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

// This is the inverse of the Fourier transform.

matrix ptoq[31][19] =
1/256,3/128,3/256,3/128,7/128,3/32,7/64,9/128,7/256,3/64,9/128,5/256,3/64,9/128,3/32,7/64,7/128,3/64,3/128,
3/128,3/64,-3/128,-3/64,15/64,0,3/32,-15/64,9/128,3/32,-3/64,-9/128,-3/32,15/64,0,-3/32,-15/64,3/32,-3/64,
3/256,-3/128,9/256,-3/128,9/128,3/32,-15/64,15/128,21/256,-3/64,-9/128,15/256,-3/64,15/128,3/32,-15/64,9/128,-3/64,-3/128,
3/128,3/64,-3/128,-3/64,9/64,-3/16,3/32,3/64,-3/128,-3/32,9/64,3/128,-3/32,3/64,-3/16,3/32,9/64,-3/32,-3/64,
3/64,-3/32,-3/64,3/32,3/32,0,-3/16,-3/32,9/64,-3/16,3/32,-9/64,3/16,3/32,0,3/16,-3/32,-3/16,3/32,
3/128,-3/64,-3/128,3/64,9/64,-3/16,-3/32,3/64,-3/128,3/32,-9/64,3/128,3/32,3/64,-3/16,-3/32,9/64,3/32,3/64,
3/256,-3/128,9/256,-3/128,-3/128,-3/32,9/64,3/128,21/256,-3/64,-9/128,15/256,-3/64,3/128,-3/32,9/64,-3/128,-3/64,-3/128,
1/128,3/64,3/128,3/64,1/64,0,3/32,3/64,-5/128,-3/32,-3/64,1/128,3/32,-3/64,0,-3/32,-1/64,-3/32,3/64,
3/128,3/64,-3/128,-3/64,-3/64,3/16,3/32,-9/64,-3/128,-3/32,9/64,3/128,-3/32,-9/64,3/16,3/32,-3/64,-3/32,-3/64,
3/128,-3/64,9/128,-3/64,3/64,0,-3/32,9/64,-15/128,3/32,3/64,3/128,-3/32,-9/64,0,3/32,-3/64,3/32,-3/64,
3/128,3/64,-3/128,-3/64,-9/64,0,3/32,9/64,9/128,3/32,-3/64,-9/128,-3/32,-9/64,0,-3/32,9/64,3/32,-3/64,
3/128,-3/64,-3/128,3/64,-3/64,3/16,-3/32,-9/64,-3/128,3/32,-9/64,3/128,3/32,-9/64,3/16,-3/32,-3/64,3/32,3/64,
1/256,3/128,3/256,3/128,-5/128,-3/32,-1/64,-3/128,7/256,3/64,9/128,5/256,3/64,-3/128,-3/32,-1/64,-5/128,3/64,3/128,
1/64,3/32,3/64,3/32,3/32,3/16,1/8,3/32,-1/64,0,-3/32,-3/64,0,-3/32,-3/16,-1/8,-3/32,0,-3/32,
3/64,3/32,-3/64,-3/32,9/32,-3/16,0,-3/32,-3/64,0,-3/32,3/64,0,-9/32,3/16,0,3/32,0,3/32,
3/64,-3/32,9/64,-3/32,3/32,3/16,-3/8,3/32,-3/64,0,3/32,-9/64,0,-3/32,-3/16,3/8,-3/32,0,3/32,
