Part A: monomial ideals L_delta = x m^2 + z^delta (y,z)^(4-delta) + m^5; the triples printed in the proof of prop:LD
   delta = 0: (x^3 - z^4 - z^5, x*y^2 - y^3*z, x*z^2 - y^4)
      in L_delta: True; local length by local standard basis: 36; dim S/(C + m^N) = 36 for N = 9 and N = 10; 36 + delta = 36
  ok   delta = 0: the triple lies in L_delta and has local length 36 = 36 + delta (both certificates)
   delta = 1: (x^3 + y^3*z - y^5, x*y^2 + z^4, x*z^2 - y^3*z)
      in L_delta: True; local length by local standard basis: 37; dim S/(C + m^N) = 37 for N = 10 and N = 11; 36 + delta = 37
  ok   delta = 1: the triple lies in L_delta and has local length 37 = 36 + delta (both certificates)
   delta = 2: (x^3 - z^4 + y^5, x*y^2 - y*z^3 - z^4, x*z^2 - y^2*z^2 + y*z^3)
      in L_delta: True; local length by local standard basis: 38; dim S/(C + m^N) = 38 for N = 10 and N = 11; 36 + delta = 38
  ok   delta = 2: the triple lies in L_delta and has local length 38 = 36 + delta (both certificates)
   delta = 3: (x^3 - z^4 + y^5, x*y^2 - y*z^3 + y^5, x*z^2 - y*z^3 + z^4)
      in L_delta: True; local length by local standard basis: 39; dim S/(C + m^N) = 39 for N = 10 and N = 11; 36 + delta = 39
  ok   delta = 3: the triple lies in L_delta and has local length 39 = 36 + delta (both certificates)
Part B (sanity): S_1 Q = S_3 iff x o G is not a square
   x o G = X*Y (rank 2): dim S_1 Q = 10
   x o G = Y*Z (rank 2): dim S_1 Q = 10
   x o G = Y^2 + Z^2 (rank 2): dim S_1 Q = 10
   x o G = X^2 + Y*Z (rank 3): dim S_1 Q = 10
   x o G = Y^2 + X*Z (rank 3): dim S_1 Q = 10
   x o G = X^2 + Y^2 + Z^2 (rank 3): dim S_1 Q = 10
   x o G = Y^2 (rank 1): dim S_1 Q = 9
   x o G = X^2 (rank 1): dim S_1 Q = 9
   x o G = X^2 + 2*X*Y + Y^2 + 4*X*Z + 4*Y*Z + 4*Z^2 (rank 1): dim S_1 Q = 9
  ok   S_1 Q = S_3 exactly when x o G has rank >= 2
Part C: members for fixed graded data (36 cases); random.seed(5) is set before the first case
