//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),dp;

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

// This is the inverse of the Fourier transform.

matrix ptoq[20][20] =
1/64,3/64,3/64,3/64,3/32,3/64,3/32,1/64,3/64,3/64,3/64,3/32,3/64,1/64,3/64,3/64,3/32,3/64,3/64,1/64,
3/64,9/64,-3/64,-3/64,-3/32,9/64,-3/32,3/64,-3/64,-3/64,3/64,3/32,3/64,-3/64,-9/64,3/64,3/32,3/64,-9/64,-3/64,
3/64,-3/64,9/64,-3/64,-3/32,3/64,3/32,-3/64,3/64,-9/64,9/64,-3/32,-3/64,3/64,-3/64,3/64,3/32,-9/64,3/64,-3/64,
3/64,-3/64,-3/64,9/64,-3/32,3/64,3/32,-3/64,-9/64,3/64,3/64,3/32,-9/64,-3/64,3/64,9/64,-3/32,-3/64,-3/64,3/64,
3/64,9/64,-3/64,-3/64,-3/32,9/64,-3/32,3/64,-3/64,-3/64,-3/64,-3/32,-3/64,3/64,9/64,-3/64,-3/32,-3/64,9/64,3/64,
3/32,-3/32,-3/32,-3/32,3/16,3/32,-3/16,-3/32,3/32,3/32,3/32,-3/16,3/32,-3/32,3/32,-3/32,3/16,-3/32,-3/32,3/32,
3/32,-3/32,-3/32,-3/32,3/16,3/32,-3/16,-3/32,3/32,3/32,-3/32,3/16,-3/32,3/32,-3/32,3/32,-3/16,3/32,3/32,-3/32,
3/64,-3/64,9/64,-3/64,-3/32,-3/64,-3/32,3/64,-3/64,9/64,9/64,-3/32,-3/64,3/64,-3/64,-3/64,-3/32,9/64,-3/64,3/64,
3/32,-3/32,-3/32,-3/32,3/16,-3/32,3/16,3/32,-3/32,-3/32,3/32,-3/16,3/32,-3/32,3/32,3/32,-3/16,3/32,3/32,-3/32,
3/64,-3/64,-3/64,9/64,-3/32,-3/64,-3/32,3/64,9/64,-3/64,-3/64,-3/32,9/64,3/64,-3/64,9/64,-3/32,-3/64,-3/64,3/64,
1/64,3/64,3/64,3/64,3/32,3/64,3/32,1/64,3/64,3/64,-3/64,-3/32,-3/64,-1/64,-3/64,-3/64,-3/32,-3/64,-3/64,-1/64,
3/64,-3/64,-3/64,9/64,-3/32,3/64,3/32,-3/64,-9/64,3/64,-3/64,-3/32,9/64,3/64,-3/64,-9/64,3/32,3/64,3/64,-3/64,
3/64,-3/64,9/64,-3/64,-3/32,3/64,3/32,-3/64,3/64,-9/64,-9/64,3/32,3/64,-3/64,3/64,-3/64,-3/32,9/64,-3/64,3/64,
3/64,-3/64,-3/64,9/64,-3/32,-3/64,-3/32,3/64,9/64,-3/64,3/64,3/32,-9/64,-3/64,3/64,-9/64,3/32,3/64,3/64,-3/64,
3/32,-3/32,-3/32,-3/32,3/16,-3/32,3/16,3/32,-3/32,-3/32,-3/32,3/16,-3/32,3/32,-3/32,-3/32,3/16,-3/32,-3/32,3/32,
3/64,-3/64,9/64,-3/64,-3/32,-3/64,-3/32,3/64,-3/64,9/64,-9/64,3/32,3/64,-3/64,3/64,3/64,3/32,-9/64,3/64,-3/64,
1/64,3/64,3/64,3/64,3/32,-3/64,-3/32,-1/64,-3/64,-3/64,3/64,3/32,3/64,1/64,3/64,-3/64,-3/32,-3/64,-3/64,-1/64,
3/64,9/64,-3/64,-3/64,-3/32,-9/64,3/32,-3/64,3/64,3/64,3/64,3/32,3/64,-3/64,-9/64,-3/64,-3/32,-3/64,9/64,3/64,
3/64,9/64,-3/64,-3/64,-3/32,-9/64,3/32,-3/64,3/64,3/64,-3/64,-3/32,-3/64,3/64,9/64,3/64,3/32,3/64,-9/64,-3/64,
1/64,3/64,3/64,3/64,3/32,-3/64,-3/32,-1/64,-3/64,-3/64,-3/64,-3/32,-3/64,-1/64,-3/64,3/64,3/32,3/64,3/64,1/64;

// 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),dp;

//This is the Fourier transform.

