//This is the ideal Fourier invariants.

ring rQ = 0,(q1,q2,q3,q4,q5,q6,q7,q8,q9,q10,q11,q12,q13,q14),dp;

ideal Invariants = 
q11*q13-q12*q14,
q9*q13-q8*q14,
q3*q13-q2*q14,
q8*q11-q9*q12,
q7*q11-q10*q13,
q2*q11-q3*q12,
q9*q10-q5*q11,
q8*q10-q5*q12,
q7*q10-q6*q12,
q5*q10-q4*q11,
q7*q9-q5*q13,
q6*q9-q4*q14,
q6*q8-q4*q13,
q3*q8-q2*q9,
q5*q7-q6*q8,
q3*q6-q1*q14,
q2*q6-q1*q13,
q5^2-q4*q9,
q3*q4-q1*q9,
q2*q4-q1*q8,
q6*q13^2-q7^2*q14,
q6*q11^2-q10^2*q14,
q6*q9^2-q3*q11^2,
q6*q8*q9-q3*q11*q12,
q5*q6*q9-q3*q10*q11,
q6*q8^2-q3*q12^2,
q4*q7*q8-q2*q10*q12,
q5*q6*q8-q3*q10*q12,
q4*q6*q8-q1*q11*q12,
q4*q6*q8-q2*q10^2,
q3*q7^2-q2*q6*q13,
q4^2*q7-q1*q10*q12,
q5^2*q6-q1*q11^2,
q5^2*q6-q3*q10^2,
q4*q5*q6-q1*q10*q11,
q4^2*q6-q1*q10^2,
q7^2*q8^2-q2*q12^2*q13;

// This is the inverse of the Fourier transform.

matrix ptoq[18][14] =
1/64,1/32,1/64,5/32,1/8,5/64,1/16,1/16,1/32,3/16,1/8,1/16,1/32,1/64,
1/16,0,-1/16,3/8,-1/4,1/16,0,0,-1/8,1/4,-1/4,0,0,-1/16,
1/32,-1/16,1/32,1/16,0,5/32,-1/8,-1/8,1/16,1/8,0,-1/8,-1/16,1/32,
1/32,1/16,1/32,1/16,0,-3/32,-1/8,1/8,1/16,-1/8,0,-1/8,1/16,1/32,
1/16,0,-1/16,-1/8,1/4,1/16,0,0,-1/8,-1/4,1/4,0,0,-1/16,
1/32,-1/16,1/32,1/16,0,-3/32,1/8,-1/8,1/16,-1/8,0,1/8,-1/16,1/32,
1/64,1/32,1/64,-3/32,-1/8,5/64,1/16,1/16,1/32,-1/16,-1/8,1/16,1/32,1/64,
1/16,1/8,1/16,1/4,1/4,-1/16,0,0,0,-1/4,-1/4,0,-1/8,-1/16,
1/16,0,-1/16,1/4,-1/4,-1/16,0,0,0,-1/4,1/4,0,0,1/16,
1/8,-1/4,1/8,0,0,-1/8,0,0,0,0,0,0,1/4,-1/8,
1/8,0,-1/8,0,0,-1/8,0,0,0,0,0,0,0,1/8,
1/16,1/8,1/16,-1/4,-1/4,-1/16,0,0,0,1/4,1/4,0,-1/8,-1/16,
1/16,0,-1/16,-1/4,1/4,-1/16,0,0,0,1/4,-1/4,0,0,1/16,
1/32,1/16,1/32,-1/16,0,5/32,1/8,-1/8,-1/16,-1/8,0,-1/8,1/16,1/32,
1/8,0,-1/8,-1/4,0,1/8,0,0,1/4,0,0,0,0,-1/8,
1/32,-1/16,1/32,-1/16,0,5/32,-1/8,1/8,-1/16,-1/8,0,1/8,-1/16,1/32,
1/32,1/16,1/32,-1/16,0,-3/32,-1/8,-1/8,-1/16,1/8,0,1/8,1/16,1/32,
1/32,-1/16,1/32,-1/16,0,-3/32,1/8,1/8,-1/16,1/8,0,-1/8,-1/16,1/32;

// 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),dp;

//This is the Fourier transform.

