Part A: Gamma_0, pure top, the bad graded data (i), (i'), (iii)
  (i) D_3 = X^2T_1 + C XY^2: 172 closure equations in 120 unknowns; 1 remain after elimination with constant pivots (74 solved)
     remaining: (-1/3) * a17 * (-3*a13*a14 + 5*b5)
     minimal prime [a17] -- solved by constant pivots: True
     minimal prime [3*a13*a14 - 5*b5] -- solved by constant pivots: True
  ok   (U) (i) D_3 = X^2T_1 + C XY^2: the closure equations define 2 components, each a graph over an affine space (irreducible)  [1.1s]
  ok   (A) (i) D_3 = X^2T_1 + C XY^2, component a17 = 0: member (seed 101) solves the closure equations, H = [1, 3, 6, 4, 2, 2, 2, 1]; three elements of its ideal: local length 41 (standard basis), 41 (stabilization at N = 13)  [2.2s]
     member: F = X^7 - X^6 + 2*X^5*Y - 21/10*X^5*Z - 2*X^5 + X^4*Y + 2*X^4*Z - 2*X^3*Y*Z + 21/20*X^3*Z^2 - X^4 + X^3*Y - X*Y^3 + 2*X^3*Z - X^2*Y*Z + 2*X^2*Z^2 + 3/10*X*Y*Z^2 - 21/200*X*Z^3 + 2*X^2*Y + X*Y^2 - 2*Y^3 - X^2*Z - X*Y*Z - 2*X*Z^2 + Y*Z^2
             T = X^5*Y + X^5 + X^4*Y - X^3*Y^2 + X^4*Z - X^3*Y*Z - 2*X^4 - X^3*Y + X*Y^3 + X^3*Z - 2*X^2*Y*Z + 3/10*X*Y^2*Z - X^2*Z^2 + 3/20*X*Y*Z^2 + X^3 + X^2*Y - 2*X*Y^2 + Y^3 - X^2*Z + X*Y*Z + Y^2*Z + 2*Z^3
     triple: ['2*x^8 - 3*x^7*y - 3*x^6*y^2 - x^5*y^3 - 2*x^4*y^4 + 2*x^3*y^5 - 3*x*y^7 + 3*y^8 - 3*x^7*z - 3*x^6*y*z - 3*x^5*y^2*z - x^4*y^3*z + 3*x^3*y^4*z + 2*x^2*y^5*z + x*y^6*z + y^7*z - 3*x^2*y^4*z^2 + 3*x*y^5*z^2 - x^5*z^3 - 3*x^4*y*z^3 + 3*x^3*y^2*z^3 - x^2*y^3*z^3 - 2*x*y^4*z^3 + x^4*z^4 + x^3*y*z^4 + x^2*y^2*z^4 - x*y^3*z^4 - 3*y^4*z^4 - x^3*z^5 + x*y^2*z^5 - 2*y^3*z^5 + x^2*z^6 + x*y*z^6 + 3*y^2*z^6 + x*z^7 + z^8 - 611587/3528000*x^7 + 3*x^6*y - x^5*y^2 + 3*x^4*y^3 - 3*x^3*y^4 - x*y^6 + y^7 + 2*x^5*y*z + 3*x^3*y^3*z + 3*x^2*y^4*z - 2*y^6*z - x^5*z^2 + 3*x^4*y*z^2 - x^3*y^2*z^2 + 3*x^2*y^3*z^2 + x*y^4*z^2 + 2*y^5*z^2 - x^4*z^3 + x^3*y*z^3 - x^2*y^2*z^3 - x*y^3*z^3 + 2*y^4*z^3 - 3*x^2*y*z^4 - 3*x*y^2*z^4 - 2*y^3*z^4 - 2*x^2*z^5 - x*z^6 + 2*y*z^6 + z^7 - 1927/56000*x^6 + 2219/6000*x^5*y - 2*x^4*y^2 + x^2*y^4 + 3*x*y^5 + y^6 - 3*x^5*z - 2*x^4*y*z + 3*x^3*y^2*z - x*y^4*z + 2*y^5*z + 2*x^4*z^2 + 2*x^3*y*z^2 - 3*x^2*y^2*z^2 - 3*x*y^3*z^2 - 3*y^4*z^2 - 2*x^3*z^3 + 2*x^2*y*z^3 - 2*x*y^2*z^3 + 3*y^3*z^3 + 3*x^2*z^4 + 2*y^2*z^4 - 3*x*z^5 + y*z^5 - z^6 + 3/20*x^5 - 779/1200*x^4*y + x^3*y^2 + x^2*y^3 + x*y^4 - 3*y^5 - x^4*z - 3*x^3*y*z - x^2*y^2*z + 3*x*y^3*z - 2*y^4*z - 2*x^3*z^2 + 3*x^2*y*z^2 + 2*x*y^2*z^2 + y^3*z^2 + x^2*z^3 - x*y*z^3 + y^2*z^3 + 3*x*z^4 - 3*z^5 - 1/10*x^3*y - x^2*y^2 - 2*x*y^3 - 2*y^4 + 3*x^3*z + 2*x*y^2*z + y^3*z + 2*x^2*z^2 - 2*x*y*z^2 + y^2*z^2 + 2*x*z^3 - 2*y*z^3 - 2*z^4 - 2*y^3 - 2*x*y*z - y^2*z + 3*y*z^2 - z^3', '-2*x^8 - 3*x^6*y^2 + 3*x^5*y^3 - 3*x^3*y^5 - 2*x*y^7 - y^8 + 3*x^7*z + 2*x^5*y^2*z + x^4*y^3*z - 3*x^3*y^4*z - 3*x^2*y^5*z + 3*x*y^6*z - 2*y^7*z + 3*x^6*z^2 - 2*x^4*y^2*z^2 + x^2*y^4*z^2 + 3*x*y^5*z^2 - x^5*z^3 + 2*x^4*y*z^3 - 2*x^3*y^2*z^3 + x*y^4*z^3 - 3*y^5*z^3 + x^4*z^4 - 2*x^2*y^2*z^4 + x*y^3*z^4 + y^4*z^4 - 2*x^3*z^5 - 3*x^2*y*z^5 + 3*x*y^2*z^5 - y^3*z^5 + 2*x^2*z^6 - 2*y^2*z^6 + 3*x*z^7 + y*z^7 + 24881/504000*x^7 + 2*x^5*y^2 + 2*x^4*y^3 + x^3*y^4 - 3*x^2*y^5 - x*y^6 - 2*y^7 + 3*x^6*z + x^5*y*z + 2*x^4*y^2*z - 2*x^3*y^3*z - 3*x^2*y^4*z - 3*x*y^5*z - y^6*z - 2*x^5*z^2 - 2*x^2*y^3*z^2 + x*y^4*z^2 + y^5*z^2 - 3*x^3*y*z^3 - 2*x^2*y^2*z^3 - x*y^3*z^3 + 3*y^4*z^3 - x^3*z^4 + 3*x^2*y*z^4 + 3*y^3*z^4 + 3*x^2*z^5 - 3*x*y*z^5 - 2*y^2*z^5 + 2*x*z^6 - 2*y*z^6 - z^7 + 10327/72000*x^6 - 2293/2000*x^5*y + 2*x^4*y^2 - 2*x^3*y^3 - 3*x^2*y^4 + 2*x*y^5 + 2*x^3*y^2*z + 3*x^2*y^3*z + 3*x*y^4*z + 3*y^5*z + 2*x^4*z^2 + 3*x^3*y*z^2 - 3*x^2*y^2*z^2 + x*y^3*z^2 - 3*y^4*z^2 + 2*x^3*z^3 + 3*x^2*y*z^3 - 3*x*y^2*z^3 + x^2*z^4 - 2*x*y*z^4 - y^2*z^4 - 3*x*z^5 + 3*y*z^5 + z^6 - 1/200*x^5 - 27/400*x^4*y - 3*x^3*y^2 + x^2*y^3 + 3*x*y^4 + y^5 + 3*x^4*z + 3*x^3*y*z + 3*x^2*y^2*z + 3*x*y^3*z + 2*y^4*z + 3*x^3*z^2 + 2*x^2*y*z^2 + 3*y^3*z^2 + 3*x^2*z^3 + x*y*z^3 + y^2*z^3 + 2*x*z^4 + y*z^4 - 3*z^5 + 1/20*x^3*y - x^2*y^2 + 3*x*y^3 - y^4 - y^3*z + 3*x^2*z^2 - 3*x*y*z^2 + 2*y^2*z^2 + 2*x*z^3 - z^4 + y^3 + x*y*z + y^2*z + 2*x*z^2 + y*z^2 + 3*z^3', '-2*x^8 + 3*x^7*y - 3*x^5*y^3 - 3*x^4*y^4 - x^3*y^5 - x*y^7 - 3*x^7*z + x^6*y*z - x^4*y^3*z - 3*x^3*y^4*z - x^2*y^5*z - 2*x*y^6*z - 3*x^6*z^2 + 3*x^5*y*z^2 + 3*x^4*y^2*z^2 - 3*x^2*y^4*z^2 - 2*x*y^5*z^2 - y^6*z^2 - x^5*z^3 + x^4*y*z^3 + x^3*y^2*z^3 + x^2*y^3*z^3 + x*y^4*z^3 - y^5*z^3 - 2*x^3*y*z^4 + x^2*y^2*z^4 + 2*x*y^3*z^4 - y^4*z^4 + 2*x^3*z^5 - 2*x^2*y*z^5 - 3*x*y^2*z^5 + 2*y^3*z^5 + x^2*z^6 - 3*x*y*z^6 + 2*y^2*z^6 + 2*x*z^7 + 3*y*z^7 - 3*z^8 + 27893/1764000*x^7 + 2*x^6*y - 3*x^5*y^2 - 2*x^4*y^3 - x^2*y^5 - x*y^6 + y^7 + 3*x^6*z - 2*x^5*y*z - 3*x^4*y^2*z + 2*x^3*y^3*z - x^2*y^4*z - x*y^5*z - 3*y^6*z - 3*x^5*z^2 + 3*x^4*y*z^2 + x^3*y^2*z^2 - 3*x^2*y^3*z^2 + y^5*z^2 + 3*x^3*y*z^3 - 2*x^2*y^2*z^3 + 3*y^4*z^3 + 2*x^3*z^4 - 3*x^2*y*z^4 + x*y^2*z^4 + 2*y^3*z^4 + x^2*z^5 - 3*x*y*z^5 + 3*x*z^6 - y*z^6 - z^7 - 257/2800*x^6 + 111/250*x^5*y - 3*x^4*y^2 + 2*x^3*y^3 + 3*x^2*y^4 - 3*x*y^5 + y^6 + x^5*z - 2*x^4*y*z + 2*x^3*y^2*z - 2*x^2*y^3*z + 3*x*y^4*z - 2*y^5*z + 2*x^4*z^2 + 3*x^3*y*z^2 - 2*x^2*y^2*z^2 + 2*x*y^3*z^2 - 3*x^3*z^3 + y^3*z^3 - 3*x^2*z^4 - x*y*z^4 + 3*y^2*z^4 - 3*x*z^5 - 2*y*z^5 + 3*z^6 - 3/20*x^5 + 467/400*x^4*y + x^3*y^2 + 2*x^2*y^3 - 2*x*y^4 + 3*y^5 - x^4*z - 3*x^3*y*z - 3*x^2*y^2*z - 3*y^4*z - 3*x^3*z^2 - 3*x^2*y*z^2 - 3*x*y^2*z^2 - 2*y^3*z^2 - 3*x^2*z^3 - 3*x*z^4 - 2*y*z^4 + 3*z^5 + 3/20*x^3*y + 2*x^2*y^2 - x*y^3 - y^4 - 3*x^3*z + 3*x^2*y*z + 2*x*y^2*z + 3*y^3*z - x^2*z^2 - 2*x*y*z^2 + 2*x*z^3 + y*z^3 - 3*y^3 + 3*x*y*z + y^2*z - y*z^2']
  ok   (A) (i) D_3 = X^2T_1 + C XY^2, component 5 b5 = 3 a13 a14: member (seed 102) solves the closure equations, H = [1, 3, 6, 4, 2, 2, 2, 1]; three elements of its ideal: local length 39 (standard basis), 39 (stabilization at N = 11)  [2.7s]
     member: F = X^7 + 2*X^6 - X^5*Y - 21*X^4*Y^2 - 21/10*X^5*Z + 2*X^5 + 2*X^4*Y - X^3*Y^2 + 52/5*X^2*Y^3 + 42/5*X*Y^4 + 341/5*X^3*Y*Z + 63/5*X^2*Y^2*Z + 21/20*X^3*Z^2 + 2*X^4 + 2*X^3*Y + X^2*Y^2 + 2*X*Y^3 - 189/25*Y^4 + X^2*Y*Z - 1569/50*X*Y^2*Z - 362/25*Y^3*Z - X^2*Z^2 - 2031/100*X*Y*Z^2 - 63/100*Y^2*Z^2 - 21/200*X*Z^3 + 2*X^3 - X^2*Y + X*Y^2 + Y^3 - X^2*Z + 2*X*Y*Z + 2*Y^2*Z + 2*X*Z^2 + 2*Y*Z^2 - 2*Z^3
             T = X^5*Y + 2*X^5 + 2*X^4*Y - X^3*Y^2 - 2*X^2*Y^3 - 2*X^4*Z - X^3*Y*Z + X^4 - X^3*Y + 2*X^2*Y^2 + 2*X*Y^3 + 1/5*Y^4 - X^3*Z + 2*X^2*Y*Z + 51/10*X*Y^2*Z + 1/5*Y^3*Z + 6/5*X^2*Z^2 + 3/20*X*Y*Z^2 - X^3 + X^2*Y - 2*Y^3 + 2*X^2*Z - X*Y*Z - 2*Y^2*Z + X*Z^2 + Y*Z^2 + 2*Z^3
     triple: ['-3*x^7*y + x^6*y^2 - 2*x^4*y^4 + 2*x^3*y^5 - 2*x*y^7 + 3*x^7*z + 3*x^6*y*z - 2*x^4*y^3*z - x^3*y^4*z - 3*y^7*z - x^6*z^2 + 3*x^5*y*z^2 + 3*x^4*y^2*z^2 + 2*x^3*y^3*z^2 - 3*x^2*y^4*z^2 - x*y^5*z^2 - x^4*y*z^3 - x^3*y^2*z^3 - 3*x*y^4*z^3 - y^5*z^3 + 2*x^4*z^4 - x^3*y*z^4 - x^2*y^2*z^4 + 2*x*y^3*z^4 - 2*y^4*z^4 - x^3*z^5 - 3*x^2*y*z^5 - x*y^2*z^5 - 2*y^3*z^5 - 2*x^2*z^6 + 3*x*y*z^6 - y^2*z^6 - 2*x*z^7 + y*z^7 + z^8 - 660449/882000*x^7 + 3*x^6*y - 3*x^5*y^2 - x^4*y^3 - 3*x^3*y^4 + 3*x^2*y^5 + x*y^6 + 3*y^7 - x^6*z + 2*x^5*y*z + 2*x^4*y^2*z - 3*x^2*y^4*z - x*y^5*z - 2*y^6*z - 2*x^5*z^2 + 3*x^4*y*z^2 + 2*x^3*y^2*z^2 + 2*x^2*y^3*z^2 - x*y^4*z^2 - y^5*z^2 - x^4*z^3 + 2*x^3*y*z^3 - 3*x^2*y^2*z^3 + 2*x*y^3*z^3 - 3*y^4*z^3 + x^3*z^4 + x^2*y*z^4 - 3*x*y^2*z^4 - 2*y^3*z^4 + x^2*z^5 + 3*x*y*z^5 - 3*y^2*z^5 - 3*x*z^6 + z^7 + 37043/252000*x^6 - 213/250*x^5*y - 2*x^4*y^2 + x^3*y^3 + 3*x^2*y^4 - x*y^5 + 3*y^6 + x^5*z - 3*x^4*y*z + 3*x^3*y^2*z + 3*y^5*z + x^4*z^2 + x^3*y*z^2 - 2*x^2*y^2*z^2 - 2*x*y^3*z^2 + 2*y^4*z^2 + 3*x^3*z^3 - x^2*y*z^3 + x*y^2*z^3 + y^3*z^3 + 3*x^2*z^4 - 3*x*y*z^4 - 3*y^2*z^4 - 3*x*z^5 - 3*y*z^5 - 2*z^6 + 61/200*x^5 + 631/1200*x^4*y + 2*x^3*y^2 + 3*x^2*y^3 + x*y^4 - y^5 - 3*x^4*z + 2*x^3*y*z - x^2*y^2*z + 3*x*y^3*z + 3*y^4*z + 3*x^3*z^2 + 2*x^2*y*z^2 - 3*x*y^2*z^2 + 3*x^2*z^3 - x*y*z^3 + 2*y^2*z^3 + y*z^4 + z^5 + 3/10*x^3*y + 2*x^2*y^2 - x*y^3 - 2*y^4 + 2*x^3*z + 2*x^2*y*z - x*y^2*z + 3*y^3*z - 2*x^2*z^2 - 2*x*y*z^2 + y*z^3 + 3*z^4 + y^3 + 2*x*y*z + 2*y^2*z - 2*x*z^2 - y*z^2 + 2*z^3', 'x^8 - 3*x^7*y - 2*x^6*y^2 + x^5*y^3 + 2*x^4*y^4 + x^3*y^5 + x^2*y^6 - 2*x*y^7 - y^8 + x^7*z - 2*x^6*y*z + 3*x^5*y^2*z + 2*x^4*y^3*z + 3*x^3*y^4*z + x^2*y^5*z - x*y^6*z - y^7*z + 3*x^6*z^2 + x^5*y*z^2 - 3*x^4*y^2*z^2 + x^3*y^3*z^2 - 3*x^2*y^4*z^2 + x*y^5*z^2 - 3*y^6*z^2 + 3*x^4*y*z^3 + 3*x^3*y^2*z^3 - 2*x*y^4*z^3 - 3*x^4*z^4 + 3*x^3*y*z^4 - 3*x^2*y^2*z^4 - x*y^3*z^4 + 3*y^4*z^4 - 3*x^2*y*z^5 + 2*x*y^2*z^5 + 2*y^3*z^5 + 3*x^2*z^6 + 3*x*y*z^6 - 3*y^2*z^6 + 3*z^8 - 79339/252000*x^7 + x^6*y - x^5*y^2 - 2*x^2*y^5 - 3*x*y^6 + 2*x^6*z + 2*x^5*y*z - x^4*y^2*z - 2*x^3*y^3*z - x^2*y^4*z - x*y^5*z + 2*y^6*z - 3*x^5*z^2 + 2*x^4*y*z^2 - 3*x^3*y^2*z^2 + 2*x^2*y^3*z^2 - 2*x*y^4*z^2 - 2*y^5*z^2 - 2*x^4*z^3 - 2*x^3*y*z^3 - x^2*y^2*z^3 - x*y^3*z^3 - x^3*z^4 + x^2*y*z^4 + 3*x*y^2*z^4 + y^3*z^4 + x^2*z^5 - 2*x*y*z^5 - 3*x*z^6 - z^7 - 1069/4500*x^6 + 1147/2000*x^5*y + x^3*y^3 - x^2*y^4 + 3*x*y^5 + 3*y^6 + 3*x^5*z + x^4*y*z - x^3*y^2*z + x*y^4*z - 2*y^5*z - x^4*z^2 - 2*x^3*y*z^2 + x^2*y^2*z^2 + x*y^3*z^2 - y^4*z^2 + x^3*z^3 - 3*x^2*y*z^3 - 3*x*y^2*z^3 + 2*y^3*z^3 - 3*x^2*z^4 - x*y*z^4 + 2*y^2*z^4 - y*z^5 - 3*z^6 + 53/200*x^5 + 811/600*x^4*y - x^3*y^2 + 3*x^2*y^3 + 3*x*y^4 - 3*y^5 + 2*x^4*z + 3*x^3*y*z - 2*x*y^3*z - 3*y^4*z - 2*x^3*z^2 + 2*x^2*y*z^2 + 2*x*y^2*z^2 - y^3*z^2 + x^2*z^3 + 2*x*y*z^3 - 3*y^2*z^3 - 2*x*z^4 + 3*y*z^4 + 3*z^5 + 1/10*x^3*y + 2*x*y^3 - y^4 + 2*x^3*z + x^2*y*z - 3*x*y^2*z + y^3*z - 2*x^2*z^2 - 3*x*y*z^2 + 3*y^2*z^2 - 2*x*z^3 + y*z^3 + 2*z^4 + y^3 - 2*x*y*z - 3*y^2*z + 2*x*z^2', '2*x^8 - 2*x^7*y - 2*x^5*y^3 + x^4*y^4 + 3*x^3*y^5 - x^2*y^6 - 2*x*y^7 + 3*y^8 - 2*x^7*z - 2*x^6*y*z - 2*x^5*y^2*z + 3*x^4*y^3*z + 3*x^3*y^4*z + x^2*y^5*z - x*y^6*z - 3*y^7*z - x^6*z^2 + 2*x^4*y^2*z^2 - 2*x^3*y^3*z^2 + 2*x^2*y^4*z^2 + x^5*z^3 + x^4*y*z^3 + 3*x^2*y^3*z^3 + 3*y^5*z^3 - 3*x^4*z^4 - 3*x^3*y*z^4 + 2*x^2*y^2*z^4 - 2*x*y^3*z^4 - y^4*z^4 + 3*x^3*z^5 + 3*x^2*y*z^5 + 2*x*y^2*z^5 - y^3*z^5 + 2*x^2*z^6 + 2*x*y*z^6 - 3*y^2*z^6 + 3*x*z^7 - 2*y*z^7 - 2*z^8 - 457697/882000*x^7 + x^6*y + 2*x^5*y^2 - x^4*y^3 - 3*x^3*y^4 - 3*x^2*y^5 + x*y^6 + 2*y^7 + 2*x^6*z - 3*x^5*y*z - 2*x^4*y^2*z - 3*x^3*y^3*z - 2*x*y^5*z - 3*y^6*z + 2*x^5*z^2 + 2*x^4*y*z^2 + x^3*y^2*z^2 + 3*x^2*y^3*z^2 + x*y^4*z^2 - 3*y^5*z^2 - 2*x^4*z^3 - 3*x^3*y*z^3 - 2*y^4*z^3 + 3*x^3*z^4 + x^2*y*z^4 - 2*x*y^2*z^4 + 2*y^3*z^4 + 3*x^2*z^5 - 3*y^2*z^5 + 2*x*z^6 + 3*y*z^6 + 14281/28000*x^6 + 409/2000*x^5*y - 2*x^4*y^2 - 3*x^3*y^3 + 2*x^2*y^4 - x*y^5 + y^6 - 2*x^5*z - x^4*y*z + 2*x^3*y^2*z + 2*x^2*y^3*z - x*y^4*z - 2*y^5*z - 2*x^4*z^2 - 2*x^3*y*z^2 - 2*x^2*y^2*z^2 + x*y^3*z^2 + y^4*z^2 - x^3*z^3 - 3*x^2*y*z^3 + y^3*z^3 + 3*x^2*z^4 - x*y*z^4 + 3*y^2*z^4 - x*z^5 + y*z^5 - 2*z^6 - 1/5*x^5 - 147/400*x^4*y + 2*x^3*y^2 + 2*x^2*y^3 + 3*y^5 - 2*x^4*z - 3*x^3*y*z - 3*x^2*y^2*z - 2*x*y^3*z + 2*y^4*z - 3*x^3*z^2 + 3*x^2*y*z^2 + 3*x*y^2*z^2 + 3*y^3*z^2 - x^2*z^3 - x*y*z^3 + y^2*z^3 - 2*x*z^4 + 2*y*z^4 - 11/20*x^3*y - 2*x^2*y^2 - 2*y^4 - 2*x^3*z - x^2*y*z + x*y^2*z - 3*y^3*z - 2*x^2*z^2 - 2*x*y*z^2 + 2*x*z^3 + 2*z^4 - 2*y^3 - 3*x*y*z - 2*y^2*z + 3*y*z^2 - 2*z^3']
  (i') D_3 = X^2T_1 + C XZ^2: 172 closure equations in 120 unknowns; 2 remain after elimination with constant pivots (74 solved)
     remaining: (1/15) * a17 * (15*p27*a13 + 10*p26*a14 - 42*a14*a18 - 21*a13*a19 - 10*q14 + 35*b8)
     remaining: b2 * a17
     minimal prime [42*a14*a18 + 21*a13*a19 - 10*a14*p26 - 15*a13*p27 - 35*b8 + 10*q14, b2] -- solved by constant pivots: True
     minimal prime [a17] -- solved by constant pivots: True
  ok   (U) (i') D_3 = X^2T_1 + C XZ^2: the closure equations define 2 components, each a graph over an affine space (irreducible)  [4.0s]
  ok   (A) (i') D_3 = X^2T_1 + C XZ^2, component a17 = 0: member (seed 103) solves the closure equations, H = [1, 3, 6, 4, 2, 2, 2, 1]; three elements of its ideal: local length 38 (standard basis), 38 (stabilization at N = 12)  [4.7s]