   (1,3,6,5,4,1), general G, pure Y^5: attempt 1 used (local lengths of the attempts made: [37]); the triple:
      f_1 = 2*x^5 - 2*x^4*y + 3*x^3*y^2 - 2*x^2*y^3 - x*y^4 - 3*y^5 + x^4*z - 4*x^3*y*z - 5*x^2*y^2*z - 3*x*y^3*z + 4*y^4*z + 4*x^3*z^2 - 2*x^2*y*z^2 + x*y^2*z^2 + 4*y^3*z^2 + x^2*z^3 - 3*x*y*z^3 + 2*y^2*z^3 - 4*x*z^4 - 3*y*z^4 - 3*z^5 - 5*y^3*z - 3*y^2*z^2 - 4*y*z^3 + 3*x^3 - 5*x^2*y - 2*x*y^2 + 2*x^2*z + 5*x*y*z + 1/6*x*z^2
      f_2 = 3*x^5 + 5*x^4*y - 2*x^3*y^2 + x^2*y^3 + x*y^4 + 5*x^4*z - 3*x^3*y*z - x^2*y^2*z - 3*x*y^3*z - y^4*z - 2*x^3*z^2 - 5*x^2*y*z^2 - 3*x*y^2*z^2 + 4*y^3*z^2 - x*y*z^3 + 4*y^2*z^3 - 4*x*z^4 - 5*y*z^4 + 4*z^5 - 3*y^3*z - y^2*z^2 - 2*z^4 - 5*x^3 - 5*x^2*y - 2*x*y^2 - 2*x^2*z - 3*x*y*z + 49/6*x*z^2
      f_3 = 2*x^5 + 2*x^4*y - 5*x^3*y^2 + x*y^4 + 2*y^5 - 3*x^4*z - x^3*y*z + x^2*y^2*z - 5*y^4*z - 5*x^3*z^2 - 5*x^2*y*z^2 + x*y^2*z^2 - 3*y^3*z^2 + 3*x^2*z^3 + 4*x*y*z^3 + 4*y^2*z^3 - 5*x*z^4 - 2*y*z^4 - 4*z^5 - y^3*z + 2*y^2*z^2 - 3*z^4 + 5*x^3 - x*y^2 - 4*x^2*z - 19/3*x*z^2
  ok   (1,3,6,5,4,1), general G, pure Y^5: H = [1, 3, 6, 5, 4, 1], closed True, f_i in Ann(M) True, deg gcd K = 1, CI local length 37 (stable pair N = 10, 11; attempt 1) <= 37
   (1,3,6,5,4,1), general G, nonpure (3,4): attempt 1 used (local lengths of the attempts made: [36]); the triple:
      f_1 = -x^5 - 5*x^4*y - 4*x^3*y^2 + 3*x^2*y^3 + 5*y^5 + x^4*z - 2*x^3*y*z + 4*x^2*y^2*z - x*y^3*z + 3*x*y^2*z^2 - y^3*z^2 - 3*x^2*z^3 - 4*x*y*z^3 - 3*y^2*z^3 - x*z^4 - 4*y*z^4 + 5*z^5 - 4*y^4 + 4*y^3*z + 40*y*z^3 + 3*x^3 - x*y^2 + 3*x^2*z + 2*x*y*z - 7/3*x*z^2
      f_2 = x^5 - 5*x^4*y + 4*x^3*y^2 + 2*x^2*y^3 - 5*x*y^4 - 2*y^5 - 2*x^4*z + 5*x^2*y^2*z + 5*x*y^3*z - 3*y^4*z - x^3*z^2 + x^2*y*z^2 - 5*x*y^2*z^2 + 4*y^3*z^2 - 3*x^2*z^3 + 2*x*y*z^3 - 3*y^2*z^3 + 5*y*z^4 + z^5 - 4*y^4 + 4*y^3*z + 3*y^2*z^2 + 40*y*z^3 - 3/10*z^4 - 3*x^3 - x^2*y - 3*x*y^2 + 2*x^2*z - 5*x*y*z + 31/6*x*z^2