matrix qtop[14][18] =
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,0,-1,1,0,-1,1,1,0,-1,0,1,0,1,0,-1,1,-1,
1,-1,1,1,-1,1,1,1,-1,1,-1,1,-1,1,-1,1,1,1,
1,3/5,1/5,1/5,-1/5,1/5,-3/5,2/5,2/5,0,0,-2/5,-2/5,-1/5,-1/5,-1/5,-1/5,-1/5,
1,-1/2,0,0,1/2,0,-1,1/2,-1/2,0,0,-1/2,1/2,0,0,0,0,0,
1,1/5,1,-3/5,1/5,-3/5,1,-1/5,-1/5,-1/5,-1/5,-1/5,-1/5,1,1/5,1,-3/5,-3/5,
1,0,-1,-1,0,1,1,0,0,0,0,0,0,1,0,-1,-1,1,
1,0,-1,1,0,-1,1,0,0,0,0,0,0,-1,0,1,-1,1,
1,-1,1,1,-1,1,1,0,0,0,0,0,0,-1,1,-1,-1,-1,
1,1/3,1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,0,0,1/3,1/3,-1/3,0,-1/3,1/3,1/3,
1,-1/2,0,0,1/2,0,-1,-1/2,1/2,0,0,1/2,-1/2,0,0,0,0,0,
1,0,-1,-1,0,1,1,0,0,0,0,0,0,-1,0,1,1,-1,
1,0,-1,1,0,-1,1,-1,0,1,0,-1,0,1,0,-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,(c0,c1,c2,e0,e1,e2),dp;

ideal P = 
c0^3*e0^4+2*c0^3*e1^4+c0^3*e2^4+4*c0^2*c1*e0^3*e1+4*c0^2*c1*e0^2*e1^2+4*c0^2*c1*e0*e1^3+4*c0^2*c1*e1^3*e2+4*c0^2*c1*e1^2*e2^2+4*c0^2*c1*e1*e2^3+2*c0^2*c2*e0^3*e2+2*c0^2*c2*e0^2*e2^2+2*c0^2*c2*e0*e2^3+6*c0^2*c2*e1^4+4*c0*c1^2*e0^3*e1+6*c0*c1^2*e0^2*e1^2+4*c0*c1^2*e0^2*e1*e2+8*c0*c1^2*e0*e1^3+4*c0*c1^2*e0*e1^2*e2+4*c0*c1^2*e0*e1*e2^2+8*c0*c1^2*e1^3*e2+6*c0*c1^2*e1^2*e2^2+4*c0*c1^2*e1*e2^3+4*c0*c1*c2*e0^2*e1^2+8*c0*c1*c2*e0^2*e1*e2+8*c0*c1*c2*e0*e1^3+8*c0*c1*c2*e0*e1^2*e2+8*c0*c1*c2*e0*e1*e2^2+8*c0*c1*c2*e1^3*e2+4*c0*c1*c2*e1^2*e2^2+2*c0*c2^2*e0^3*e2+2*c0*c2^2*e0^2*e2^2+2*c0*c2^2*e0*e2^3+6*c0*c2^2*e1^4+2*c1^3*e0^4+4*c1^3*e0^3*e2+4*c1^3*e0^2*e2^2+4*c1^3*e0*e2^3+16*c1^3*e1^4+2*c1^3*e2^4+4*c1^2*c2*e0^3*e1+6*c1^2*c2*e0^2*e1^2+4*c1^2*c2*e0^2*e1*e2+8*c1^2*c2*e0*e1^3+4*c1^2*c2*e0*e1^2*e2+4*c1^2*c2*e0*e1*e2^2+8*c1^2*c2*e1^3*e2+6*c1^2*c2*e1^2*e2^2+4*c1^2*c2*e1*e2^3+4*c1*c2^2*e0^3*e1+4*c1*c2^2*e0^2*e1^2+4*c1*c2^2*e0*e1^3+4*c1*c2^2*e1^3*e2+4*c1*c2^2*e1^2*e2^2+4*c1*c2^2*e1*e2^3+c2^3*e0^4+2*c2^3*e1^4+c2^3*e2^4,
