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e = SymmetricFunctionAlgebra(QQ, basis='elementary')
R.<x,z1,z2,z3,z4,z5,a1,a2,a3,a4,a5> = PolynomialRing(QQ,11,order='lex(1),lex(5),deglex(5)')
asym1 = -e([1]).expand(5).subs(x0=z1,x1=z2,x2=z3,x3=z4,x4=z5)
asym2 = +e([2]).expand(5).subs(x0=z1,x1=z2,x2=z3,x3=z4,x4=z5)
asym3 = -e([3]).expand(5).subs(x0=z1,x1=z2,x2=z3,x3=z4,x4=z5)
asym4 = +e([4]).expand(5).subs(x0=z1,x1=z2,x2=z3,x3=z4,x4=z5)
asym5 = -e([5]).expand(5).subs(x0=z1,x1=z2,x2=z3,x3=z4,x4=z5)
Isym = R.ideal([a1-asym1, a2-asym2, a3-asym3, a4-asym4, a5-asym5])
p = z1^2*(z2*z5 + z3*z4) + z2^2*(z1*z3 + z4*z5) + z3^2*(z1*z5 + z2*z4) + z4^2*(z1*z2 + z3*z5) + z5^2*(z1*z4 + z2*z3)
tmp = (x-p)*(x-p).subs({z4:z5,z5:z4})
res = tmp*tmp.subs({z1:z2,z2:z3,z3:z1})*tmp.subs({z1:z3,z2:z1,z3:z2})
B = Isym.groebner_basis()
res0 = res.reduce(B)