      f_3 = -3*x^5 + 2*x^4*y - 3*x^3*y^2 + 2*x^2*y^3 - x*y^4 + y^5 + 4*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 + x*y^2*z^2 - 3*y^3*z^2 - x^2*z^3 - 3*x*y*z^3 + 5*y^2*z^3 - 4*x*z^4 + 2*y*z^4 - 5*y^4 + y^3*z - y^2*z^2 + 50*y*z^3 + 1/10*z^4 - 4*x^3 - 3*x^2*y + x^2*z - 3*x*y*z + 41/6*x*z^2
  ok   (1,3,6,5,4,1), general G, nonpure (3,4): H = [1, 3, 6, 5, 4, 1], closed True, f_i in Ann(M) True, deg gcd K = 0, CI local length 36 (stable pair N = 9, 10; attempt 1) <= 36
   (1,3,6,5,4,1), general G, nonpure (2,5): attempt 1 used (local lengths of the attempts made: [38]); the triple:
      f_1 = 5*x^5 - 5*x^4*y - x^2*y^3 + 2*y^5 - x^4*z + 3*x^3*y*z + 2*x^2*y^2*z + 5*x*y^3*z + 3*y^4*z + 5*x^3*z^2 - x^2*y*z^2 - 3*x*y^2*z^2 - y^3*z^2 - x^2*z^3 + 4*x*y*z^3 - 5*x*z^4 + 5*y*z^4 - z^5 - 4*y^3*z + 4*y*z^3 - x^3 + 3*x^2*y + 3*x*y^2 + 3*x^2*z
      f_2 = 5*x^5 + 3*x*y^4 - y^5 + x^4*z - 3*x^3*y*z - 3*x*y^3*z - 4*y^4*z + 2*x^3*z^2 + 3*x^2*y*z^2 - 2*x*y^2*z^2 + 5*y^3*z^2 - 3*x^2*z^3 + y^2*z^3 + 2*x*z^4 - 3*y*z^4 + 4*z^5 - y*z^3 + 2*x^3 - x^2*y + 5*x*y^2 + 3*x^2*z + 5*x*y*z - 7/6*x*z^2
      f_3 = -x^5 - 5*x^4*y - 3*x^3*y^2 + 4*x^2*y^3 + 5*x*y^4 - 2*y^5 - 2*x^4*z + 4*x^3*y*z + 5*x^2*y^2*z - 3*x*y^3*z - 3*y^4*z + 2*x^3*z^2 - 3*x^2*y*z^2 + 5*x*y^2*z^2 - 5*y^3*z^2 - 2*x^2*z^3 - 5*x*y*z^3 - 3*y^2*z^3 + 2*x*z^4 - 4*y*z^4 + 4*y^3*z + 3*y^2*z^2 + 5*y*z^3 + 4*x^3 + 3*x^2*y + x*y^2 + 5*x^2*z - x*y*z - 25/6*x*z^2
  ok   (1,3,6,5,4,1), general G, nonpure (2,5): H = [1, 3, 6, 5, 4, 1], closed True, f_i in Ann(M) True, deg gcd K = 2, CI local length 38 (stable pair N = 10, 11; attempt 1) <= 38
   (1,3,6,5,4,1), general G, X-free Y^4 Z: attempt 1 used (local lengths of the attempts made: [38]); the triple:
      f_1 = -4*x^5 + 4*x^4*y - 2*x^3*y^2 - 4*x*y^4 + 4*y^5 - 2*x^4*z + x^2*y^2*z + 3*x*y^3*z - y^4*z - 5*x^3*z^2 - x^2*y*z^2 - 4*y^3*z^2 + x^2*z^3 - 5*x*y*z^3 - y^2*z^3 + 3*x*z^4 + 3*y*z^4 + 3*z^5 + y^2*z^2 + 5*y*z^3 + 4*z^4 + 3*x^3 - 5*x^2*y + 2*x*y^2 + x^2*z + 2/3*x*z^2