matrix qtop[20][20] =
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,-1/3,-1/3,-1/3,-1/3,1,-1/3,-1/3,-1/3,-1/3,-1/3,1,1,1,1,
1,-1/3,1,-1/3,-1/3,-1/3,-1/3,1,-1/3,-1/3,1,-1/3,1,-1/3,-1/3,1,1,-1/3,-1/3,1,
1,-1/3,-1/3,1,-1/3,-1/3,-1/3,-1/3,-1/3,1,1,1,-1/3,1,-1/3,-1/3,1,-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,-1/3,-1/3,-1/3,1/3,-1/3,1,-1/3,-1/3,1,
1,1,1/3,1/3,1,1/3,1/3,-1/3,-1/3,-1/3,1,1/3,1/3,-1/3,-1/3,-1/3,-1,-1,-1,-1,
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,-1,1/3,1/3,-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,-1/3,1/3,1/3,-1/3,-1/3,1,1,-1,1/3,1,-1/3,-1/3,-1,1/3,1/3,-1,
1,-1/3,-1,1/3,-1/3,1/3,1/3,1,-1/3,-1/3,1,1/3,-1,-1/3,-1/3,1,-1,1/3,1/3,-1,
1,1/3,1,1/3,-1/3,1/3,-1/3,1,1/3,-1/3,-1,-1/3,-1,1/3,-1/3,-1,1,1/3,-1/3,-1,
1,1/3,-1/3,1/3,-1/3,-1/3,1/3,-1/3,-1/3,-1/3,-1,-1/3,1/3,1/3,1/3,1/3,1,1/3,-1/3,-1,
1,1/3,-1/3,-1,-1/3,1/3,-1/3,-1/3,1/3,1,-1,1,1/3,-1,-1/3,1/3,1,1/3,-1/3,-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,-1/3,-1/3,1/3,-1/3,-1,-1/3,1/3,1/3,-1/3,1/3,1,-1,1,-1,
1,1/3,1/3,1,-1/3,-1/3,1/3,-1/3,1/3,1,-1,-1,-1/3,-1,-1/3,1/3,-1,-1/3,1/3,1,
1,1/3,1/3,-1/3,-1/3,1/3,-1/3,-1/3,-1/3,-1/3,-1,1/3,-1/3,1/3,1/3,1/3,-1,-1/3,1/3,1,
1,1/3,-1,-1/3,-1/3,-1/3,1/3,1,1/3,-1/3,-1,1/3,1,1/3,-1/3,-1,-1,-1/3,1/3,1,
1,-1,1/3,-1/3,1,-1/3,1/3,-1/3,1/3,-1/3,-1,1/3,-1/3,1/3,-1/3,1/3,-1,1,-1,1,
1,-1,-1,1,1,1,-1,1,-1,1,-1,-1,1,-1,1,-1,-1,1,-1,1;


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,(b0,b1,b2,b3,e0,e1,e2,e3),dp;

ideal P = 
b0^2*e0^4+b0^2*e1^4+b0^2*e2^4+b0^2*e3^4+2*b0*b1*e0^3*e1+2*b0*b1*e0*e1^3+2*b0*b1*e2^3*e3+2*b0*b1*e2*e3^3+2*b0*b2*e0^3*e2+2*b0*b2*e0*e2^3+2*b0*b2*e1^3*e3+2*b0*b2*e1*e3^3+2*b0*b3*e0^3*e3+2*b0*b3*e0*e3^3+2*b0*b3*e1^3*e2+2*b0*b3*e1*e2^3+b1^2*e0^4+b1^2*e1^4+b1^2*e2^4+b1^2*e3^4+2*b1*b2*e0^3*e3+2*b1*b2*e0*e3^3+2*b1*b2*e1^3*e2+2*b1*b2*e1*e2^3+2*b1*b3*e0^3*e2+2*b1*b3*e0*e2^3+2*b1*b3*e1^3*e3+2*b1*b3*e1*e3^3+b2^2*e0^4+b2^2*e1^4+b2^2*e2^4+b2^2*e3^4+2*b2*b3*e0^3*e1+2*b2*b3*e0*e1^3+2*b2*b3*e2^3*e3+2*b2*b3*e2*e3^3+b3^2*e0^4+b3^2*e1^4+b3^2*e2^4+b3^2*e3^4,