4*c0^3*e0^3*e1+4*c0^3*e0*e1^3+4*c0^3*e1^3*e2+4*c0^3*e1*e2^3+4*c0^2*c1*e0^3*e1+24*c0^2*c1*e0^2*e1^2+4*c0^2*c1*e0^2*e1*e2+8*c0^2*c1*e0*e1^3+16*c0^2*c1*e0*e1^2*e2+4*c0^2*c1*e0*e1*e2^2+8*c0^2*c1*e1^3*e2+24*c0^2*c1*e1^2*e2^2+4*c0^2*c1*e1*e2^3+12*c0^2*c2*e0^2*e1*e2+12*c0^2*c2*e0*e1^3+12*c0^2*c2*e0*e1*e2^2+12*c0^2*c2*e1^3*e2+4*c0*c1^2*e0^3*e1+32*c0*c1^2*e0^2*e1^2+12*c0*c1^2*e0^2*e1*e2+16*c0*c1^2*e0*e1^3+64*c0*c1^2*e0*e1^2*e2+12*c0*c1^2*e0*e1*e2^2+16*c0*c1^2*e1^3*e2+32*c0*c1^2*e1^2*e2^2+4*c0*c1^2*e1*e2^3+16*c0*c1*c2*e0^2*e1^2+16*c0*c1*c2*e0^2*e1*e2+16*c0*c1*c2*e0*e1^3+96*c0*c1*c2*e0*e1^2*e2+16*c0*c1*c2*e0*e1*e2^2+16*c0*c1*c2*e1^3*e2+16*c0*c1*c2*e1^2*e2^2+12*c0*c2^2*e0^2*e1*e2+12*c0*c2^2*e0*e1^3+12*c0*c2^2*e0*e1*e2^2+12*c0*c2^2*e1^3*e2+8*c1^3*e0^3*e1+24*c1^3*e0^2*e1*e2+32*c1^3*e0*e1^3+24*c1^3*e0*e1*e2^2+32*c1^3*e1^3*e2+8*c1^3*e1*e2^3+4*c1^2*c2*e0^3*e1+32*c1^2*c2*e0^2*e1^2+12*c1^2*c2*e0^2*e1*e2+16*c1^2*c2*e0*e1^3+64*c1^2*c2*e0*e1^2*e2+12*c1^2*c2*e0*e1*e2^2+16*c1^2*c2*e1^3*e2+32*c1^2*c2*e1^2*e2^2+4*c1^2*c2*e1*e2^3+4*c1*c2^2*e0^3*e1+24*c1*c2^2*e0^2*e1^2+4*c1*c2^2*e0^2*e1*e2+8*c1*c2^2*e0*e1^3+16*c1*c2^2*e0*e1^2*e2+4*c1*c2^2*e0*e1*e2^2+8*c1*c2^2*e1^3*e2+24*c1*c2^2*e1^2*e2^2+4*c1*c2^2*e1*e2^3+4*c2^3*e0^3*e1+4*c2^3*e0*e1^3+4*c2^3*e1^3*e2+4*c2^3*e1*e2^3,
2*c0^3*e0^3*e2+2*c0^3*e0*e2^3+4*c0^3*e1^4+4*c0^2*c1*e0^2*e1^2+8*c0^2*c1*e0^2*e1*e2+8*c0^2*c1*e0*e1^3+8*c0^2*c1*e0*e1^2*e2+8*c0^2*c1*e0*e1*e2^2+8*c0^2*c1*e1^3*e2+4*c0^2*c1*e1^2*e2^2+2*c0^2*c2*e0^3*e2+8*c0^2*c2*e0^2*e2^2+2*c0^2*c2*e0*e2^3+12*c0^2*c2*e1^4+4*c0*c1^2*e0^2*e1^2+16*c0*c1^2*e0^2*e1*e2+16*c0*c1^2*e0*e1^3+24*c0*c1^2*e0*e1^2*e2+16*c0*c1^2*e0*e1*e2^2+16*c0*c1^2*e1^3*e2+4*c0*c1^2*e1^2*e2^2+16*c0*c1*c2*e0^2*e1*e2+16*c0*c1*c2*e0*e1^3+32*c0*c1*c2*e0*e1^2*e2+16*c0*c1*c2*e0*e1*e2^2+16*c0*c1*c2*e1^3*e2+2*c0*c2^2*e0^3*e2+8*c0*c2^2*e0^2*e2^2+2*c0*c2^2*e0*e2^3+12*c0*c2^2*e1^4+8*c1^3*e0^3*e2+16*c1^3*e0^2*e2^2+8*c1^3*e0*e2^3+32*c1^3*e