3/64,3/32,-3/64,-3/32,3/32,3/16,0,-9/32,-3/64,0,-3/32,3/64,0,-3/32,-3/16,0,9/32,0,3/32,
3/32,-3/16,-3/32,3/16,3/16,-3/8,0,3/16,-3/32,0,3/16,3/32,0,-3/16,3/8,0,-3/16,0,-3/16,
3/64,-3/32,9/64,-3/32,-3/32,-3/16,3/8,-3/32,-3/64,0,3/32,-9/64,0,3/32,3/16,-3/8,3/32,0,3/32,
3/32,-3/16,-3/32,3/16,-3/16,3/8,0,-3/16,-3/32,0,3/16,3/32,0,3/16,-3/8,0,3/16,0,-3/16,
3/64,3/32,-3/64,-3/32,-3/32,-3/16,0,9/32,-3/64,0,-3/32,3/64,0,3/32,3/16,0,-9/32,0,3/32,
1/64,3/32,3/64,3/32,-3/32,-3/16,-1/8,-3/32,-1/64,0,-3/32,-3/64,0,3/32,3/16,1/8,3/32,0,-3/32,
3/64,3/32,-3/64,-3/32,-9/32,3/16,0,3/32,-3/64,0,-3/32,3/64,0,9/32,-3/16,0,-3/32,0,3/32,
1/128,3/64,3/128,3/64,-1/64,0,-3/32,-3/64,7/128,3/32,9/64,5/128,-3/32,-3/64,0,-3/32,-1/64,-3/32,3/64,
3/64,3/32,-3/64,-3/32,-3/32,0,-3/16,3/32,9/64,3/16,-3/32,-9/64,3/16,-3/32,0,3/16,3/32,-3/16,-3/32,
3/128,-3/64,9/128,-3/64,-3/64,0,3/32,-9/64,21/128,-3/32,-9/64,15/128,3/32,-9/64,0,3/32,-3/64,3/32,-3/64,
3/64,3/32,-3/64,-3/32,-3/32,0,-3/16,3/32,-3/64,-3/16,9/32,3/64,3/16,3/32,0,-3/16,-3/32,3/16,-3/32,
3/64,-3/32,-3/64,3/32,-3/32,0,3/16,3/32,9/64,-3/16,3/32,-9/64,-3/16,-3/32,0,-3/16,3/32,3/16,3/32,
3/64,-3/32,-3/64,3/32,-3/32,0,3/16,3/32,-3/64,3/16,-9/32,3/64,-3/16,3/32,0,3/16,-3/32,-3/16,3/32,
1/128,3/64,3/128,3/64,-1/64,0,-3/32,-3/64,-5/128,-3/32,-3/64,1/128,-3/32,3/64,0,3/32,1/64,3/32,3/64,
3/128,-3/64,9/128,-3/64,-3/64,0,3/32,-9/64,-15/128,3/32,3/64,3/128,3/32,9/64,0,-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),dp;

//This is the Fourier transform.

matrix qtop[19][31] =
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,-1/3,1/3,-1/3,-1/3,-1/3,1,1/3,-1/3,1/3,-1/3,1,1,1/3,-1/3,1/3,-1/3,-1/3,-1/3,1/3,1,1/3,1,1/3,-1/3,1/3,-1/3,-1/3,1,-1/3,