3*b0^2*e0^3*e1+3*b0^2*e0*e1^3+3*b0^2*e2^3*e3+3*b0^2*e2*e3^3+12*b0*b1*e0^2*e1^2+12*b0*b1*e2^2*e3^2+6*b0*b2*e0^2*e1*e2+6*b0*b2*e0*e1^2*e3+6*b0*b2*e0*e2^2*e3+6*b0*b2*e1*e2*e3^2+6*b0*b3*e0^2*e1*e3+6*b0*b3*e0*e1^2*e2+6*b0*b3*e0*e2*e3^2+6*b0*b3*e1*e2^2*e3+3*b1^2*e0^3*e1+3*b1^2*e0*e1^3+3*b1^2*e2^3*e3+3*b1^2*e2*e3^3+6*b1*b2*e0^2*e1*e3+6*b1*b2*e0*e1^2*e2+6*b1*b2*e0*e2*e3^2+6*b1*b2*e1*e2^2*e3+6*b1*b3*e0^2*e1*e2+6*b1*b3*e0*e1^2*e3+6*b1*b3*e0*e2^2*e3+6*b1*b3*e1*e2*e3^2+3*b2^2*e0^3*e1+3*b2^2*e0*e1^3+3*b2^2*e2^3*e3+3*b2^2*e2*e3^3+12*b2*b3*e0^2*e1^2+12*b2*b3*e2^2*e3^2+3*b3^2*e0^3*e1+3*b3^2*e0*e1^3+3*b3^2*e2^3*e3+3*b3^2*e2*e3^3,
3*b0^2*e0^3*e2+3*b0^2*e0*e2^3+3*b0^2*e1^3*e3+3*b0^2*e1*e3^3+6*b0*b1*e0^2*e1*e2+6*b0*b1*e0*e1^2*e3+6*b0*b1*e0*e2^2*e3+6*b0*b1*e1*e2*e3^2+12*b0*b2*e0^2*e2^2+12*b0*b2*e1^2*e3^2+6*b0*b3*e0^2*e2*e3+6*b0*b3*e0*e1*e2^2+6*b0*b3*e0*e1*e3^2+6*b0*b3*e1^2*e2*e3+3*b1^2*e0^3*e2+3*b1^2*e0*e2^3+3*b1^2*e1^3*e3+3*b1^2*e1*e3^3+6*b1*b2*e0^2*e2*e3+6*b1*b2*e0*e1*e2^2+6*b1*b2*e0*e1*e3^2+6*b1*b2*e1^2*e2*e3+12*b1*b3*e0^2*e2^2+12*b1*b3*e1^2*e3^2+3*b2^2*e0^3*e2+3*b2^2*e0*e2^3+3*b2^2*e1^3*e3+3*b2^2*e1*e3^3+6*b2*b3*e0^2*e1*e2+6*b2*b3*e0*e1^2*e3+6*b2*b3*e0*e2^2*e3+6*b2*b3*e1*e2*e3^2+3*b3^2*e0^3*e2+3*b3^2*e0*e2^3+3*b3^2*e1^3*e3+3*b3^2*e1*e3^3,
3*b0^2*e0^3*e3+3*b0^2*e0*e3^3+3*b0^2*e1^3*e2+3*b0^2*e1*e2^3+6*b0*b1*e0^2*e1*e3+6*b0*b1*e0*e1^2*e2+6*b0*b1*e0*e2*e3^2+6*b0*b1*e1*e2^2*e3+6*b0*b2*e0^2*e2*e3+6*b0*b2*e0*e1*e2^2+6*b0*b2*e0*e1*e3^2+6*b0*b2*e1^2*e2*e3+12*b0*b3*e0^2*e3^2+12*b0*b3*e1^2*e2^2+3*b1^2*e0^3*e3+3*b1^2*e0*e3^3+3*b1^2*e1^3*e2+3*b1^2*e1*e2^3+12*b1*b2*e0^2*e3^2+12*b1*b2*e1^2*e2^2+6*b1*b3*e0^2*e2*e3+6*b1*b3*e0*e1*e2^2+6*b1*b3*e0*e1*e3^2+6*b1*b3*e1^2*e2*e3+3*b2^2*e0^3*e3+3*b2^2*e0*e3^3+3*b2^2*e1^3*e2+3*b2^2*e1*e2^3+6*b2*b3*e0^2*e1*e3+6*b2*b3*e0*e1^2*e2+6*b2*b3*e0*e2*e3^2+6*b2*b3*e1*e2^2*e3+3*b3^2*e0^3*e3+3*b3^2*e0*e3^3+3*b3^2*e1^3*e2+3*b3^2*e1*e2^3,