      f_2 = 5*x^5 + 3*x^4*y + 2*x^3*y^2 + 4*x^2*y^3 - 2*x*y^4 + 5*y^5 - 3*x^3*y*z + x^2*y^2*z - 3*x*y^3*z - 5*y^4*z - 3*x^3*z^2 + 4*x^2*y*z^2 + 4*x*y^2*z^2 + y^3*z^2 + 2*x^2*z^3 - 4*x*y*z^3 - 4*y^2*z^3 + 4*z^5 + y^2*z^2 + y*z^3 + 4*z^4 + 4*x^2*y - x^2*z - 3*x*y*z - 5/2*x*z^2
      f_3 = x^5 - 2*x^4*y - x^3*y^2 + x^2*y^3 - x*y^4 - 5*y^5 + 2*x^4*z + 2*x^3*y*z - 3*x^2*y^2*z - x*y^3*z + x^3*z^2 + 4*x^2*y*z^2 + 2*x*y^2*z^2 - y^3*z^2 + 4*x^2*z^3 - x*y*z^3 + 2*y^2*z^3 + x*z^4 - y*z^4 - 3*z^5 + 4*y^2*z^2 + y*z^3 + z^4 + 4*x^3 + 3*x^2*y - 3*x^2*z + 5*x*y*z - 47/6*x*z^2
  ok   (1,3,6,5,4,1), general G, X-free Y^4 Z: H = [1, 3, 6, 5, 4, 1], closed True, f_i in Ann(M) True, deg gcd K = 2, CI local length 38 (stable pair N = 10, 11; attempt 1) <= 38
   (1,3,6,5,3,2), general G, r = 3 generic: attempt 1 used (local lengths of the attempts made: [36]); the triple:
      f_1 = -x^5 - 5*x^4*y + x^3*y^2 + 3*x^2*y^3 + 2*x*y^4 - 3*y^5 - 3*x^4*z - 5*x^3*y*z + 3*x^2*y^2*z - 5*x*y^3*z - y^4*z + 2*x^3*z^2 - x^2*y*z^2 - 2*x*y^2*z^2 - y^3*z^2 - 2*x^2*z^3 + 2*x*y*z^3 + 2*y^2*z^3 - x*z^4 + 4*y*z^4 + 2*z^5 + 2*y^4 - 2*y^3*z - 20*y*z^3 + 1/5*z^4 + 2*x^3 - 5*x^2*y + 5*x*y^2 - 4*x^2*z + 4*x*y*z - 2/3*x*z^2
      f_2 = -x^5 + 3*x^4*y + 4*x^3*y^2 - 5*x^2*y^3 + 5*x*y^4 - y^5 - x^3*y*z + 2*x^2*y^2*z + 4*x*y^3*z + 3*y^4*z + 3*x^3*z^2 - 3*x*y^2*z^2 + 2*y^3*z^2 - 5*x*y*z^3 + 5*y^2*z^3 - 5*x*z^4 - 4*y*z^4 + 5*z^5 - 3*y^4 - y^3*z + 30*y*z^3 + 1/10*z^4 + 3*x^3 + 5*x^2*y - 4*x*y^2 + 4*x^2*z - 5*x*y*z - 25/6*x*z^2
      f_3 = 5*x^5 + 5*x^4*y + x^3*y^2 + 5*x^2*y^3 - 4*x*y^4 - y^5 + 2*x^4*z + 2*x^3*y*z - 2*x^2*y^2*z - x*y^3*z + 5*y^4*z + x^3*z^2 - x^2*y*z^2 - 5*x*y^2*z^2 + 2*y^3*z^2 + 2*x^2*z^3 + x*y*z^3 - y^2*z^3 + 4*x*z^4 - 3*y*z^4 - 3*z^5 + 4*y^4 + 4*y^3*z - 40*y*z^3 - 2/5*z^4 + 3*x^3 - 2*x^2*y + x*y^2 - 5*x^2*z + x*y*z - 7/2*x*z^2
  ok   (1,3,6,5,3,2), general G, r = 3 generic: H = [1, 3, 6, 5, 3, 2], closed True, f_i in Ann(M) True, deg gcd K = 0, CI local length 36 (stable pair N = 9, 10; attempt 1) <= 36