1,-1/3,1,-1/3,-1/3,-1/3,1,1,-1/3,1,-1/3,-1/3,1,1,-1/3,1,-1/3,-1/3,1,-1/3,-1/3,1,-1/3,1,-1/3,1,-1/3,-1/3,-1/3,1,1,
1,-1/3,-1/3,-1/3,1/3,1/3,-1/3,1,-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,-1/3,1,-1/3,-1/3,-1/3,1/3,1/3,1,-1/3,
1,5/7,3/7,3/7,1/7,3/7,-1/7,1/7,-1/7,1/7,-3/7,-1/7,-5/7,3/7,3/7,1/7,1/7,1/7,-1/7,-1/7,-1/7,-3/7,-3/7,-1/7,-1/7,-1/7,-1/7,-1/7,-1/7,-1/7,-1/7,
1,0,1/3,-1/3,0,-1/3,-1/3,0,1/3,0,0,1/3,-1,1/2,-1/6,1/6,1/6,-1/6,-1/6,1/6,-1/6,-1/2,1/6,0,0,0,0,0,0,0,0,
1,1/7,-5/7,1/7,-1/7,-1/7,3/7,3/7,1/7,-1/7,1/7,-1/7,-1/7,2/7,0,-2/7,0,0,2/7,0,0,-2/7,0,-3/7,-1/7,1/7,-1/7,1/7,1/7,-3/7,1/7,
1,-5/9,5/9,1/9,-1/9,1/9,1/9,1/3,-1/3,1/3,1/3,-1/3,-1/3,1/3,-1/9,1/9,-1/3,1/9,-1/9,-1/9,1/3,-1/3,1/9,-1/3,1/9,-1/3,1/9,1/9,1/9,-1/3,-1/3,
1,3/7,1,-1/7,3/7,-1/7,1,-5/7,-1/7,-5/7,3/7,-1/7,1,-1/7,-1/7,-1/7,-1/7,-1/7,-1/7,-1/7,-1/7,-1/7,-1/7,1,3/7,1,-1/7,3/7,-1/7,-5/7,-5/7,
1,1/3,-1/3,-1/3,-1/3,1/3,-1/3,-1,-1/3,1/3,1/3,1/3,1,0,0,0,0,0,0,0,0,0,0,1,1/3,-1/3,-1/3,-1/3,1/3,-1,1/3,
1,-1/9,-1/3,1/3,1/9,-1/3,-1/3,-1/3,1/3,1/9,-1/9,-1/3,1,-1/3,-1/9,1/9,-1/9,1/9,1/9,1/9,-1/9,-1/3,-1/9,1,-1/9,-1/3,1/3,1/9,-1/3,-1/3,1/9,
1,-3/5,1,1/5,-3/5,1/5,1,1/5,1/5,1/5,-3/5,1/5,1,-3/5,1/5,-3/5,1/5,1/5,-3/5,1/5,1/5,-3/5,1/5,1,-3/5,1,1/5,-3/5,1/5,1/5,1/5,
1,-1/3,-1/3,-1/3,1/3,1/3,-1/3,1,-1/3,-1/3,-1/3,1/3,1,0,0,0,0,0,0,0,0,0,0,-1,1/3,1/3,1/3,-1/3,-1/3,-1,1/3,
1,5/9,5/9,1/9,1/9,1/9,1/9,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/9,-1/9,-1/9,1/9,1/9,1/9,1/3,1/3,-1/3,-1/9,-1/3,1/9,-1/9,1/9,1/3,1/3,
1,0,1/3,-1/3,0,-1/3,-1/3,0,1/3,0,0,1/3,-1,-1/2,1/6,-1/6,-1/6,1/6,1/6,-1/6,1/6,1/2,-1/6,0,0,0,0,0,0,0,0,
1,-1/7,-5/7,1/7,1/7,-1/7,3/7,-3/7,1/7,1/7,-1/7,-1/7,-1/7,-2/7,0,2/7,0,0,-2/7,0,0,2/7,0,-3/7,1/7,1/7,-1/7,-1/7,1/7,3/7,-1/7,
1,-5/7,3/7,3/7,-1/7,3/7,-1/7,-1/7,-1/7,-1/7,3/7,-1/7,-5/7,-3/7,1/7,-1/7,3/7,-1/7,1/7,1/7,-3/7,3/7,-1/7,-1/7,1/7,-1/7,-1/7,1/7,-1/7,1/7,1/7,