6*b0^2*e0^2*e1^2+6*b0^2*e2^2*e3^2+6*b0*b1*e0^3*e1+6*b0*b1*e0*e1^3+6*b0*b1*e2^3*e3+6*b0*b1*e2*e3^3+6*b0*b2*e0^2*e1*e3+6*b0*b2*e0*e1^2*e2+6*b0*b2*e0*e2*e3^2+6*b0*b2*e1*e2^2*e3+6*b0*b3*e0^2*e1*e2+6*b0*b3*e0*e1^2*e3+6*b0*b3*e0*e2^2*e3+6*b0*b3*e1*e2*e3^2+6*b1^2*e0^2*e1^2+6*b1^2*e2^2*e3^2+6*b1*b2*e0^2*e1*e2+6*b1*b2*e0*e1^2*e3+6*b1*b2*e0*e2^2*e3+6*b1*b2*e1*e2*e3^2+6*b1*b3*e0^2*e1*e3+6*b1*b3*e0*e1^2*e2+6*b1*b3*e0*e2*e3^2+6*b1*b3*e1*e2^2*e3+6*b2^2*e0^2*e1^2+6*b2^2*e2^2*e3^2+6*b2*b3*e0^3*e1+6*b2*b3*e0*e1^3+6*b2*b3*e2^3*e3+6*b2*b3*e2*e3^3+6*b3^2*e0^2*e1^2+6*b3^2*e2^2*e3^2,
6*b0^2*e0^2*e1*e2+6*b0^2*e0*e1^2*e3+6*b0^2*e0*e2^2*e3+6*b0^2*e1*e2*e3^2+12*b0*b1*e0^2*e1*e3+12*b0*b1*e0*e1^2*e2+12*b0*b1*e0*e2*e3^2+12*b0*b1*e1*e2^2*e3+12*b0*b2*e0^2*e2*e3+12*b0*b2*e0*e1*e2^2+12*b0*b2*e0*e1*e3^2+12*b0*b2*e1^2*e2*e3+48*b0*b3*e0*e1*e2*e3+6*b1^2*e0^2*e1*e2+6*b1^2*e0*e1^2*e3+6*b1^2*e0*e2^2*e3+6*b1^2*e1*e2*e3^2+48*b1*b2*e0*e1*e2*e3+12*b1*b3*e0^2*e2*e3+12*b1*b3*e0*e1*e2^2+12*b1*b3*e0*e1*e3^2+12*b1*b3*e1^2*e2*e3+6*b2^2*e0^2*e1*e2+6*b2^2*e0*e1^2*e3+6*b2^2*e0*e2^2*e3+6*b2^2*e1*e2*e3^2+12*b2*b3*e0^2*e1*e3+12*b2*b3*e0*e1^2*e2+12*b2*b3*e0*e2*e3^2+12*b2*b3*e1*e2^2*e3+6*b3^2*e0^2*e1*e2+6*b3^2*e0*e1^2*e3+6*b3^2*e0*e2^2*e3+6*b3^2*e1*e2*e3^2,
6*b0^2*e0^2*e1*e3+6*b0^2*e0*e1^2*e2+6*b0^2*e0*e2*e3^2+6*b0^2*e1*e2^2*e3+12*b0*b1*e0^2*e1*e2+12*b0*b1*e0*e1^2*e3+12*b0*b1*e0*e2^2*e3+12*b0*b1*e1*e2*e3^2+48*b0*b2*e0*e1*e2*e3+12*b0*b3*e0^2*e2*e3+12*b0*b3*e0*e1*e2^2+12*b0*b3*e0*e1*e3^2+12*b0*b3*e1^2*e2*e3+6*b1^2*e0^2*e1*e3+6*b1^2*e0*e1^2*e2+6*b1^2*e0*e2*e3^2+6*b1^2*e1*e2^2*e3+12*b1*b2*e0^2*e2*e3+12*b1*b2*e0*e1*e2^2+12*b1*b2*e0*e1*e3^2+12*b1*b2*e1^2*e2*e3+48*b1*b3*e0*e1*e2*e3+6*b2^2*e0^2*e1*e3+6*b2^2*e0*e1^2*e2+6*b2^2*e0*e2*e3^2+6*b2^2*e1*e2^2*e3+12*b2*b3*e0^2*e1*e2+12*b2*b3*e0*e1^2*e3+12*b2*b3*e0*e2^2*e3+12*b2*b3*e1*e2*e3^2+6*b3^2*e0^2*e1*e3+6*b3^2*e0*e1^2*e2+6*b3^2*e0*e2*e3^2+6*b3^2*e1*e2^2*e3,