   (1,3,6,5,3,2), general G, r = 3 with a fifth power: attempt 1 used (local lengths of the attempts made: [37]); the triple:
      f_1 = 4*x^5 + 3*x^4*y - 4*x^3*y^2 - 4*x^2*y^3 - 5*y^5 - 5*x^4*z - 2*x^3*y*z - 4*x^2*y^2*z + x*y^3*z - 4*y^4*z - x^3*z^2 + 5*x^2*y*z^2 - 4*x*y^2*z^2 - 4*y^3*z^2 - 5*x^2*z^3 + x*y*z^3 - 2*y^2*z^3 + 2*x*z^4 + 4*y*z^4 + 5*z^5 - 3*y^3*z + y^2*z^2 - 1/10*z^4 + 2*x^3 + 3*x^2*y + 2*x^2*z + 3*x*y*z - 23/6*x*z^2
      f_2 = -3*x^5 + x^4*y - 4*x^3*y^2 - 5*x^2*y^3 + x*y^4 + 4*y^5 - 3*x^4*z + 4*x^3*y*z + 3*x^2*y^2*z - 4*x*y^3*z - 4*y^4*z - 2*x^3*z^2 - 4*x^2*y*z^2 - x*y^2*z^2 + y^3*z^2 + 5*x^2*z^3 - 5*x*y*z^3 + 4*y^2*z^3 + 4*x*z^4 - 3*y*z^4 - 5*z^5 - 3*y^3*z + 4*y^2*z^2 - 2/5*z^4 - 4*x^3 - 3*x^2*y + 2*x*y^2 - x^2*z - 3*x*y*z + 37/6*x*z^2
      f_3 = 5*x^4*y - 5*x^3*y^2 - 3*x^2*y^3 + 3*x*y^4 - 2*y^5 - 3*x^3*y*z + 4*x^2*y^2*z - x*y^3*z - y^4*z - x^3*z^2 + 5*x^2*y*z^2 + 3*x*y^2*z^2 - 4*y^3*z^2 + 5*x^2*z^3 + 3*x*y*z^3 + 3*y^2*z^3 + 4*x*z^4 - 2*y*z^4 - z^5 + 2*y^3*z + 2*y^2*z^2 - 1/5*z^4 + x^3 - 4*x^2*y + 5*x*y^2 + 4*x*y*z + x*z^2
  ok   (1,3,6,5,3,2), general G, r = 3 with a fifth power: H = [1, 3, 6, 5, 3, 2], closed True, f_i in Ann(M) True, deg gcd K = 1, CI local length 37 (stable pair N = 10, 11; attempt 1) <= 37
   (1,3,6,5,3,2), general G, r = 2 secant: attempt 1 used (local lengths of the attempts made: [38]); the triple:
      f_1 = 4*x^5 + 5*x^4*y - 4*x^3*y^2 + 4*x^2*y^3 + 2*x*y^4 - 5*y^5 - 5*x^4*z - 4*x^3*y*z + 2*x^2*y^2*z + 2*x*y^3*z + 3*y^4*z + 3*x^3*z^2 - 2*x^2*y*z^2 + y^3*z^2 + x^2*z^3 - 4*x*y*z^3 - 5*y^2*z^3 + 3*x*z^4 - 3*y*z^4 + z^5 + 5*y^3*z - 4*y*z^3 + 2*x^3 + x^2*y - 3*x*y^2 - 3*x^2*z - 5*x*y*z - 17/6*x*z^2
      f_2 = 4*x^5 + x^4*y - 2*x^3*y^2 + 4*x^2*y^3 - 3*x*y^4 + y^5 - x^3*y*z + x^2*y*z^2 + 5*x*y^2*z^2 + 3*y^3*z^2 + x^2*z^3 - x*y*z^3 - 5*y^2*z^3 + 5*x*z^4 + 4*y*z^4 + 4*z^5 + 2*y^3*z - 2*y^2*z^2 - 3*y*z^3 + 5*x^3 - 2*x^2*y - 4*x*y^2 - 4*x^2*z + 5*x*y*z - 35/6*x*z^2