1,1/3,-1/3,-1/3,-1/3,1/3,-1/3,-1,-1/3,1/3,1/3,1/3,1,0,0,0,0,0,0,0,0,0,0,-1,-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,-1/3,-1/3,-1/3,1/3,1,-1,1/3,1/3,1/3,-1/3,1/3,-1/3,1/3,-1,1/3,1,-1/3,-1/3,-1/3,1/3,1/3,1,-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,(c0,c1,c2,f0,f1,f2),dp;

ideal P = 
c0^3*f0^5+2*c0^3*f1^5+c0^3*f2^5+4*c0^2*c1*f0^4*f1+2*c0^2*c1*f0^3*f1^2+2*c0^2*c1*f0^2*f1^3+4*c0^2*c1*f0*f1^4+4*c0^2*c1*f1^4*f2+2*c0^2*c1*f1^3*f2^2+2*c0^2*c1*f1^2*f2^3+4*c0^2*c1*f1*f2^4+2*c0^2*c2*f0^4*f2+c0^2*c2*f0^3*f2^2+c0^2*c2*f0^2*f2^3+2*c0^2*c2*f0*f2^4+6*c0^2*c2*f1^5+4*c0*c1^2*f0^4*f1+4*c0*c1^2*f0^3*f1^2+4*c0*c1^2*f0^3*f1*f2+2*c0*c1^2*f0^2*f1^3+8*c0*c1^2*f0*f1^4+4*c0*c1^2*f0*f1^3*f2+4*c0*c1^2*f0*f1*f2^3+8*c0*c1^2*f1^4*f2+2*c0*c1^2*f1^3*f2^2+4*c0*c1^2*f1^2*f2^3+4*c0*c1^2*f1*f2^4+4*c0*c1*c2*f0^3*f1^2+8*c0*c1*c2*f0^3*f1*f2+8*c0*c1*c2*f0*f1^4+8*c0*c1*c2*f0*f1^3*f2+8*c0*c1*c2*f0*f1*f2^3+8*c0*c1*c2*f1^4*f2+4*c0*c1*c2*f1^2*f2^3+2*c0*c2^2*f0^4*f2+c0*c2^2*f0^3*f2^2+c0*c2^2*f0^2*f2^3+2*c0*c2^2*f0*f2^4+6*c0*c2^2*f1^5+2*c1^3*f0^5+4*c1^3*f0^4*f2+2*c1^3*f0^3*f2^2+2*c1^3*f0^2*f2^3+4*c1^3*f0*f2^4+16*c1^3*f1^5+2*c1^3*f2^5+4*c1^2*c2*f0^4*f1+4*c1^2*c2*f0^3*f1^2+4*c1^2*c2*f0^3*f1*f2+2*c1^2*c2*f0^2*f1^3+8*c1^2*c2*f0*f1^4+4*c1^2*c2*f0*f1^3*f2+4*c1^2*c2*f0*f1*f2^3+8*c1^2*c2*f1^4*f2+2*c1^2*c2*f1^3*f2^2+4*c1^2*c2*f1^2*f2^3+4*c1^2*c2*f1*f2^4+4*c1*c2^2*f0^4*f1+2*c1*c2^2*f0^3*f1^2+2*c1*c2^2*f0^2*f1^3+4*c1*c2^2*f0*f1^4+4*c1*c2^2*f1^4*f2+2*c1*c2^2*f1^3*f2^2+2*c1*c2^2*f1^2*f2^3+4*c1*c2^2*f1*f2^4+c2^3*f0^5+2*c2^3*f1^5+c2^3*f2^5,
