The realization space is
  [1   x1 + 1   1   0   0   1   x1 + 1   x1^2 + x1 + 1               0      x1^2 + x1 + 1      x1^2 + x1 + 1]
  [1       x1   0   1   0   1       x1               0   x1^2 + x1 + 1   x1^3 + x1^2 + x1   x1^3 + x1^2 + x1]
  [0        0   0   0   1   1       x1       x1^2 + x1               1          x1^2 + x1                 x1]
in the multivariate polynomial ring in 1 variable over ZZ
within the vanishing set of the ideal
Ideal (x1^13 + 4*x1^12 + 8*x1^11 + 10*x1^10 + 8*x1^9 + 4*x1^8 + x1^7)
avoiding the zero loci of the polynomials
RingElem[x1 - 1, x1, x1 + 1, x1^2 + x1 + 1, x1^3 + x1^2 + x1 - 1, x1^2 + 1, x1^3 + x1^2 - 1, x1^2 + 2*x1 + 2]