     member: F = X^7 - X^6 + X^5*Y - 21/10*X^5*Z - 2*X^5 + 2*X^4*Y - 121/5*X^3*Y^2 - 2*X^4*Z - X^3*Y*Z + 21/20*X^3*Z^2 - X^4 - 2*X^3*Y + X^2*Y^2 - 151/25*X*Y^3 - 2*X^3*Z + 363/50*X*Y^2*Z + X^2*Z^2 + 3/20*X*Y*Z^2 + X*Z^3 + 2*X^2*Y + 2*X*Y^2 - Y^3 - 2*X^2*Z - X*Y*Z - 2*Y^2*Z - 2*X*Z^2 + 2*Y*Z^2 - Z^3
             T = X^5*Y + 2*X^5 - X^4*Y + X^3*Y^2 + X^4*Z - X^3*Y*Z + 2*X^4 - X^3*Y + 2*X^2*Y^2 - X*Y^3 - 2*X^2*Y*Z - 3/10*X*Y^2*Z - X^2*Z^2 + 3/20*X*Y*Z^2 + 2*X*Z^3 + 2*X^3 - X^2*Y + 2*X*Y^2 + Y^3 + 2*X^2*Z - X*Y*Z + Y*Z^2 + 2*Z^3
     triple: ['-2*x^8 - x^7*y - x^6*y^2 + x^5*y^3 + 2*x^4*y^4 + x^2*y^6 + 2*x*y^7 + 2*y^8 + 3*x^5*y^2*z - x^4*y^3*z + 2*x^3*y^4*z - 2*x^2*y^5*z + 3*x*y^6*z + 3*y^7*z - 2*x^6*z^2 + x^5*y*z^2 + 2*x^4*y^2*z^2 + x^3*y^3*z^2 - 2*x^2*y^4*z^2 - 2*x*y^5*z^2 + 2*y^6*z^2 + 2*x^5*z^3 - 2*x^4*y*z^3 + 3*x^3*y^2*z^3 + 3*x^2*y^3*z^3 + 3*x*y^4*z^3 - y^5*z^3 - 2*x^4*z^4 + 2*x^3*y*z^4 + x^2*y^2*z^4 - 3*x*y^3*z^4 - 3*y^4*z^4 - x^3*z^5 - 2*x^2*y*z^5 - 3*x^2*z^6 - x*y*z^6 + 3*y^2*z^6 + 2*x*z^7 + 2*y*z^7 - 3*z^8 + 36427/441000*x^7 - 3*x^6*y + 2*x^5*y^2 - 2*x^4*y^3 - 2*x^3*y^4 + x^2*y^5 + x*y^6 - y^7 + 3*x^6*z - 2*x^5*y*z + 2*x^4*y^2*z - 2*x^3*y^3*z - x*y^5*z - x^5*z^2 + 3*x^4*y*z^2 - x^3*y^2*z^2 - 3*x^2*y^3*z^2 - y^5*z^2 + 3*x^4*z^3 - 2*x^3*y*z^3 - 3*x^2*y^2*z^3 + 3*y^4*z^3 - 2*x^3*z^4 + 2*x^2*y*z^4 - x*y^2*z^4 - y^3*z^4 - 2*x^2*z^5 - 2*x*y*z^5 - y^2*z^5 - x*z^6 - 3*z^7 + 1291/5040*x^6 - 5809/6000*x^5*y + 2*x^4*y^2 - 2*x^3*y^3 + x^2*y^4 + 2*x*y^5 - 3*y^6 + 3*x^5*z + 3*x^4*y*z - 2*x^3*y^2*z + 2*x^2*y^3*z + 3*x*y^4*z + 2*y^5*z - 3*x^4*z^2 + 3*x^3*y*z^2 + x^2*y^2*z^2 - y^4*z^2 + 2*x^3*z^3 - 2*x^2*y*z^3 - 3*x*y^2*z^3 + x^2*z^4 - 3*x*y*z^4 + 2*y*z^5 - z^6 + 33/100*x^5 - 587/600*x^4*y - 3*x^3*y^2 + 2*x^2*y^3 - 2*x*y^4 + 2*y^5 + x^4*z + 3*x^3*y*z - y^4*z - 2*x^3*z^2 + 3*x*y^2*z^2 + 2*y^3*z^2 + 2*x^2*z^3 - 3*x*y*z^3 + y^2*z^3 - x*z^4 - 2*y*z^4 - z^5 - 1/5*x^3*y + 2*x^2*y^2 + y^4 + 3*x^3*z - 2*x*y^2*z - 2*y^3*z + 3*x^2*z^2 - 3*x*y*z^2 - 2*y^2*z^2 + 2*x*z^3 + 2*y*z^3 - 2*z^4 + 3*x*y^2 + 2*x*y*z + 2*y*z^2', '3*x^8 - 2*x^7*y + x^6*y^2 - 3*x^5*y^3 - 2*x^4*y^4 + x^3*y^5 + x^2*y^6 + 2*x*y^7 - 2*y^8 - 2*x^7*z - x^6*y*z + x^5*y^2*z - 3*x^4*y^3*z + x^3*y^4*z - 3*x^2*y^5*z + 2*x*y^6*z - 2*y^7*z + 3*x^6*z^2 + 2*x^5*y*z^2 + 2*x^3*y^3*z^2 + 2*x^2*y^4*z^2 + 3*x*y^5*z^2 - y^6*z^2 - x^5*z^3 + x^3*y^2*z^3 - 2*x^2*y^3*z^3 - 2*x*y^4*z^3 - 3*y^5*z^3 - 3*x^4*z^4 - 3*x^3*y*z^4 + x^2*y^2*z^4 + x*y^3*z^4 + 2*y^4*z^4 + 2*x^3*z^5 + 3*x^2*y*z^5 + 3*x*y^2*z^5 + y^3*z^5 + 3*x^2*z^6 + 2*x*y*z^6 - 3*y^2*z^6 + 2*y*z^7 - 2*z^8 - 8341/126000*x^7 - 3*x^6*y + 3*x^5*y^2 + x^4*y^3 - 2*x^3*y^4 + 3*x^2*y^5 + 2*x*y^6 - y^7 - 2*x^6*z + 2*x^5*y*z - 3*x^4*y^2*z - 3*x^2*y^4*z + x*y^5*z - 2*y^6*z - 2*x^5*z^2 + 2*x^4*y*z^2 - 2*x^3*y^2*z^2 - x^2*y^3*z^2 - 2*x*y^4*z^2 + 2*y^5*z^2 - x^4*z^3 + 3*x^3*y*z^3 - 2*x*y^3*z^3 - y^4*z^3 - 2*x^3*z^4 + x^2*y*z^4 + 2*y^3*z^4 - 3*x*y*z^5 + 3*y^2*z^5 - 3*x*z^6 + 2*y*z^6 - 1127/18000*x^6 + 173/400*x^5*y + 3*x^4*y^2 - x^3*y^3 + 3*x*y^5 - y^6 + x^5*z - x^4*y*z + x^3*y^2*z - x^2*y^3*z + x*y^4*z + y^5*z - x^4*z^2 + 2*x^3*y*z^2 + x^2*y^2*z^2 - 3*x*y^3*z^2 + 2*y^4*z^2 - x^3*z^3 - x^2*y*z^3 - 3*x*y^2*z^3 - 3*y^3*z^3 - y^2*z^4 + z^6 + 3/20*x^5 - 1/120*x^4*y - 2*x^3*y^2 - 2*x^2*y^3 - x*y^4 + y^5 + x^3*y*z + 2*x^2*y^2*z - 3*x*y^3*z - 2*y^4*z + 2*x^3*z^2 - x^2*y*z^2 + 3*x*y^2*z^2 + y^3*z^2 + x^2*z^3 + 3*x*y*z^3 - 3*y^2*z^3 + 2*x*z^4 - 3*y*z^4 - z^5 + 1/10*x^3*y - 2*x^2*y^2 + 2*x*y^3 - y^4 + 3*x^3*z + x^2*y*z + 2*x*y^2*z + 2*y^3*z + 3*x^2*z^2 - x*y*z^2 + y^2*z^2 + x*z^3 + y*z^3 - z^4 + y^3 + 2*x*y*z - 2*y*z^2 - 2*z^3', '3*x^8 + x^7*y + 3*x^5*y^3 + x^4*y^4 - 3*x^2*y^6 + 3*x*y^7 - y^8 + 3*x^7*z - 3*x^6*y*z - 3*x^5*y^2*z + 3*x^4*y^3*z + x^3*y^4*z - x^2*y^5*z + 3*x*y^6*z + 2*y^7*z - 3*x^3*y^3*z^2 + x^2*y^4*z^2 - y^6*z^2 + 2*x^5*z^3 - 2*x^4*y*z^3 + 3*x^2*y^3*z^3 - 2*x*y^4*z^3 + 2*y^5*z^3 + 2*x^4*z^4 - 2*x^3*y*z^4 + 3*x^2*y^2*z^4 + y^4*z^4 - x^3*z^5 + 3*x^2*y*z^5 + x*y^2*z^5 + 3*y^3*z^5 - 2*x^2*z^6 - x*y*z^6 - 3*y^2*z^6 - 2*x*z^7 + 3*y*z^7 + z^8 - 10373/49000*x^7 - 3*x^6*y + 3*x^5*y^2 + x^4*y^3 - x^3*y^4 - 2*x^2*y^5 - 2*x*y^6 - 2*y^7 - 3*x^6*z - x^5*y*z + 2*x^4*y^2*z + 3*x^3*y^3*z + 2*x*y^5*z + 3*y^6*z + x^5*z^2 - 3*x^4*y*z^2 - x^3*y^2*z^2 - 2*x^2*y^3*z^2 + x*y^4*z^2 + 2*y^5*z^2 + 2*x^3*y*z^3 - x^2*y^2*z^3 + 3*x*y^3*z^3 + y^4*z^3 - x^2*y*z^4 - x*y^2*z^4 + 2*y^3*z^4 + x^2*z^5 + 2*x*y*z^5 + 3*y^2*z^5 - 2*x*z^6 + 2*y*z^6 - z^7 + 731/6300*x^6 + 77/2000*x^5*y + x^4*y^2 - 3*x^3*y^3 - 3*x^2*y^4 + 2*y^6 - 2*x^5*z - x^4*y*z + x^3*y^2*z - x*y^4*z - 2*y^5*z + 2*x^4*z^2 + x^2*y^2*z^2 - x*y^3*z^2 + y^4*z^2 - 2*x^3*z^3 + 3*x^2*y*z^3 - 3*x*y^2*z^3 - 2*x^2*z^4 + 3*x*y*z^4 + 2*y^2*z^4 - 2*x*z^5 + 2*y*z^5 - 1/100*x^5 - 139/200*x^4*y - 2*x^3*y^2 - x^2*y^3 - 2*y^5 - 2*x^4*z + x^3*y*z + 3*x^2*y^2*z + 2*x*y^3*z - 3*y^4*z - 3*x^3*z^2 + 3*y^3*z^2 + 2*x^2*z^3 + 3*x*y*z^3 - 2*y^2*z^3 + 2*x*z^4 + 3*y*z^4 + 3*z^5 + 3*x^2*y^2 - 2*x*y^3 + 2*y^4 + x^3*z - 3*x^2*y*z + 2*x*y^2*z + 3*y^3*z - 2*x^2*z^2 + x*y*z^2 - 3*y^2*z^2 - 3*x*z^3 - 2*y*z^3 + 3*z^4 - x*y^2 + 3*y^3 - 2*x*y*z + y^2*z + 2*z^3']
  ok   (A) (i') D_3 = X^2T_1 + C XZ^2, component b2 = 0, b8 = ...: member (seed 104) solves the closure equations, H = [1, 3, 6, 4, 2, 2, 2, 1]; three elements of its ideal: local length 36 (standard basis), 36 (stabilization at N = 11)  [5.2s]
     member: F = X^7 - X^6 - 2*X^5*Y + 21/2*X^4*Y^2 - 21/10*X^5*Z + X^4*Y + 74/5*X^3*Y^2 - 52/5*X^2*Y^3 + 21/10*X*Y^4 - X^4*Z + 2*X^3*Y*Z - 63/10*X^2*Y^2*Z + 21/20*X^3*Z^2 - 2*X^4 - X^3*Y - 2*X^2*Y^2 - 444/25*X*Y^3 + 131/50*Y^4 - X^3*Z - 2*X^2*Y*Z - 186/25*X*Y^2*Z + 26/25*Y^3*Z - X^2*Z^2 - 3/10*X*Y*Z^2 + 63/200*Y^2*Z^2 - 2*X*Z^3 + 2*X^3 + X*Y^2 + 2*X^2*Z - X*Y*Z - 2*Y^2*Z - 2*Z^3
             T = X^5*Y + X^5 + X^4*Y - 2*X^3*Y^2 + X^2*Y^3 - X^3*Y*Z - 2*X^4 - 2*X^3*Y - 2*X^2*Y^2 + 2*X*Y^3 - 1/5*Y^4 - X^3*Z - 46/35*X^2*Y*Z + 3/5*X*Y^2*Z - 1/10*Y^3*Z - X^2*Z^2 + 3/20*X*Y*Z^2 + 2*X^3 - 2*X*Y^2 + X^2*Z + 2*X*Y*Z - X*Z^2 - 2*Y*Z^2 + 2*Z^3
     triple: ['3*x^8 - x^7*y - x^5*y^3 + 2*x^4*y^4 - x^3*y^5 + 3*x^2*y^6 - 3*x*y^7 - y^8 + x^6*y*z + x^5*y^2*z + 3*x^4*y^3*z - x^3*y^4*z + 2*x^2*y^5*z - y^7*z + 3*x^6*z^2 + 2*x^5*y*z^2 - 2*x^4*y^2*z^2 - 2*x^3*y^3*z^2 - 2*x^2*y^4*z^2 + x*y^5*z^2 + y^6*z^2 - 3*x^3*y^2*z^3 - 2*x^2*y^3*z^3 + 2*y^5*z^3 + x^4*z^4 + x^3*y*z^4 - 2*x^2*y^2*z^4 - 2*x*y^3*z^4 - 3*y^4*z^4 - 3*x^3*z^5 + 3*x^2*y*z^5 - y^3*z^5 + 3*x^2*z^6 - 3*y^2*z^6 - 2*x*z^7 + y*z^7 - 2*z^8 - 183679/4116000*x^7 - 2*x^6*y + 3*x^4*y^3 - 3*x^3*y^4 + 2*x^2*y^5 - 2*x*y^6 + y^7 - 3*x^6*z + 2*x^5*y*z + 3*x^4*y^2*z + 2*x^3*y^3*z - x^2*y^4*z - 2*x*y^5*z - y^6*z - 3*x^5*z^2 - 3*x^4*y*z^2 + 3*x^3*y^2*z^2 - x^2*y^3*z^2 + x*y^4*z^2 + 2*y^5*z^2 + 3*x^4*z^3 + x^3*y*z^3 + x^2*y^2*z^3 + x*y^3*z^3 + 3*y^4*z^3 + x*y^2*z^4 + y^3*z^4 + 3*x^2*z^5 - 3*x*y*z^5 + y^2*z^5 + 3*x*z^6 - y*z^6 + z^7 - 17069/294000*x^6 + 2721/7000*x^5*y + x^4*y^2 - 2*x^3*y^3 - x^2*y^4 - 2*x*y^5 + 2*x^5*z + 2*x^4*y*z - 3*x^3*y^2*z - 3*x^2*y^3*z - 2*x*y^4*z - 2*y^5*z - 3*x^4*z^2 + x^3*y*z^2 - 2*x^2*y^2*z^2 - x*y^3*z^2 + 2*y^4*z^2 - 3*x^3*z^3 - 2*x^2*y*z^3 - 2*x*y^2*z^3 - 3*y^3*z^3 - 3*x^2*z^4 + 3*x*y*z^4 + 3*y^2*z^4 - x*z^5 + y*z^5 + 3*z^6 - 29/200*x^5 + 1083/2800*x^4*y + x^3*y^2 - 2*x^2*y^3 - 3*x*y^4 - 2*y^5 + x^4*z - x^3*y*z - 3*x*y^3*z + 2*x^3*z^2 - y^3*z^2 - x^2*z^3 - 3*x*y*z^3 - y^2*z^3 - 3*x*z^4 + y*z^4 - 3*z^5 - 3/10*x^4 + 11/20*x^3*y - x^2*y^2 - x*y^3 - 3*y^4 - 2*x^3*z + x^2*y*z - 2*x*y^2*z - x^2*z^2 + x*y*z^2 - y^2*z^2 - 3*y*z^3 - z^4 + 3*x*y^2 - y^3 - 3*x*y*z - y^2*z - y*z^2 - z^3', 'x^8 + 2*x^7*y + x^6*y^2 + 3*x^5*y^3 + x^4*y^4 - 3*x^3*y^5 - 2*x^2*y^6 + 3*x*y^7 + y^8 - 2*x^7*z - x^6*y*z + x^5*y^2*z - x^4*y^3*z - x^3*y^4*z + x^2*y^5*z - 2*x*y^6*z - y^7*z - 2*x^6*z^2 - x^5*y*z^2 + 3*x^4*y^2*z^2 + 2*x^3*y^3*z^2 + x^2*y^4*z^2 + 2*x*y^5*z^2 - y^6*z^2 + 3*x^5*z^3 - 3*x^4*y*z^3 + x^3*y^2*z^3 + x^2*y^3*z^3 - 3*x*y^4*z^3 + 2*y^5*z^3 + x^4*z^4 - 3*x^3*y*z^4 - x^2*y^2*z^4 + 2*x*y^3*z^4 - y^4*z^4 - 2*x^3*z^5 - x^2*y*z^5 - x*y^2*z^5 + y^3*z^5 - x^2*z^6 + 2*x*y*z^6 + 2*y^2*z^6 + 2*x*z^7 + 2*y*z^7 - z^8 + 556643/2058000*x^7 + 