6*c0^3*f0^4*f1+6*c0^3*f0*f1^4+6*c0^3*f1^4*f2+6*c0^3*f1*f2^4+30*c0^2*c1*f0^3*f1^2+24*c0^2*c1*f0^2*f1^3+6*c0^2*c1*f0^2*f1^2*f2+24*c0^2*c1*f0*f1^3*f2+6*c0^2*c1*f0*f1^2*f2^2+24*c0^2*c1*f1^3*f2^2+30*c0^2*c1*f1^2*f2^3+12*c0^2*c2*f0^3*f1*f2+12*c0^2*c2*f0^2*f1*f2^2+18*c0^2*c2*f0*f1^4+12*c0^2*c2*f0*f1*f2^3+18*c0^2*c2*f1^4*f2+30*c0*c1^2*f0^3*f1^2+48*c0*c1^2*f0^2*f1^3+42*c0*c1^2*f0^2*f1^2*f2+48*c0*c1^2*f0*f1^3*f2+42*c0*c1^2*f0*f1^2*f2^2+48*c0*c1^2*f1^3*f2^2+30*c0*c1^2*f1^2*f2^3+48*c0*c1*c2*f0^2*f1^3+72*c0*c1*c2*f0^2*f1^2*f2+48*c0*c1*c2*f0*f1^3*f2+72*c0*c1*c2*f0*f1^2*f2^2+48*c0*c1*c2*f1^3*f2^2+12*c0*c2^2*f0^3*f1*f2+12*c0*c2^2*f0^2*f1*f2^2+18*c0*c2^2*f0*f1^4+12*c0*c2^2*f0*f1*f2^3+18*c0*c2^2*f1^4*f2+12*c1^3*f0^4*f1+24*c1^3*f0^3*f1*f2+24*c1^3*f0^2*f1*f2^2+48*c1^3*f0*f1^4+24*c1^3*f0*f1*f2^3+48*c1^3*f1^4*f2+12*c1^3*f1*f2^4+30*c1^2*c2*f0^3*f1^2+48*c1^2*c2*f0^2*f1^3+42*c1^2*c2*f0^2*f1^2*f2+48*c1^2*c2*f0*f1^3*f2+42*c1^2*c2*f0*f1^2*f2^2+48*c1^2*c2*f1^3*f2^2+30*c1^2*c2*f1^2*f2^3+30*c1*c2^2*f0^3*f1^2+24*c1*c2^2*f0^2*f1^3+6*c1*c2^2*f0^2*f1^2*f2+24*c1*c2^2*f0*f1^3*f2+6*c1*c2^2*f0*f1^2*f2^2+24*c1*c2^2*f1^3*f2^2+30*c1*c2^2*f1^2*f2^3+6*c2^3*f0^4*f1+6*c2^3*f0*f1^4+6*c2^3*f1^4*f2+6*c2^3*f1*f2^4,
3*c0^3*f0^4*f2+3*c0^3*f0*f2^4+6*c0^3*f1^5+12*c0^2*c1*f0^3*f1*f2+6*c0^2*c1*f0^2*f1^3+6*c0^2*c1*f0^2*f1^2*f2+12*c0^2*c1*f0*f1^4+6*c0^2*c1*f0*f1^2*f2^2+12*c0^2*c1*f0*f1*f2^3+12*c0^2*c1*f1^4*f2+6*c0^2*c1*f1^3*f2^2+9*c0^2*c2*f0^3*f2^2+9*c0^2*c2*f0^2*f2^3+18*c0^2*c2*f1^5+12*c0*c1^2*f0^3*f1*f2+6*c0*c1^2*f0^2*f1^3+12*c0*c1^2*f0^2*f1^2*f2+24*c0*c1^2*f0^2*f1*f2^2+24*c0*c1^2*f0*f1^4+12*c0*c1^2*f0*f1^3*f2+12*c0*c1^2*f0*f1^2*f2^2+12*c0*c1^2*f0*f1*f2^3+24*c0*c1^2*f1^4*f2+6*c0*c1^2*f1^3*f2^2+12*c0*c1*c2*f0^2*f1^2*f2+48*c0*c1*c2*f0^2*f1*f2^2+24*c0*c1*c2*f0*f1^4+24*c0*c1*c2*f0*f1^3*f2+12*c0*c1*c2*f0*f1^2*f2^2+24*c0*c1*c2*f1^4*f2+9*c0*c2^2*f0^3*f2^2+9*c0*c2^2*f0^2*f2^3+18*c0*c2^2*f1^5+6*c1^3*f0^4*f2+18*c1^3*f0^3*f2^2+18*c1^3*f0^2*f2^3+6*c1^3*f0*f2^4+48*c1^3*f1^5+12*c1^2*c2*f0^3*f1*f2+6*c1^2*c2*f0^2*f1^3+12*c1^2