6*b0^2*e0^2*e2^2+6*b0^2*e1^2*e3^2+6*b0*b1*e0^2*e2*e3+6*b0*b1*e0*e1*e2^2+6*b0*b1*e0*e1*e3^2+6*b0*b1*e1^2*e2*e3+6*b0*b2*e0^3*e2+6*b0*b2*e0*e2^3+6*b0*b2*e1^3*e3+6*b0*b2*e1*e3^3+6*b0*b3*e0^2*e1*e2+6*b0*b3*e0*e1^2*e3+6*b0*b3*e0*e2^2*e3+6*b0*b3*e1*e2*e3^2+6*b1^2*e0^2*e2^2+6*b1^2*e1^2*e3^2+6*b1*b2*e0^2*e1*e2+6*b1*b2*e0*e1^2*e3+6*b1*b2*e0*e2^2*e3+6*b1*b2*e1*e2*e3^2+6*b1*b3*e0^3*e2+6*b1*b3*e0*e2^3+6*b1*b3*e1^3*e3+6*b1*b3*e1*e3^3+6*b2^2*e0^2*e2^2+6*b2^2*e1^2*e3^2+6*b2*b3*e0^2*e2*e3+6*b2*b3*e0*e1*e2^2+6*b2*b3*e0*e1*e3^2+6*b2*b3*e1^2*e2*e3+6*b3^2*e0^2*e2^2+6*b3^2*e1^2*e3^2,
6*b0^2*e0^2*e2*e3+6*b0^2*e0*e1*e2^2+6*b0^2*e0*e1*e3^2+6*b0^2*e1^2*e2*e3+48*b0*b1*e0*e1*e2*e3+12*b0*b2*e0^2*e1*e2+12*b0*b2*e0*e1^2*e3+12*b0*b2*e0*e2^2*e3+12*b0*b2*e1*e2*e3^2+12*b0*b3*e0^2*e1*e3+12*b0*b3*e0*e1^2*e2+12*b0*b3*e0*e2*e3^2+12*b0*b3*e1*e2^2*e3+6*b1^2*e0^2*e2*e3+6*b1^2*e0*e1*e2^2+6*b1^2*e0*e1*e3^2+6*b1^2*e1^2*e2*e3+12*b1*b2*e0^2*e1*e3+12*b1*b2*e0*e1^2*e2+12*b1*b2*e0*e2*e3^2+12*b1*b2*e1*e2^2*e3+12*b1*b3*e0^2*e1*e2+12*b1*b3*e0*e1^2*e3+12*b1*b3*e0*e2^2*e3+12*b1*b3*e1*e2*e3^2+6*b2^2*e0^2*e2*e3+6*b2^2*e0*e1*e2^2+6*b2^2*e0*e1*e3^2+6*b2^2*e1^2*e2*e3+48*b2*b3*e0*e1*e2*e3+6*b3^2*e0^2*e2*e3+6*b3^2*e0*e1*e2^2+6*b3^2*e0*e1*e3^2+6*b3^2*e1^2*e2*e3,
6*b0^2*e0^2*e3^2+6*b0^2*e1^2*e2^2+6*b0*b1*e0^2*e2*e3+6*b0*b1*e0*e1*e2^2+6*b0*b1*e0*e1*e3^2+6*b0*b1*e1^2*e2*e3+6*b0*b2*e0^2*e1*e3+6*b0*b2*e0*e1^2*e2+6*b0*b2*e0*e2*e3^2+6*b0*b2*e1*e2^2*e3+6*b0*b3*e0^3*e3+6*b0*b3*e0*e3^3+6*b0*b3*e1^3*e2+6*b0*b3*e1*e2^3+6*b1^2*e0^2*e3^2+6*b1^2*e1^2*e2^2+6*b1*b2*e0^3*e3+6*b1*b2*e0*e3^3+6*b1*b2*e1^3*e2+6*b1*b2*e1*e2^3+6*b1*b3*e0^2*e1*e3+6*b1*b3*e0*e1^2*e2+6*b1*b3*e0*e2*e3^2+6*b1*b3*e1*e2^2*e3+6*b2^2*e0^2*e3^2+6*b2^2*e1^2*e2^2+6*b2*b3*e0^2*e2*e3+6*b2*b3*e0*e1*e2^2+6*b2*b3*e0*e1*e3^2+6*b2*b3*e1^2*e2*e3+6*b3^2*e0^2*e3^2+6*b3^2*e1^2*e2^2,
b0^2*e0^3*e1+b0^2*e0*e1^3+b0^2*e2^3*e3+b0^2*e2*e3^3+2*b0*b1*e0^4+2*b0*b1*e1^4+2*b0*b1*e2^4+2*b0*b1*e3^4+2*b0*b2*e0^3*e3+2*b0*b2*e0*e3^3+2*b0*b2*e1^3*e2+2*b0*b2*e1*e2^3+2*b0*b3*e0^3*e2+2*b0*b3*e0*e2^3+2*b0*b3*e1^3*e3+2*b0*b3*e1*e3^3+b1^2*e0^3*e1+b1^2*e0*e1^3+b1^2*e2^3*e3+b1^2*e2*e3^3+2*b1*b2*e0^3*e2+2*b1*b2*e0*e2^3+2*b1*b2*e1^3*e3+2*b1*b2*e1*e3^3+2*b1*b3*e0^3*e3+2*b1*b3*e0*e3^3+2*b1*b3*e1^3*e2+2*b1*b3*e1*e2^3+b2^2*e0^3*e1+b2^2*e0*e1^3+b2^2*e2^3*e3+b2^2*e2*e3^3+2*b2*b3*e0^4+2*b2*b3*e1^4+2*b2*b3*e2^4+2*b2*b3*e3^4+b3^2*e0^3*e1+b3^2*e0*e1^3+b3^2*e2^3*e3+b3^2*e2*e3^3,
