The realization space is
  [1                x1^2   1   0   0   1                x1^2     -x1 + 1    0     -x1^2 + x1                   x1^3]
  [1   2*x1^2 - 3*x1 + 1   0   1   0   1   2*x1^2 - 3*x1 + 1           0    1      -2*x1 + 1      4*x1^2 - 4*x1 + 1]
  [0                   0   0   0   1   1   2*x1^2 - 3*x1 + 1   -2*x1 + 1   x1   -2*x1^2 + x1   2*x1^3 - 3*x1^2 + x1]
in the multivariate polynomial ring in 1 variable over ZZ
within the vanishing set of the ideal
Ideal (16*x1^12 - 120*x1^11 + 364*x1^10 - 594*x1^9 + 576*x1^8 - 342*x1^7 + 122*x1^6 - 24*x1^5 + 2*x1^4)
avoiding the zero loci of the polynomials
RingElem[2*x1 - 1, x1 - 1, x1, x1^2 - 3*x1 + 1, x1 - 2, x1^2 + x1 - 1, x1^2 - x1 + 1, x1^3 - 2*x1^2 + 3*x1 - 1, x1^3 - 5*x1^2 + 4*x1 - 1]