1^4+4*c1^2*c2*e0^2*e1^2+16*c1^2*c2*e0^2*e1*e2+16*c1^2*c2*e0*e1^3+24*c1^2*c2*e0*e1^2*e2+16*c1^2*c2*e0*e1*e2^2+16*c1^2*c2*e1^3*e2+4*c1^2*c2*e1^2*e2^2+4*c1*c2^2*e0^2*e1^2+8*c1*c2^2*e0^2*e1*e2+8*c1*c2^2*e0*e1^3+8*c1*c2^2*e0*e1^2*e2+8*c1*c2^2*e0*e1*e2^2+8*c1*c2^2*e1^3*e2+4*c1*c2^2*e1^2*e2^2+2*c2^3*e0^3*e2+2*c2^3*e0*e2^3+4*c2^3*e1^4,
4*c0^3*e0^2*e1^2+4*c0^3*e1^2*e2^2+2*c0^2*c1*e0^4+4*c0^2*c1*e0^3*e1+4*c0^2*c1*e0^2*e1*e2+4*c0^2*c1*e0^2*e2^2+8*c0^2*c1*e0*e1^3+4*c0^2*c1*e0*e1*e2^2+8*c0^2*c1*e1^4+8*c0^2*c1*e1^3*e2+4*c0^2*c1*e1*e2^3+2*c0^2*c1*e2^4+8*c0^2*c2*e0^2*e1^2+8*c0^2*c2*e0*e1^2*e2+8*c0^2*c2*e1^2*e2^2+2*c0*c1^2*e0^4+8*c0*c1^2*e0^3*e1+4*c0*c1^2*e0^3*e2+8*c0*c1^2*e0^2*e1*e2+4*c0*c1^2*e0^2*e2^2+16*c0*c1^2*e0*e1^3+8*c0*c1^2*e0*e1*e2^2+4*c0*c1^2*e0*e2^3+16*c0*c1^2*e1^4+16*c0*c1^2*e1^3*e2+8*c0*c1^2*e1*e2^3+2*c0*c1^2*e2^4+8*c0*c1*c2*e0^3*e1+8*c0*c1*c2*e0^3*e2+8*c0*c1*c2*e0^2*e1*e2+16*c0*c1*c2*e0*e1^3+8*c0*c1*c2*e0*e1*e2^2+8*c0*c1*c2*e0*e2^3+16*c0*c1*c2*e1^4+16*c0*c1*c2*e1^3*e2+8*c0*c1*c2*e1*e2^3+8*c0*c2^2*e0^2*e1^2+8*c0*c2^2*e0*e1^2*e2+8*c0*c2^2*e1^2*e2^2+24*c1^3*e0^2*e1^2+16*c1^3*e0*e1^2*e2+24*c1^3*e1^2*e2^2+2*c1^2*c2*e0^4+8*c1^2*c2*e0^3*e1+4*c1^2*c2*e0^3*e2+8*c1^2*c2*e0^2*e1*e2+4*c1^2*c2*e0^2*e2^2+16*c1^2*c2*e0*e1^3+8*c1^2*c2*e0*e1*e2^2+4*c1^2*c2*e0*e2^3+16*c1^2*c2*e1^4+16*c1^2*c2*e1^3*e2+8*c1^2*c2*e1*e2^3+2*c1^2*c2*e2^4+2*c1*c2^2*e0^4+4*c1*c2^2*e0^3*e1+4*c1*c2^2*e0^2*e1*e2+4*c1*c2^2*e0^2*e2^2+8*c1*c2^2*e0*e1^3+4*c1*c2^2*e0*e1*e2^2+8*c1*c2^2*e1^4+8*c1*c2^2*e1^3*e2+4*c1*c2^2*e1*e2^3+2*c1*c2^2*e2^4+4*c2^3*e0^2*e1^2+4*c2^3*e1^2*e2^2,
4*c0^3*e0^2*e1*e2+4*c0^3*e0*e1^3+4*c0^3*e0*e1*e2^2+4*c0^3*e1^3*e2+4*c0^2*c1*e0^3*e1+8*c0^2*c1*e0^2*e1^2+4*c0^2*c1*e0^2*e1*e2+8*c0^2*c1*e0*e1^3+48*c0^2*c1*e0*e1^2*e2+4*c0^2*c1*e0*e1*e2^2+8*c0^2*c1*e1^3*e2+8*c0^2*c1*e1^2*e2^2+4*c0^2*c1*e1*e2^3+4*c0^2*c2*e0^3*e1+8*c0^2*c2*e0^2*e1*e2+12*c0^2*c2*e0*e1^3+8*c0^