3*b0^2*e0^2*e1*e3+3*b0^2*e0*e1^2*e2+3*b0^2*e0*e2*e3^2+3*b0^2*e1*e2^2*e3+6*b0*b1*e0^3*e3+6*b0*b1*e0*e3^3+6*b0*b1*e1^3*e2+6*b0*b1*e1*e2^3+12*b0*b2*e0^2*e3^2+12*b0*b2*e1^2*e2^2+6*b0*b3*e0^2*e2*e3+6*b0*b3*e0*e1*e2^2+6*b0*b3*e0*e1*e3^2+6*b0*b3*e1^2*e2*e3+3*b1^2*e0^2*e1*e3+3*b1^2*e0*e1^2*e2+3*b1^2*e0*e2*e3^2+3*b1^2*e1*e2^2*e3+6*b1*b2*e0^2*e2*e3+6*b1*b2*e0*e1*e2^2+6*b1*b2*e0*e1*e3^2+6*b1*b2*e1^2*e2*e3+12*b1*b3*e0^2*e3^2+12*b1*b3*e1^2*e2^2+3*b2^2*e0^2*e1*e3+3*b2^2*e0*e1^2*e2+3*b2^2*e0*e2*e3^2+3*b2^2*e1*e2^2*e3+6*b2*b3*e0^3*e3+6*b2*b3*e0*e3^3+6*b2*b3*e1^3*e2+6*b2*b3*e1*e2^3+3*b3^2*e0^2*e1*e3+3*b3^2*e0*e1^2*e2+3*b3^2*e0*e2*e3^2+3*b3^2*e1*e2^2*e3,
3*b0^2*e0^2*e1*e2+3*b0^2*e0*e1^2*e3+3*b0^2*e0*e2^2*e3+3*b0^2*e1*e2*e3^2+6*b0*b1*e0^3*e2+6*b0*b1*e0*e2^3+6*b0*b1*e1^3*e3+6*b0*b1*e1*e3^3+6*b0*b2*e0^2*e2*e3+6*b0*b2*e0*e1*e2^2+6*b0*b2*e0*e1*e3^2+6*b0*b2*e1^2*e2*e3+12*b0*b3*e0^2*e2^2+12*b0*b3*e1^2*e3^2+3*b1^2*e0^2*e1*e2+3*b1^2*e0*e1^2*e3+3*b1^2*e0*e2^2*e3+3*b1^2*e1*e2*e3^2+12*b1*b2*e0^2*e2^2+12*b1*b2*e1^2*e3^2+6*b1*b3*e0^2*e2*e3+6*b1*b3*e0*e1*e2^2+6*b1*b3*e0*e1*e3^2+6*b1*b3*e1^2*e2*e3+3*b2^2*e0^2*e1*e2+3*b2^2*e0*e1^2*e3+3*b2^2*e0*e2^2*e3+3*b2^2*e1*e2*e3^2+6*b2*b3*e0^3*e2+6*b2*b3*e0*e2^3+6*b2*b3*e1^3*e3+6*b2*b3*e1*e3^3+3*b3^2*e0^2*e1*e2+3*b3^2*e0*e1^2*e3+3*b3^2*e0*e2^2*e3+3*b3^2*e1*e2*e3^2,
3*b0^2*e0^2*e2*e3+3*b0^2*e0*e1*e2^2+3*b0^2*e0*e1*e3^2+3*b0^2*e1^2*e2*e3+12*b0*b1*e0^2*e3^2+12*b0*b1*e1^2*e2^2+6*b0*b2*e0^3*e3+6*b0*b2*e0*e3^3+6*b0*b2*e1^3*e2+6*b0*b2*e1*e2^3+6*b0*b3*e0^2*e1*e3+6*b0*b3*e0*e1^2*e2+6*b0*b3*e0*e2*e3^2+6*b0*b3*e1*e2^2*e3+3*b1^2*e0^2*e2*e3+3*b1^2*e0*e1*e2^2+3*b1^2*e0*e1*e3^2+3*b1^2*e1^2*e2*e3+6*b1*b2*e0^2*e1*e3+6*b1*b2*e0*e1^2*e2+6*b1*b2*e0*e2*e3^2+6*b1*b2*e1*e2^2*e3+6*b1*b3*e0^3*e3+6*b1*b3*e0*e3^3+6*b1*b3*e1^3*e2+6*b1*b3*e1*e2^3+3*b2^2*e0^2*e2*e3+3*b2^2*e0*e1*e2^2+3*b2^2*e0*e1*e3^2+3*b2^2*e1^2*e2*e3+12*b2*b3*e0^2*e3^2+12*b2*b3*e1^2*e2^2+3*b3^2*e0^2*e2*e3+3*b3^2*e0*e1*e2^2+3*b3^2*e0*e1*e3^2+3*b3^2*e1^2*e2*e3,