      f_3 = -2*x^5 - 2*x^4*y + 2*x^3*y^2 - x*y^4 + 2*y^5 - 5*x^4*z + 2*x^3*y*z - 5*x^2*y^2*z + 4*x*y^3*z - 4*y^4*z + 5*x^3*z^2 + x^2*y*z^2 + x*y^2*z^2 - 2*y^3*z^2 + 2*x^2*z^3 - 3*y^2*z^3 + 2*x*z^4 + 5*y*z^4 - 3*z^5 - 5*y^3*z - 5*y^2*z^2 - 3*y*z^3 - 2*x^3 - 3*x^2*y - 4*x*y^2 + x^2*z - 4*x*y*z + 5*x*z^2
  ok   (1,3,6,5,3,2), general G, r = 2 secant: H = [1, 3, 6, 5, 3, 2], closed True, f_i in Ann(M) True, deg gcd K = 2, CI local length 38 (stable pair N = 10, 11; attempt 1) <= 38
   (1,3,6,5,3,2), general G, r = 2 tangent: attempt 1 used (local lengths of the attempts made: [38]); the triple:
      f_1 = -5*x^5 + 2*x^4*y - 5*x^2*y^3 + 2*x*y^4 - y^5 - 5*x^4*z - 5*x^2*y^2*z - 3*x*y^3*z + 2*y^4*z + 5*x^2*y*z^2 - 4*x*y^2*z^2 + 5*y^3*z^2 + 2*x^2*z^3 + 5*x*y*z^3 - 2*y^2*z^3 - 3*y*z^4 - 5*z^5 - y^2*z^2 + 5*y*z^3 - 2*z^4 + 3*x^3 - x^2*y - x*y^2 - 4*x^2*z - 3*x*y*z - 19/6*x*z^2
      f_2 = 5*x^4*y - 3*x^3*y^2 - x^2*y^3 - 2*x*y^4 + 5*y^5 - x^4*z + 2*x^3*y*z - 4*x^2*y^2*z - 3*x*y^3*z + 4*y^4*z + 2*x^2*y*z^2 + 5*x*y^2*z^2 + y^3*z^2 - 3*x^2*z^3 - 2*x*y*z^3 - y^2*z^3 - 5*x*z^4 - 5*y*z^4 + 2*z^5 - 4*y^2*z^2 + 4*y*z^3 + z^4 - 2*x^3 - 5*x^2*y - 2*x*y^2 + 4*x^2*z + 5*x*y*z + 35/6*x*z^2
      f_3 = 3*x^5 + 4*x^4*y - 4*x^3*y^2 + 2*x^2*y^3 + 5*x*y^4 + y^5 - 4*x^3*y*z - 4*x^2*y^2*z - 2*x*y^3*z - 5*y^4*z - 2*x^3*z^2 + 2*x^2*y*z^2 + 5*x*y^2*z^2 - 5*y^3*z^2 - 5*x^2*z^3 - 5*x*y*z^3 - y^2*z^3 + 2*x*z^4 + y*z^4 + 2*z^5 - 5*y^2*z^2 - 5*y*z^3 + 3*z^4 - 5*x^3 + 2*x^2*y + x*y^2 + 5*x^2*z - 3*x*y*z + 35/6*x*z^2
  ok   (1,3,6,5,3,2), general G, r = 2 tangent: H = [1, 3, 6, 5, 3, 2], closed True, f_i in Ann(M) True, deg gcd K = 2, CI local length 38 (stable pair N = 10, 11; attempt 1) <= 38
   (1,3,6,5,3,2), general G, r = 3 gcd z^3: attempt 1 used (local lengths of the attempts made: [39]); the triple:
      f_1 = 5*x^5 + x^4*y + 3*x^3*y^2 - 3*x*y^4 - 5*y^5 - 2*x^4*z - 2*x^3*y*z - 4*x^2*y^2*z + 4*x*y^3*z + 2*y^4*z - 4*x^3*z^2 - 4*x^2*y*z^2 - 4*x*y^2*z^2 + 3*y^3*z^2 + 3*x^2*z^3 - 2*x*y*z^3 + 4*x*z^4 + 2*y*z^4 + z^5 - 3*y*z^3 + 3*z^4 - x^3 + 5*x^2*y - 5*x*y^2 - 5*x^2*z - 2*x*y*z - 11/3*x*z^2