2*c2*e0*e1*e2^2+12*c0^2*c2*e1^3*e2+4*c0^2*c2*e1*e2^3+4*c0*c1^2*e0^3*e1+32*c0*c1^2*e0^2*e1^2+12*c0*c1^2*e0^2*e1*e2+16*c0*c1^2*e0*e1^3+64*c0*c1^2*e0*e1^2*e2+12*c0*c1^2*e0*e1*e2^2+16*c0*c1^2*e1^3*e2+32*c0*c1^2*e1^2*e2^2+4*c0*c1^2*e1*e2^3+48*c0*c1*c2*e0^2*e1^2+16*c0*c1*c2*e0^2*e1*e2+16*c0*c1*c2*e0*e1^3+32*c0*c1*c2*e0*e1^2*e2+16*c0*c1*c2*e0*e1*e2^2+16*c0*c1*c2*e1^3*e2+48*c0*c1*c2*e1^2*e2^2+4*c0*c2^2*e0^3*e1+8*c0*c2^2*e0^2*e1*e2+12*c0*c2^2*e0*e1^3+8*c0*c2^2*e0*e1*e2^2+12*c0*c2^2*e1^3*e2+4*c0*c2^2*e1*e2^3+8*c1^3*e0^3*e1+24*c1^3*e0^2*e1*e2+32*c1^3*e0*e1^3+24*c1^3*e0*e1*e2^2+32*c1^3*e1^3*e2+8*c1^3*e1*e2^3+4*c1^2*c2*e0^3*e1+32*c1^2*c2*e0^2*e1^2+12*c1^2*c2*e0^2*e1*e2+16*c1^2*c2*e0*e1^3+64*c1^2*c2*e0*e1^2*e2+12*c1^2*c2*e0*e1*e2^2+16*c1^2*c2*e1^3*e2+32*c1^2*c2*e1^2*e2^2+4*c1^2*c2*e1*e2^3+4*c1*c2^2*e0^3*e1+8*c1*c2^2*e0^2*e1^2+4*c1*c2^2*e0^2*e1*e2+8*c1*c2^2*e0*e1^3+48*c1*c2^2*e0*e1^2*e2+4*c1*c2^2*e0*e1*e2^2+8*c1*c2^2*e1^3*e2+8*c1*c2^2*e1^2*e2^2+4*c1*c2^2*e1*e2^3+4*c2^3*e0^2*e1*e2+4*c2^3*e0*e1^3+4*c2^3*e0*e1*e2^2+4*c2^3*e1^3*e2,
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8*c0^3*e0^2*e1*e2+8*c0^3*e0*e1^3+8*c0^3*e0*e1*e2^2+8*c0^3*e1^3*e2+32*c0^2*c1*e0^2*e1^2+16*c0^2*c1*e0^2*e1*e2+16*c0^2*c1*e0*e1^3+64*c0^2*c1*e0*e1^2*e2+16*c0^2*c1*e0*e1*e2^2+16*c0^2*c1*e1^3*e2+32*c0^2*c1*e1^2*e2^2+8*c0^2*c2*e0^3*e1+16*c0^2*c2*e0^2*e1*e2+24*c0^2*c2*e0*e1^3+16*c0^2*c2*e0*e1*e2^2+24*c0^2*c2*e1^3*e2+8*c0^2*c2*e1*e2^3+8*c0*c1^2*e0^3*e1+64*c0*c1^2*e0^2*e1^2+24*c0*c1^2*e0^2*e1*e2+32*c0*c1^2*e0*e1^3+128*c0*c1^2*e0*e1^2*e2+24*c0*c1^2*e0*e1*e2^2+32*c0*c1^2*e1^3*e2+64*c0*c1^2*e1^2*e2^2+8*c0*c1^2*e1*e2^3+16*c0*c1*c2*e0^3*e1+64*c0*c1*c2*e0^2*e1^2+16*c0*c1*c2*e0^2*e1*e2+32*c0*c1*c2*e0*e1^3+128*c0*c1*c2*e0*e1^2*e2+16*c0*c1*c2*e0*e1*e2^2+32*c0*c1*c2*e1^3*e2+64*c0*c1*c2*e1^2*e2^2+16*c0*c1*c2*e1*e2^3+8*c0*c2^2*e0^3*e1+16*c0*c2^2*e0^2*e1*e2+24*c0*c2^2*e0*e1^3+16*c0