3*x^6*y - 2*x^5*y^2 + 3*x^4*y^3 - 2*x^3*y^4 - 2*x^2*y^5 + x*y^6 - 2*y^7 + 2*x^5*y*z + 2*x^3*y^3*z + x^2*y^4*z + 3*x*y^5*z - y^6*z + 3*x^5*z^2 + x^4*y*z^2 - 3*x^3*y^2*z^2 + 3*x^2*y^3*z^2 - 3*x*y^4*z^2 - 3*y^5*z^2 + 2*x^3*y*z^3 + 2*x^2*y^2*z^3 - 2*x*y^3*z^3 - 2*y^4*z^3 + 2*x^3*z^4 + 2*x^2*y*z^4 - 3*y^3*z^4 + x^2*z^5 + 3*x*y*z^5 + 2*y^2*z^5 + y*z^6 + z^7 - 29789/294000*x^6 + 907/7000*x^5*y - 2*x^4*y^2 + 3*x^3*y^3 + x^2*y^4 + 3*x*y^5 - y^6 + 3*x^5*z + 3*x^4*y*z - x^3*y^2*z + 3*x^2*y^3*z - x*y^4*z + x^4*z^2 + x^3*y*z^2 - 3*x^2*y^2*z^2 - x*y^3*z^2 + y^4*z^2 - 2*x^3*z^3 + x*y^2*z^3 + 3*y^3*z^3 + 2*x^2*z^4 - 2*x*y*z^4 - 2*y^2*z^4 - 2*y*z^5 - 3*z^6 - 63/200*x^5 + 3271/2800*x^4*y - x^3*y^2 + 2*x*y^4 - 2*x^4*z - x^3*y*z - 3*x*y^3*z + y^4*z + 3*x^3*z^2 - 3*x^2*y*z^2 + 2*x*y^2*z^2 - y^3*z^2 - 3*x^2*z^3 - 3*x*y*z^3 + 3*y^2*z^3 - 3*x*z^4 + 3*y*z^4 + z^5 - 3/10*x^4 + 11/20*x^3*y + 3*x^2*y^2 - 2*x*y^3 + 3*y^4 + x^3*z + 3*x^2*y*z + x*y^2*z - 2*y^3*z - 3*x^2*z^2 + x*y*z^2 - 2*x*z^3 + 2*y*z^3 + 3*x*y^2 + 2*y^3 + 3*x*y*z + 3*y^2*z + 3*y*z^2 + 3*z^3', '3*x^8 - 2*x^7*y + x^5*y^3 - 3*x^3*y^5 + 2*x^2*y^6 + 3*x*y^7 - 2*x^7*z + 3*x^6*y*z - 2*x^5*y^2*z + 3*x^4*y^3*z + x^3*y^4*z + 2*x^2*y^5*z + 3*x*y^6*z - y^7*z - x^6*z^2 - 3*x^5*y*z^2 - 3*x^4*y^2*z^2 - x^3*y^3*z^2 - x*y^5*z^2 + 2*y^6*z^2 + x^5*z^3 - 3*x^3*y^2*z^3 + 3*x^2*y^3*z^3 + 3*x*y^4*z^3 - y^5*z^3 + 3*x^4*z^4 + x^3*y*z^4 + 2*x^2*y^2*z^4 - 3*x*y^3*z^4 - y^4*z^4 + 3*x^3*z^5 + 2*x*y^2*z^5 + 3*y^3*z^5 + 2*x^2*z^6 + 2*x*y*z^6 - y^2*z^6 - 2*x*z^7 - 2*y*z^7 + 343447/1234800*x^7 - x^6*y - 2*x^5*y^2 + 2*x^4*y^3 - 2*x^3*y^4 + x^2*y^5 + 2*x*y^6 + y^7 + 3*x^6*z + 2*x^5*y*z + 2*x^4*y^2*z - 2*x^3*y^3*z - 3*x*y^5*z + y^6*z + x^5*z^2 + x^4*y*z^2 - x^2*y^3*z^2 - x*y^4*z^2 + 3*y^5*z^2 - 2*x^4*z^3 - x*y^3*z^3 - y^4*z^3 - 2*x^3*z^4 - 2*x^2*y*z^4 + 2*y^3*z^4 + x^2*z^5 - x*y*z^5 - 2*y^2*z^5 - 3*x*z^6 - 3*z^7 + 80503/294000*x^6 - 16813/21000*x^5*y - 2*x^4*y^2 - 2*x^2*y^4 + 3*x*y^5 - 3*y^6 - 2*x^5*z - 3*x^4*y*z + 3*x*y^4*z + 2*y^5*z + 3*x^2*y^2*z^2 - 3*x*y^3*z^2 + y^4*z^2 - x^2*y*z^3 - 2*y^3*z^3 - x^2*z^4 - 3*y^2*z^4 - 2*x*z^5 + y*z^5 + 1/25*x^5 - 349/1400*x^4*y - x^3*y^2 + 3*x^4*z - 3*x^3*y*z + 2*x^2*y^2*z + 2*y^4*z + 2*x^3*z^2 - 2*x^2*y*z^2 + 3*x*y^2*z^2 + y^3*z^2 - x*y*z^3 + y^2*z^3 + x*z^4 - 3*y*z^4 - 3*z^5 + 3/10*x^4 - 7/20*x^3*y + 2*x^2*y^2 + 3*x*y^3 - 3*y^4 + 3*x^3*z - 2*x^2*y*z - 2*y^3*z + x^2*z^2 + 2*x*y*z^2 + y^2*z^2 - 2*x*z^3 + 3*y*z^3 - 3*z^4 - 3*x*y^2 - y^3 + 3*x*y*z - 2*y^2*z + 2*y*z^2 + 3*z^3']
  (iii) D_3 = C[X,Y]_3: 172 closure equations in 120 unknowns; 0 remain after elimination with constant pivots (76 solved)
  ok   (U) (iii) D_3 = C[X,Y]_3: all closure equations are solved by constant pivots: one affine space  [5.9s]
  ok   (A) (iii) D_3 = C[X,Y]_3, component whole family: member (seed 105) solves the closure equations, H = [1, 3, 6, 4, 2, 2, 2, 1]; three elements of its ideal: local length 41 (standard basis), 41 (stabilization at N = 12)  [6.6s]
     member: F = X^7 + 2*X^5*Y - 21/2*X^4*Y^2 + 21/10*X^5*Z - 2*X^5 - 2*X^4*Y + 2*X^3*Y^2 + 11/5*X^2*Y^3 + 21/10*X*Y^4 + 9*X^4*Z - 74/5*X^3*Y*Z - 63/10*X^2*Y^2*Z + 21/20*X^3*Z^2 - X^4 + 2*X*Y^3 + 2*Y^4 + 634/25*X^2*Y*Z - 129/25*X*Y^2*Z + 19/10*Y^3*Z + 9/25*X^2*Z^2 - 237/50*X*Y*Z^2 - 63/200*Y^2*Z^2 + 21/200*X*Z^3 + X^3 + 2*X^2*Y + X*Y^2 - Y^3 + X*Y*Z - 2*Y^2*Z - X*Z^2 - Y*Z^2
             T = X^5*Y + 2*X^4*Y - X^3*Y^2 - X^2*Y^3 + X^4*Z + X^3*Y*Z - 2*X^3*Y + 2*X^2*Y^2 - 2*Y^4 + 2*X^2*Y*Z - 3/2*X*Y^2*Z - 1/10*Y^3*Z + 3/5*X^2*Z^2 + 3/20*X*Y*Z^2 + 2*X^3 + X^2*Y + 2*X*Y^2 + 2*Y^3 + X^2*Z + 2*X*Y*Z + 2*Y^2*Z + 2*X*Z^2 + Y*Z^2 - Z^3
