*d1*e0*f1+18*a1*b1*c1*d1*e1*f1,
6*a0*b0*c1*d1*e1*f1+6*a0*b1*c0*d1*e1*f1+6*a0*b1*c1*d0*e1*f0+6*a0*b1*c1*d1*e1*f1+6*a1*b0*c0*d1*e0*f1+12*a1*b0*c1*d1*e1*f1+6*a1*b1*c0*d0*e1*f0+6*a1*b1*c0*d1*e1*f1+6*a1*b1*c1*d0*e0*f0+6*a1*b1*c1*d0*e1*f0+12*a1*b1*c1*d1*e0*f1+18*a1*b1*c1*d1*e1*f1,
6*a0*b0*c1*d0*e1*f0+6*a0*b1*c0*d1*e1*f1+12*a0*b1*c1*d1*e1*f1+6*a1*b0*c0*d0*e1*f0+6*a1*b0*c1*d0*e0*f0+6*a1*b0*c1*d0*e1*f0+6*a1*b1*c0*d1*e0*f1+6*a1*b1*c0*d1*e1*f1+12*a1*b1*c1*d1*e0*f1+30*a1*b1*c1*d1*e1*f1,
6*a0*b0*c1*d1*e1*f1+6*a0*b1*c0*d1*e1*f1+6*a0*b1*c1*d0*e1*f0+6*a0*b1*c1*d1*e1*f1+6*a1*b0*c0*d1*e1*f1+6*a1*b0*c1*d1*e0*f1+6*a1*b0*c1*d1*e1*f1+6*a1*b1*c0*d0*e1*f0+6*a1*b1*c0*d1*e0*f1+6*a1*b1*c1*d0*e0*f0+6*a1*b1*c1*d0*e1*f0+6*a1*b1*c1*d1*e0*f1+24*a1*b1*c1*d1*e1*f1,
6*a0*b0*c0*d1*e1*f1+6*a0*b1*c1*d0*e1*f1+6*a0*b1*c1*d1*e1*f0+6*a0*b1*c1*d1*e1*f1+6*a1*b0*c1*d1*e0*f1+12*a1*b0*c1*d1*e1*f1+6*a1*b1*c0*d0*e1*f1+6*a1*b1*c0*d1*e0*f1+6*a1*b1*c0*d1*e1*f0+6*a1*b1*c1*d0*e0*f1+6*a1*b1*c1*d0*e1*f1+6*a1*b1*c1*d1*e0*f0+6*a1*b1*c1*d1*e1*f0+12*a1*b1*c1*d1*e1*f1,
6*a0*b0*c1*d1*e1*f1+6*a0*b1*c0*d1*e1*f1+6*a0*b1*c1*d0*e1*f1+6*a0*b1*c1*d1*e1*f0+6*a1*b0*c0*d1*e0*f1+12*a1*b0*c1*d1*e1*f1+6*a1*b1*c0*d0*e1*f1+6*a1*b1*c0*d1*e1*f0+6*a1*b1*c1*d0*e0*f1+6*a1*b1*c1*d0*e1*f1+6*a1*b1*c1*d1*e0*f0+6*a1*b1*c1*d1*e0*f1+6*a1*b1*c1*d1*e1*f0+12*a1*b1*c1*d1*e1*f1,
6*a0*b0*c1*d1*e1*f0+6*a0*b1*c0*d1*e1*f1+6*a0*b1*c1*d0*e1*f1+6*a0*b1*c1*d1*e1*f1+6*a1*b0*c0*d1*e1*f0+6*a1*b0*c1*d1*e0*f0+6*a1*b0*c1*d1*e1*f0+6*a1*b1*c0*d0*e1*f1+6*a1*b1*c0*d1*e0*f1+6*a1*b1*c1*d0*e0*f1+6*a1*b1*c1*d0*e1*f1+6*a1*b1*c1*d1*e0*f1+24*a1*b1*c1*d1*e1*f1,
6*a0*b0*c1*d0*e1*f1+6*a0*b1*c0*d1*e1*f1+6*a0*b1*c1*d1*e1*f0+6*a0*b1*c1*d1*e1*f1+6*a1*b0*c0*d0*e1*f1+6*a1*b0*c1*d0*e0*f1+6*a1*b0*c1*d0*e1*f1+6*a1*b1*c0*d1*e0*f1+6*a1*b1*c0*d1*e1*f0+6*a1*b1*c1*d1*e0*f0+6*a1*b1*c1*d1*e0*f1+6*a1*b1*c1*d1*e1*f0+24*a1*b1*c1*d1*e1*f1;

// 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;
}