*c2^2*e0*e1*e2^2+24*c0*c2^2*e1^3*e2+8*c0*c2^2*e1*e2^3+16*c1^3*e0^3*e1+48*c1^3*e0^2*e1*e2+64*c1^3*e0*e1^3+48*c1^3*e0*e1*e2^2+64*c1^3*e1^3*e2+16*c1^3*e1*e2^3+8*c1^2*c2*e0^3*e1+64*c1^2*c2*e0^2*e1^2+24*c1^2*c2*e0^2*e1*e2+32*c1^2*c2*e0*e1^3+128*c1^2*c2*e0*e1^2*e2+24*c1^2*c2*e0*e1*e2^2+32*c1^2*c2*e1^3*e2+64*c1^2*c2*e1^2*e2^2+8*c1^2*c2*e1*e2^3+32*c1*c2^2*e0^2*e1^2+16*c1*c2^2*e0^2*e1*e2+16*c1*c2^2*e0*e1^3+64*c1*c2^2*e0*e1^2*e2+16*c1*c2^2*e0*e1*e2^2+16*c1*c2^2*e1^3*e2+32*c1*c2^2*e1^2*e2^2+8*c2^3*e0^2*e1*e2+8*c2^3*e0*e1^3+8*c2^3*e0*e1*e2^2+8*c2^3*e1^3*e2,
4*c0^3*e0^2*e2^2+4*c0^3*e1^4+8*c0^2*c1*e0^2*e1*e2+8*c0^2*c1*e0*e1^3+16*c0^2*c1*e0*e1^2*e2+8*c0^2*c1*e0*e1*e2^2+8*c0^2*c1*e1^3*e2+4*c0^2*c2*e0^3*e2+4*c0^2*c2*e0^2*e2^2+4*c0^2*c2*e0*e2^3+12*c0^2*c2*e1^4+4*c0*c1^2*e0^2*e1^2+16*c0*c1^2*e0^2*e1*e2+16*c0*c1^2*e0*e1^3+24*c0*c1^2*e0*e1^2*e2+16*c0*c1^2*e0*e1*e2^2+16*c0*c1^2*e1^3*e2+4*c0*c1^2*e1^2*e2^2+8*c0*c1*c2*e0^2*e1^2+16*c0*c1*c2*e0^2*e1*e2+16*c0*c1*c2*e0*e1^3+16*c0*c1*c2*e0*e1^2*e2+16*c0*c1*c2*e0*e1*e2^2+16*c0*c1*c2*e1^3*e2+8*c0*c1*c2*e1^2*e2^2+4*c0*c2^2*e0^3*e2+4*c0*c2^2*e0^2*e2^2+4*c0*c2^2*e0*e2^3+12*c0*c2^2*e1^4+8*c1^3*e0^3*e2+16*c1^3*e0^2*e2^2+8*c1^3*e0*e2^3+32*c1^3*e1^4+4*c1^2*c2*e0^2*e1^2+16*c1^2*c2*e0^2*e1*e2+16*c1^2*c2*e0*e1^3+24*c1^2*c2*e0*e1^2*e2+16*c1^2*c2*e0*e1*e2^2+16*c1^2*c2*e1^3*e2+4*c1^2*c2*e1^2*e2^2+8*c1*c2^2*e0^2*e1*e2+8*c1*c2^2*e0*e1^3+16*c1*c2^2*e0*e1^2*e2+8*c1*c2^2*e0*e1*e2^2+8*c1*c2^2*e1^3*e2+4*c2^3*e0^2*e2^2+4*c2^3*e1^4,
2*c0^3*e0^2*e1^2+4*c0^3*e0*e1^2*e2+2*c0^3*e1^2*e2^2+4*c0^2*c1*e0^3*e1+4*c0^2*c1*e0^3*e2+4*c0^2*c1*e0^2*e1*e2+8*c0^2*c1*e0*e1^3+4*c0^2*c1*e0*e1*e2^2+4*c0^2*c1*e0*e2^3+8*c0^2*c1*e1^4+8*c0^2*c1*e1^3*e2+4*c0^2*c1*e1*e2^3+10*c0^2*c2*e0^2*e1^2+4*c0^2*c2*e0*e1^2*e2+10*c0^2*c2*e1^2*e2^2+2*c0*c1^2*e0^4+8*c0*c1^2*e0^3*e1+4*c0*c1^2*e0^3*e2+8*c0*c1^2*e0^2*e1*e2+4*c0*c1^2*e0^2*e2^2+16*c0*c1^2*e0*e1^3+8*c0*c