*c2*f0^2*f1^2*f2+24*c1^2*c2*f0^2*f1*f2^2+24*c1^2*c2*f0*f1^4+12*c1^2*c2*f0*f1^3*f2+12*c1^2*c2*f0*f1^2*f2^2+12*c1^2*c2*f0*f1*f2^3+24*c1^2*c2*f1^4*f2+6*c1^2*c2*f1^3*f2^2+12*c1*c2^2*f0^3*f1*f2+6*c1*c2^2*f0^2*f1^3+6*c1*c2^2*f0^2*f1^2*f2+12*c1*c2^2*f0*f1^4+6*c1*c2^2*f0*f1^2*f2^2+12*c1*c2^2*f0*f1*f2^3+12*c1*c2^2*f1^4*f2+6*c1*c2^2*f1^3*f2^2+3*c2^3*f0^4*f2+3*c2^3*f0*f2^4+6*c2^3*f1^5,
6*c0^3*f0^3*f1^2+6*c0^3*f0^2*f1^3+6*c0^3*f1^3*f2^2+6*c0^3*f1^2*f2^3+6*c0^2*c1*f0^4*f1+12*c0^2*c1*f0^3*f1^2+24*c0^2*c1*f0^2*f1^3+12*c0^2*c1*f0^2*f1^2*f2+12*c0^2*c1*f0^2*f1*f2^2+12*c0^2*c1*f0*f1^4+12*c0^2*c1*f0*f1^2*f2^2+12*c0^2*c1*f1^4*f2+24*c0^2*c1*f1^3*f2^2+12*c0^2*c1*f1^2*f2^3+6*c0^2*c1*f1*f2^4+18*c0^2*c2*f0^2*f1^3+18*c0^2*c2*f0^2*f1^2*f2+18*c0^2*c2*f0*f1^2*f2^2+18*c0^2*c2*f1^3*f2^2+6*c0*c1^2*f0^4*f1+24*c0*c1^2*f0^3*f1^2+12*c0*c1^2*f0^3*f1*f2+24*c0*c1^2*f0^2*f1^3+24*c0*c1^2*f0^2*f1^2*f2+12*c0*c1^2*f0^2*f1*f2^2+24*c0*c1^2*f0*f1^4+48*c0*c1^2*f0*f1^3*f2+24*c0*c1^2*f0*f1^2*f2^2+12*c0*c1^2*f0*f1*f2^3+24*c0*c1^2*f1^4*f2+24*c0*c1^2*f1^3*f2^2+24*c0*c1^2*f1^2*f2^3+6*c0*c1^2*f1*f2^4+24*c0*c1*c2*f0^3*f1^2+24*c0*c1*c2*f0^3*f1*f2+24*c0*c1*c2*f0^2*f1^2*f2+24*c0*c1*c2*f0*f1^4+96*c0*c1*c2*f0*f1^3*f2+24*c0*c1*c2*f0*f1^2*f2^2+24*c0*c1*c2*f0*f1*f2^3+24*c0*c1*c2*f1^4*f2+24*c0*c1*c2*f1^2*f2^3+18*c0*c2^2*f0^2*f1^3+18*c0*c2^2*f0^2*f1^2*f2+18*c0*c2^2*f0*f1^2*f2^2+18*c0*c2^2*f1^3*f2^2+12*c1^3*f0^3*f1^2+48*c1^3*f0^2*f1^3+36*c1^3*f0^2*f1^2*f2+36*c1^3*f0*f1^2*f2^2+48*c1^3*f1^3*f2^2+12*c1^3*f1^2*f2^3+6*c1^2*c2*f0^4*f1+24*c1^2*c2*f0^3*f1^2+12*c1^2*c2*f0^3*f1*f2+24*c1^2*c2*f0^2*f1^3+24*c1^2*c2*f0^2*f1^2*f2+12*c1^2*c2*f0^2*f1*f2^2+24*c1^2*c2*f0*f1^4+48*c1^2*c2*f0*f1^3*f2+24*c1^2*c2*f0*f1^2*f2^2+12*c1^2*c2*f0*f1*f2^3+24*c1^2*c2*f1^4*f2+24*c1^2*c2*f1^3*f2^2+24*c1^2*c2*f1^2*f2^3+6*c1^2*c2*f1*f2^4+6*c1*c2^2*f0^