24*b0^2*e0*e1*e2*e3+12*b0*b1*e0^2*e2*e3+12*b0*b1*e0*e1*e2^2+12*b0*b1*e0*e1*e3^2+12*b0*b1*e1^2*e2*e3+12*b0*b2*e0^2*e1*e3+12*b0*b2*e0*e1^2*e2+12*b0*b2*e0*e2*e3^2+12*b0*b2*e1*e2^2*e3+12*b0*b3*e0^2*e1*e2+12*b0*b3*e0*e1^2*e3+12*b0*b3*e0*e2^2*e3+12*b0*b3*e1*e2*e3^2+24*b1^2*e0*e1*e2*e3+12*b1*b2*e0^2*e1*e2+12*b1*b2*e0*e1^2*e3+12*b1*b2*e0*e2^2*e3+12*b1*b2*e1*e2*e3^2+12*b1*b3*e0^2*e1*e3+12*b1*b3*e0*e1^2*e2+12*b1*b3*e0*e2*e3^2+12*b1*b3*e1*e2^2*e3+24*b2^2*e0*e1*e2*e3+12*b2*b3*e0^2*e2*e3+12*b2*b3*e0*e1*e2^2+12*b2*b3*e0*e1*e3^2+12*b2*b3*e1^2*e2*e3+24*b3^2*e0*e1*e2*e3,
3*b0^2*e0^2*e2*e3+3*b0^2*e0*e1*e2^2+3*b0^2*e0*e1*e3^2+3*b0^2*e1^2*e2*e3+12*b0*b1*e0^2*e2^2+12*b0*b1*e1^2*e3^2+6*b0*b2*e0^2*e1*e2+6*b0*b2*e0*e1^2*e3+6*b0*b2*e0*e2^2*e3+6*b0*b2*e1*e2*e3^2+6*b0*b3*e0^3*e2+6*b0*b3*e0*e2^3+6*b0*b3*e1^3*e3+6*b0*b3*e1*e3^3+3*b1^2*e0^2*e2*e3+3*b1^2*e0*e1*e2^2+3*b1^2*e0*e1*e3^2+3*b1^2*e1^2*e2*e3+6*b1*b2*e0^3*e2+6*b1*b2*e0*e2^3+6*b1*b2*e1^3*e3+6*b1*b2*e1*e3^3+6*b1*b3*e0^2*e1*e2+6*b1*b3*e0*e1^2*e3+6*b1*b3*e0*e2^2*e3+6*b1*b3*e1*e2*e3^2+3*b2^2*e0^2*e2*e3+3*b2^2*e0*e1*e2^2+3*b2^2*e0*e1*e3^2+3*b2^2*e1^2*e2*e3+12*b2*b3*e0^2*e2^2+12*b2*b3*e1^2*e3^2+3*b3^2*e0^2*e2*e3+3*b3^2*e0*e1*e2^2+3*b3^2*e0*e1*e3^2+3*b3^2*e1^2*e2*e3,
b0^2*e0^3*e2+b0^2*e0*e2^3+b0^2*e1^3*e3+b0^2*e1*e3^3+2*b0*b1*e0^3*e3+2*b0*b1*e0*e3^3+2*b0*b1*e1^3*e2+2*b0*b1*e1*e2^3+2*b0*b2*e0^4+2*b0*b2*e1^4+2*b0*b2*e2^4+2*b0*b2*e3^4+2*b0*b3*e0^3*e1+2*b0*b3*e0*e1^3+2*b0*b3*e2^3*e3+2*b0*b3*e2*e3^3+b1^2*e0^3*e2+b1^2*e0*e2^3+b1^2*e1^3*e3+b1^2*e1*e3^3+2*b1*b2*e0^3*e1+2*b1*b2*e0*e1^3+2*b1*b2*e2^3*e3+2*b1*b2*e2*e3^3+2*b1*b3*e0^4+2*b1*b3*e1^4+2*b1*b3*e2^4+2*b1*b3*e3^4+b2^2*e0^3*e2+b2^2*e0*e2^3+b2^2*e1^3*e3+b2^2*e1*e3^3+2*b2*b3*e0^3*e3+2*b2*b3*e0*e3^3+2*b2*b3*e1^3*e2+2*b2*b3*e1*e2^3+b3^2*e0^3*e2+b3^2*e0*e2^3+b3^2*e1^3*e3+b3^2*e1*e3^3,