     triple: ['3*x^8 + x^7*y - x^5*y^3 + 3*x^4*y^4 - x^3*y^5 + 2*x^2*y^6 - 3*x*y^7 + y^8 - 2*x^7*z - 2*x^6*y*z + 2*x^5*y^2*z + 2*x^3*y^4*z + 3*x^2*y^5*z - x^6*z^2 - 3*x^5*y*z^2 - 2*x^4*y^2*z^2 + 2*x^3*y^3*z^2 - 3*y^6*z^2 + x^5*z^3 - 3*x^4*y*z^3 + 2*x^3*y^2*z^3 + x^2*y^3*z^3 - 3*x*y^4*z^3 + y^5*z^3 - 2*x^4*z^4 + 2*x^3*y*z^4 - 3*x^2*y^2*z^4 - 3*y^4*z^4 - x^3*z^5 - x^2*y*z^5 + 2*x*y^2*z^5 + 2*y^3*z^5 + x^2*z^6 + 2*x*y*z^6 + 3*x*z^7 - y*z^7 - 2*z^8 + 147619/252000*x^7 + 3*x^6*y + 3*x^5*y^2 - 2*x^4*y^3 - 2*x^3*y^4 - 3*x^2*y^5 - 2*x*y^6 + y^7 - x^5*y*z + 2*x^4*y^2*z + 2*x^3*y^3*z + 3*x*y^5*z + 3*y^6*z - 3*x^3*y^2*z^2 + 2*x*y^4*z^2 + 2*y^5*z^2 + x^4*z^3 + 3*x^2*y^2*z^3 - 3*x*y^3*z^3 + y^4*z^3 + x^3*z^4 + 2*x^2*y*z^4 - 3*x*y^2*z^4 + 3*y^3*z^4 - x^2*z^5 - x*y*z^5 + y^2*z^5 + 3*x*z^6 - 2*y*z^6 - 2*z^7 - 187/33600*x^6 + 679/1500*x^5*y + 3*x^4*y^2 - 3*x^3*y^3 + 3*x^2*y^4 + x*y^5 + 2*y^6 - 3*x^5*z + 2*x^4*y*z - 3*x^3*y^2*z - 3*x^2*y^3*z - 3*x*y^4*z + y^5*z - 3*x^4*z^2 - x^3*y*z^2 - x^2*y^2*z^2 + x*y^3*z^2 - x^3*z^3 - 3*x^2*y*z^3 - 3*x*y^2*z^3 - 2*y^3*z^3 - 3*x^2*z^4 + 3*x*y*z^4 - 3*y^2*z^4 - y*z^5 + 89/400*x^5 - 313/1200*x^4*y - x^3*y^2 - 2*x^2*y^3 - 3*x^4*z + 2*x^3*y*z + 3*x^2*y^2*z - 2*x*y^3*z + 2*y^4*z + 2*x^3*z^2 - 2*x*y^2*z^2 - 3*y^3*z^2 - x^2*z^3 - 3*x*y*z^3 - 2*y^2*z^3 + 3*x*z^4 - 2*y*z^4 + 3/20*x^4 + 11/20*x^3*y + 2*x^2*y^2 + 3*x*y^3 + 2*x^3*z + 3*x^2*y*z + y^3*z + 2*x^2*z^2 + 2*x*y*z^2 - 2*y^2*z^2 + 2*x*z^3 - 2*y*z^3 - 3*z^4 - 3*x^2*z + x*y*z - y^2*z - 3*x*z^2 - 3*y*z^2 + 3*z^3', 'x^8 + 2*x^7*y + 2*x^5*y^3 - x^4*y^4 - 3*x^3*y^5 - x^2*y^6 + 2*x*y^7 + y^8 - x^7*z + x^6*y*z - x^5*y^2*z - 2*x^4*y^3*z + 2*x^3*y^4*z - 2*x^2*y^5*z + 2*x^6*z^2 + x^5*y*z^2 - x^4*y^2*z^2 - x^3*y^3*z^2 - 3*x^2*y^4*z^2 + x*y^5*z^2 - y^6*z^2 - 2*x^5*z^3 + x^4*y*z^3 + 3*x^3*y^2*z^3 - x^2*y^3*z^3 - x*y^4*z^3 - 2*y^5*z^3 + 3*x^4*z^4 - x^3*y*z^4 - x*y^3*z^4 + y^4*z^4 - x^2*y*z^5 + 3*x*y^2*z^5 - 2*y^3*z^5 - 3*x^2*z^6 - x*y*z^6 + y^2*z^6 - 2*x*z^7 - 2*y*z^7 + 331/800*x^7 + x^6*y - 3*x^4*y^3 + 2*x^2*y^5 - 3*x*y^6 - y^7 - x^5*y*z - x^4*y^2*z - 3*x^3*y^3*z - 3*x^2*y^4*z - x*y^5*z - y^6*z - 3*x^5*z^2 - x^4*y*z^2 - 2*x^3*y^2*z^2 - 3*x*y^4*z^2 + y^5*z^2 + x^4*z^3 + 3*x^2*y^2*z^3 - 3*x*y^3*z^3 + y^4*z^3 + 3*x^3*z^4 - x^2*y*z^4 - 2*x*y^2*z^4 + 2*y^3*z^4 - x^2*z^5 + 3*x*y*z^5 - 3*y^2*z^5 - x*z^6 - z^7 + 2989/12000*x^6 + 71/100*x^5*y + 2*x^4*y^2 + 2*x^3*y^3 - 2*x^2*y^4 - 2*x*y^5 - 3*y^6 - x^5*z + 3*x^3*y^2*z + x^2*y^3*z + y^5*z + 3*x^4*z^2 + x^3*y*z^2 + 2*x^2*y^2*z^2 + 3*x*y^3*z^2 + 3*y^4*z^2 - 3*x^3*z^3 + 2*x^2*y*z^3 - 2*x*y^2*z^3 + 3*y^3*z^3 - x^2*z^4 - 2*x*y*z^4 - 3*y^2*z^4 - 3*x*z^5 + y*z^5 + 39/400*x^5 + 27/80*x^4*y - x*y^4 - y^5 - 3*x^4*z + 3*x^3*y*z + 3*x^2*y^2*z - 3*x*y^3*z + y^4*z - 3*x^3*z^2 - 2*x^2*y*z^2 - 2*x*y^2*z^2 - 3*y^3*z^2 - 2*x^2*z^3 + y^2*z^3 + x*z^4 + z^5 - 3/20*x^4 - 3/5*x^3*y + x^2*y^2 - 3*x*y^3 + y^4 - 2*x^3*z + 2*x^2*y*z + 2*x*y^2*z - 2*x*y*z^2 - 2*x*z^3 + 3*y*z^3 + 2*z^4 + 3*x^2*z - y^2*z - x*z^2 - y*z^2 + 2*z^3', '-3*x^8 - 3*x^7*y + 3*x^6*y^2 + 2*x^5*y^3 - x^4*y^4 - x^3*y^5 - 3*x*y^7 + y^8 + 3*x^7*z + 2*x^6*y*z - x^5*y^2*z + 2*x^4*y^3*z - x^3*y^4*z + 2*x^2*y^5*z - 3*x*y^6*z + 2*y^7*z - 3*x^6*z^2 - 3*x^5*y*z^2 - 2*x^4*y^2*z^2 + x^2*y^4*z^2 + 3*x*y^5*z^2 - y^6*z^2 - 2*x^4*y*z^3 - 2*x^3*y^2*z^3 - x*y^4*z^3 - x^4*z^4 + x^3*y*z^4 - x^2*y^2*z^4 + 3*x*y^3*z^4 - x^2*y*z^5 + 3*x*y^2*z^5 + 2*y^3*z^5 + 2*x^2*z^6 - 2*x*y*z^6 - 3*z^8 - 69593/504000*x^7 + x^6*y + 3*x^5*y^2 + 3*x^4*y^3 - 3*x^3*y^4 + x^2*y^5 - 2*x*y^6 - y^7 - 2*x^6*z - 3*x^5*y*z + 2*x^4*y^2*z + x^3*y^3*z + 2*x^2*y^4*z - 3*x*y^5*z - 3*y^6*z + x^5*z^2 + x^4*y*z^2 - 2*x^3*y^2*z^2 - 2*x*y^4*z^2 - y^5*z^2 + 2*x^4*z^3 + 3*x^3*y*z^3 - 3*x^2*y^2*z^3 - 3*x*y^3*z^3 - y^4*z^3 + 2*x^3*z^4 + 2*x^2*y*z^4 - x*y^2*z^4 + 3*x^2*z^5 - x*y*z^5 + y^2*z^5 + x*z^6 + 3*y*z^6 - z^7 + 32609/84000*x^6 + 211/6000*x^5*y - 2*x^4*y^2 + x^2*y^4 + 2*x*y^5 - 3*y^6 + 3*x^4*y*z + x^3*y^2*z - 2*x^2*y^3*z + 3*x^4*z^2 + 2*x^3*y*z^2 + x^2*y^2*z^2 + 2*x*y^3*z^2 + 2*x^3*z^3 - x^2*y*z^3 - x*y^2*z^3 + 3*y^3*z^3 + 3*x^2*z^4 + 2*x*y*z^4 + y^2*z^4 - 2*x*z^5 + y*z^5 + 131/240*x^5 + 33/50*x^4*y + 2*x^3*y^2 + 3*x^2*y^3 - 3*x*y^4 + 3*y^5 - x^4*z + 3*x^2*y^2*z + 3*x*y^3*z - 3*y^4*z + 2*x^3*z^2 + 2*x^2*y*z^2 + x*y^2*z^2 - 2*y^3*z^2 - 3*x^2*z^3 + 2*x*y*z^3 - y^2*z^3 - 3*x*z^4 + 3*y*z^4 + 1/20*x^4 + 1/20*x^3*y + 3*x^2*y^2 - y^4 - 3*x^3*z + 3*x^2*y*z + 2*x*y^2*z + 3*y^3*z - 3*x^2*z^2 - x*y*z^2 - y^2*z^2 + x*z^3 - y*z^3 + 2*z^4 - x^2*z + 3*x*y*z - y^2*z - 3*x*z^2 + 2*y*z^2 + 2*z^3']