4*f1+12*c1*c2^2*f0^3*f1^2+24*c1*c2^2*f0^2*f1^3+12*c1*c2^2*f0^2*f1^2*f2+12*c1*c2^2*f0^2*f1*f2^2+12*c1*c2^2*f0*f1^4+12*c1*c2^2*f0*f1^2*f2^2+12*c1*c2^2*f1^4*f2+24*c1*c2^2*f1^3*f2^2+12*c1*c2^2*f1^2*f2^3+6*c1*c2^2*f1*f2^4+6*c2^3*f0^3*f1^2+6*c2^3*f0^2*f1^3+6*c2^3*f1^3*f2^2+6*c2^3*f1^2*f2^3,
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6*c0^3*f0^2*f1^2*f2+12*c0^3*f0*f1^3*f2+6*c0^3*f0*f1^2*f2^2+12*c0^2*c1*f0^3*f1*f2+12*c0^2*c1*f0^3*f2^2+24*c0^2*c1*f0^2*f1*f2^2+12*c0^2*c1*f0^2*f2^3+24*c0^2*c1*f0*f1^4+12*c0^2*c1*f0*f1*f2^3+24*c0^2*c1*f1^5+24*c0^2*c1*f1^4*f2+12*c0^2*c2*f0^2*f1^3+18*c0^2*c2*f0^2*f1^2*f2+12*c0^2*c2*f0*f1^3*f2+18*c0^2*c2*f0*f1^2*f2^2+12*c0^2*c2*f1^3*f2^2+6*c0*c1^2*f0^4*f2+24*c0*c1^2*f0^3*f1*f2+18*c0*c1^2*f0^3*f2^2+48*c0*c1^2*f0^2*f1*f2^2+18*c0*c1^2*f0^2*f2^3+48*c0*c1^2*f0*f1^4+24*c0*c1^2*f0*f1*f2^3+6*c0*c1^2*f0*f2^4+48*c0*c1^2*f1^5+48*c0*c1^2*f1^4*f2+12*c0*c1*c2*f0^4*f2+24*c0*c1*c2*f0^3*f1*f2+12*c0*c1*c2*f0^3*f2^2+48*c0*c1*c2*f0^2*f1*f2^2+12*c0*c1*c2*f0^2*f2^3+48*c0*c1*c2*f0*f1^4+24*c0*c1*c2*f0*f1*f2^3+12*c0*c1*c2*f0*f2^4+48*c0*c1*c2*f1^5+48*c0*c1*c2*f1^4*f2+12*c0*c2^2*f0^2*f1^3+18*c0*c2^2*f0^2*f1^2*f2+12*c0*c2^2*f0*f1^3*f2+18*c0*c2^2*f0*f1^2*f2^2+12*c0*c2^2*f1^3*f2^2+24*c1^3*f0^2*f1^3+48*c1^3*f0^2*f1^2*f2+48*c1^3*f0*f1^3*f2+48*c1^3*f0*f1^2*f2^2+24*c1^3*f1^3*f2^2+6*c1^2*c2*f0^4*f2+24*c1^2*c2*f0^3*f1*f2+18*c1^2*c2*f0^3*f2^2+48*c1^2*c2*f0^2*f1*f2^2+18*c1^2*c2*f0^2*f2^3+48*c1^2*c2*f0*f1^4+24*c1^2*c2*f0*f1*f2^3+6*c1^2*c2*f0*f2^4+48*c1^2*c2*f1^5+48*c1^2*c2*f1^4*f2+12*c1*c2^2*f0^3*f1*f2+12*c1*c2^2*f0^3*f2^2+24*c1*c2^2*f0^2*f1*f2^2+12*c1*c2^2*f0^2*f2^3+24*c1*c2^2*f0*f1^4+12*c1*c2^2*f0*f1*f2^3+24*c1*c2^2*f1^5+24*c1*c2^2*f1^4*f2+6*c2^3*f0^2*f1^2*f2+12*c2^3*f0*f1^3*f2+6*c2^3*f0*f1^2*f2^2;

// 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;
}