3*b0^2*e0^2*e1*e2+3*b0^2*e0*e1^2*e3+3*b0^2*e0*e2^2*e3+3*b0^2*e1*e2*e3^2+6*b0*b1*e0^2*e1*e3+6*b0*b1*e0*e1^2*e2+6*b0*b1*e0*e2*e3^2+6*b0*b1*e1*e2^2*e3+6*b0*b2*e0^3*e1+6*b0*b2*e0*e1^3+6*b0*b2*e2^3*e3+6*b0*b2*e2*e3^3+12*b0*b3*e0^2*e1^2+12*b0*b3*e2^2*e3^2+3*b1^2*e0^2*e1*e2+3*b1^2*e0*e1^2*e3+3*b1^2*e0*e2^2*e3+3*b1^2*e1*e2*e3^2+12*b1*b2*e0^2*e1^2+12*b1*b2*e2^2*e3^2+6*b1*b3*e0^3*e1+6*b1*b3*e0*e1^3+6*b1*b3*e2^3*e3+6*b1*b3*e2*e3^3+3*b2^2*e0^2*e1*e2+3*b2^2*e0*e1^2*e3+3*b2^2*e0*e2^2*e3+3*b2^2*e1*e2*e3^2+6*b2*b3*e0^2*e1*e3+6*b2*b3*e0*e1^2*e2+6*b2*b3*e0*e2*e3^2+6*b2*b3*e1*e2^2*e3+3*b3^2*e0^2*e1*e2+3*b3^2*e0*e1^2*e3+3*b3^2*e0*e2^2*e3+3*b3^2*e1*e2*e3^2,
3*b0^2*e0^2*e1*e3+3*b0^2*e0*e1^2*e2+3*b0^2*e0*e2*e3^2+3*b0^2*e1*e2^2*e3+6*b0*b1*e0^2*e1*e2+6*b0*b1*e0*e1^2*e3+6*b0*b1*e0*e2^2*e3+6*b0*b1*e1*e2*e3^2+12*b0*b2*e0^2*e1^2+12*b0*b2*e2^2*e3^2+6*b0*b3*e0^3*e1+6*b0*b3*e0*e1^3+6*b0*b3*e2^3*e3+6*b0*b3*e2*e3^3+3*b1^2*e0^2*e1*e3+3*b1^2*e0*e1^2*e2+3*b1^2*e0*e2*e3^2+3*b1^2*e1*e2^2*e3+6*b1*b2*e0^3*e1+6*b1*b2*e0*e1^3+6*b1*b2*e2^3*e3+6*b1*b2*e2*e3^3+12*b1*b3*e0^2*e1^2+12*b1*b3*e2^2*e3^2+3*b2^2*e0^2*e1*e3+3*b2^2*e0*e1^2*e2+3*b2^2*e0*e2*e3^2+3*b2^2*e1*e2^2*e3+6*b2*b3*e0^2*e1*e2+6*b2*b3*e0*e1^2*e3+6*b2*b3*e0*e2^2*e3+6*b2*b3*e1*e2*e3^2+3*b3^2*e0^2*e1*e3+3*b3^2*e0*e1^2*e2+3*b3^2*e0*e2*e3^2+3*b3^2*e1*e2^2*e3,
b0^2*e0^3*e3+b0^2*e0*e3^3+b0^2*e1^3*e2+b0^2*e1*e2^3+2*b0*b1*e0^3*e2+2*b0*b1*e0*e2^3+2*b0*b1*e1^3*e3+2*b0*b1*e1*e3^3+2*b0*b2*e0^3*e1+2*b0*b2*e0*e1^3+2*b0*b2*e2^3*e3+2*b0*b2*e2*e3^3+2*b0*b3*e0^4+2*b0*b3*e1^4+2*b0*b3*e2^4+2*b0*b3*e3^4+b1^2*e0^3*e3+b1^2*e0*e3^3+b1^2*e1^3*e2+b1^2*e1*e2^3+2*b1*b2*e0^4+2*b1*b2*e1^4+2*b1*b2*e2^4+2*b1*b2*e3^4+2*b1*b3*e0^3*e1+2*b1*b3*e0*e1^3+2*b1*b3*e2^3*e3+2*b1*b3*e2*e3^3+b2^2*e0^3*e3+b2^2*e0*e3^3+b2^2*e1^3*e2+b2^2*e1*e2^3+2*b2*b3*e0^3*e2+2*b2*b3*e0*e2^3+2*b2*b3*e1^3*e3+2*b2*b3*e1*e3^3+b3^2*e0^3*e3+b3^2*e0*e3^3+b3^2*e1^3*e2+b3^2*e1*e2^3;

// 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(3,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;
}