Part B: the graded data (ii), (iv), (v): I* test
  ok   (A) (ii) X^2T_1 + C XYZ: the sparse triple [x^3*z + y^3, x^8 + y^3 + x*z^2, x^5*z + x*y^2 + z^3] lies in I*  [6.7s]
  ok   (A) (ii) X^2T_1 + C XYZ, D_7 = C X^7: the sparse triple of I* cuts out a complete intersection of local length 41 (standard basis), (41, 13) (stabilization)  [6.7s]
  ok   (A) (ii) X^2T_1 + C XZ(Y+Z): the sparse triple [x^8 + y^3 - 2*x*y*z + x*z^2, x^3*z + y*z^2, x^4*z + x*y^2 + z^3] lies in I*  [6.7s]
  ok   (A) (ii) X^2T_1 + C XZ(Y+Z), D_7 = C X^7: the sparse triple of I* cuts out a complete intersection of local length 41 (standard basis), (41, 13) (stabilization)  [6.8s]
  ok   (A) (iv) nu_3(C4), C4 = V(v - u^2, u^4): the sparse triple [-x*y^2 + x^2*z, x^8 + y^2*z, x^2*y^2 - y^3 + x*y*z + z^3] lies in I*  [6.8s]
  ok   (A) (iv) nu_3(C4), C4 = V(v - u^2, u^4), D_7 = C X^7: the sparse triple of I* cuts out a complete intersection of local length 41 (standard basis), (41, 13) (stabilization)  [6.8s]
  ok   (A) (v) nu_3(C4), C4 = V(v - u^3, u^4): the sparse triple [-y^3 + x^2*z, x^8 + x*y*z, x^2*y^2 + z^3] lies in I*  [6.9s]
  ok   (A) (v) nu_3(C4), C4 = V(v - u^3, u^4), D_7 = C X^7: the sparse triple of I* cuts out a complete intersection of local length 41 (standard basis), (41, 13) (stabilization)  [6.9s]
  ok   (S) other points of the orbits (ii), (iv), (v): three elements of I* give [('(ii) g2 = (Y+2Z)(Y-3Z)', 41), ('(ii) g2 = Y(2Y+Z)', 41), ('(iv) (a,b) = (3,-2)', 41), ('(iv) (a,b) = (-1,5)', 41), ('(v) (a,b) = (0,7)', 41)]  [10.8s]
Part C: the rank condition of Lemma D.3
  ok   (S) g3 != 0: rank condition of Lemma D.3 holds on X^2T_1 + C(Xg2 + g3): ranks [3, 3, 3, 3, 3, 3, 3]  [11.0s]
  ok   (S) g3 = 0: rank 3 for g2 a square, rank 2 otherwise: [3, 3, 3, 2, 2, 2]  [11.1s]
  ok   (S) curvilinear C4 (z = y^2, y^3, 0 and random): the rank condition does not hold on nu_3(C4) (ranks [2, 2, 1, 2, 2, 2]), so these flags are treated in Part B  [11.2s]
Part D: Gamma = {X, Y}: rays
   pure top, seed 401: H = [1, 3, 6, 4, 2, 2, 2, 1], x^4 b, x^4 c in I: True, b = 2/735*x^3 - 1/21*x^2 + y, c = -19/1470*x^3 - 1/42*x^2 + z, (I : L) not in I + L: True
   nonpure top, seed 402: H = [1, 3, 6, 4, 2, 2, 2, 1], x^4 b, x^4 c in I: True, b = 4/245*x^3 - 1/21*x^2 + y, c = 1/490*x^3 - 1/21*x^2 + z, (I : L) not in I + L: True
   pure top, seed 403: H = [1, 3, 6, 4, 2, 2, 2, 1], x^4 b, x^4 c in I: True, b = -1/210*x^3 + 1/21*x^2 + y, c = 1/105*x^3 + 1/42*x^2 + z, (I : L) not in I + L: True
   nonpure top, seed 404: H = [1, 3, 6, 4, 2, 2, 2, 1], x^4 b, x^4 c in I: True, b = 2/735*x^3 - 1/21*x^2 + y, c = 1/490*x^3 + 1/21*x^2 + z, (I : L) not in I + L: True
  ok   (S) rays along (y - beta(x), z - gamma(x)) on 4 random actual algebras over Gamma = {X, Y}  [15.0s]
Part E: nonpure top X^6 Y over Gamma_0
   trial 0: dim C = 3, conditions True, closure dim 21, H = [1, 3, 6, 4, 2, 2, 2, 1]; perturbed: conditions False, closure dim 22; base scheme length 3
   trial 1: dim C = 3, conditions True, closure dim 21, H = [1, 3, 6, 4, 2, 2, 2, 1]; perturbed: conditions False, closure dim 22; base scheme length 3
   trial 2: dim C = 3, conditions True, closure dim 21, H = [1, 3, 6, 4, 2, 2, 2, 1]; perturbed: conditions False, closure dim 22; base scheme length 3
   trial 3: dim C = 3, conditions True, closure dim 21, H = [1, 3, 6, 4, 2, 2, 2, 1]; perturbed: conditions False, closure dim 22; base scheme length 3
   trial 4: dim C = 3, conditions True, closure dim 21, H = [1, 3, 6, 4, 2, 2, 2, 1]; perturbed: conditions False, closure dim 22; base scheme length 3
   trial 5: dim C = 3, conditions True, closure dim 21, H = [1, 3, 6, 4, 2, 2, 2, 1]; perturbed: conditions False, closure dim 22; base scheme length 3
  ok   (S) nonpure criterion agrees with direct closedness, H = (1,3,6,4,2,2,2,1), and a random D_3 containing C has base scheme of length dim C  [16.5s]
total time 16.5 s
ALL CHECKS PASSED