1^2*e0*e1*e2^2+4*c0*c1^2*e0*e2^3+16*c0*c1^2*e1^4+16*c0*c1^2*e1^3*e2+8*c0*c1^2*e1*e2^3+2*c0*c1^2*e2^4+4*c0*c1*c2*e0^4+8*c0*c1*c2*e0^3*e1+8*c0*c1*c2*e0^2*e1*e2+8*c0*c1*c2*e0^2*e2^2+16*c0*c1*c2*e0*e1^3+8*c0*c1*c2*e0*e1*e2^2+16*c0*c1*c2*e1^4+16*c0*c1*c2*e1^3*e2+8*c0*c1*c2*e1*e2^3+4*c0*c1*c2*e2^4+10*c0*c2^2*e0^2*e1^2+4*c0*c2^2*e0*e1^2*e2+10*c0*c2^2*e1^2*e2^2+24*c1^3*e0^2*e1^2+16*c1^3*e0*e1^2*e2+24*c1^3*e1^2*e2^2+2*c1^2*c2*e0^4+8*c1^2*c2*e0^3*e1+4*c1^2*c2*e0^3*e2+8*c1^2*c2*e0^2*e1*e2+4*c1^2*c2*e0^2*e2^2+16*c1^2*c2*e0*e1^3+8*c1^2*c2*e0*e1*e2^2+4*c1^2*c2*e0*e2^3+16*c1^2*c2*e1^4+16*c1^2*c2*e1^3*e2+8*c1^2*c2*e1*e2^3+2*c1^2*c2*e2^4+4*c1*c2^2*e0^3*e1+4*c1*c2^2*e0^3*e2+4*c1*c2^2*e0^2*e1*e2+8*c1*c2^2*e0*e1^3+4*c1*c2^2*e0*e1*e2^2+4*c1*c2^2*e0*e2^3+8*c1*c2^2*e1^4+8*c1*c2^2*e1^3*e2+4*c1*c2^2*e1*e2^3+2*c2^3*e0^2*e1^2+4*c2^3*e0*e1^2*e2+2*c2^3*e1^2*e2^2,
8*c0^3*e0*e1^2*e2+8*c0^2*c1*e0^2*e1*e2+8*c0^2*c1*e0^2*e2^2+8*c0^2*c1*e0*e1^3+8*c0^2*c1*e0*e1*e2^2+8*c0^2*c1*e1^4+8*c0^2*c1*e1^3*e2+4*c0^2*c2*e0^2*e1^2+16*c0^2*c2*e0*e1^2*e2+4*c0^2*c2*e1^2*e2^2+4*c0*c1^2*e0^3*e2+16*c0*c1^2*e0^2*e1*e2+8*c0*c1^2*e0^2*e2^2+16*c0*c1^2*e0*e1^3+16*c0*c1^2*e0*e1*e2^2+4*c0*c1^2*e0*e2^3+16*c0*c1^2*e1^4+16*c0*c1^2*e1^3*e2+8*c0*c1*c2*e0^3*e2+16*c0*c1*c2*e0^2*e1*e2+16*c0*c1*c2*e0*e1^3+16*c0*c1*c2*e0*e1*e2^2+8*c0*c1*c2*e0*e2^3+16*c0*c1*c2*e1^4+16*c0*c1*c2*e1^3*e2+4*c0*c2^2*e0^2*e1^2+16*c0*c2^2*e0*e1^2*e2+4*c0*c2^2*e1^2*e2^2+8*c1^3*e0^2*e1^2+48*c1^3*e0*e1^2*e2+8*c1^3*e1^2*e2^2+4*c1^2*c2*e0^3*e2+16*c1^2*c2*e0^2*e1*e2+8*c1^2*c2*e0^2*e2^2+16*c1^2*c2*e0*e1^3+16*c1^2*c2*e0*e1*e2^2+4*c1^2*c2*e0*e2^3+16*c1^2*c2*e1^4+16*c1^2*c2*e1^3*e2+8*c1*c2^2*e0^2*e1*e2+8*c1*c2^2*e0^2*e2^2+8*c1*c2^2*e0*e1^3+8*c1*c2^2*e0*e1*e2^2+8*c1*c2^2*e1^4+8*c1*c2^2*e1^3*e2+8*c2^3*e0*e1^2*e2;

// 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(2,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;
}