      f_2 = 2*x^5 - 2*x^4*y - 4*x^3*y^2 + 2*x^2*y^3 + 2*y^5 + 3*x^4*z - 3*x^3*y*z - 4*x^2*y^2*z - 3*x*y^3*z - 2*y^4*z - x^3*z^2 - x^2*y*z^2 - 3*x*y^2*z^2 - 4*y^3*z^2 + x^2*z^3 + 3*x*y*z^3 + y^2*z^3 - 4*x*z^4 + 4*y*z^4 + 5*z^5 - 5*y*z^3 - z^4 + 4*x^3 + x^2*y + 5*x*y^2 + 3*x^2*z - 3*x*y*z - 19/6*x*z^2
      f_3 = -x^5 - 4*x^3*y^2 + x^2*y^3 + 5*x*y^4 - 3*y^5 - 4*x^4*z + x^3*y*z + 3*x^2*y^2*z - 3*x*y^3*z + 5*y^4*z - 5*x^2*y*z^2 + 4*x*y^2*z^2 - 5*y^3*z^2 - x^2*z^3 - 5*x*y*z^3 - 5*y^2*z^3 + 2*x*z^4 + y*z^4 - z^5 - 5*y*z^3 - 2*z^4 - 2*x^3 - x^2*y + 2*x*y^2 - 5*x^2*z - 2*x*y*z + 4/3*x*z^2
  ok   (1,3,6,5,3,2), general G, r = 3 gcd z^3: H = [1, 3, 6, 5, 3, 2], closed True, f_i in Ann(M) True, deg gcd K = 3, CI local length 39 (stable pair N = 10, 11; attempt 1) <= 39
   (1,3,6,5,4,1), x o G binary (G = X(Y^2 + YZ)), pure Y^5: attempt 1 used (local lengths of the attempts made: [37]); the triple:
      f_1 = -x^5 - 5*x^4*y + 3*x^3*y^2 - 5*x^2*y^3 - 2*y^5 - 4*x^4*z + 3*x^3*y*z - 4*x^2*y^2*z + x*y^3*z + 3*y^4*z - 5*x^3*z^2 - 5*x^2*y*z^2 - x*y^2*z^2 + y^3*z^2 - 3*x^2*z^3 + 4*x*y*z^3 - 3*y^2*z^3 - 5*x*z^4 - 2*z^5 - y^3*z + 2*y^2*z^2 + 5*y*z^3 - 4*x^2*y - 3*x*y^2 - x^2*z + 6*x*y*z - 3*x*z^2
      f_2 = 3*x^5 + 4*x^4*y + 5*x^3*y^2 + x^2*y^3 + y^5 - 3*x^4*z - 3*x^3*y*z + x^2*y^2*z + 2*x*y^3*z - 3*y^4*z + 2*x^3*z^2 - 4*x^2*y*z^2 - 2*x*y^2*z^2 + 2*y^3*z^2 + 2*x*y*z^3 - 5*y^2*z^3 + 4*x*z^4 + 5*y*z^4 - 3*z^5 + 4*y^3*z - 3*y^2*z^2 - 4*y*z^3 - z^4 - 3*x*y^2 - 5*x^2*z + 6*x*y*z + 4*x*z^2
      f_3 = -5*x^5 - 2*x^4*y - 5*x^3*y^2 + 2*x^2*y^3 - 2*x*y^4 + 3*y^5 - 5*x^4*z + 4*x^3*y*z - 2*x*y^3*z + 3*y^4*z - 2*x^3*z^2 + x^2*y*z^2 - x*y^2*z^2 + 2*x^2*z^3 - 2*x*y*z^3 - 2*y^2*z^3 - x*z^4 - 4*y^3*z - 3*y^2*z^2 + y*z^3 + 5*z^4 + 3*x^3 + 2*x^2*y + 4*x*y^2 + 2*x^2*z - 8*x*y*z - 5*x*z^2
  ok   (1,3,6,5,4,1), x o G binary (G = X(Y^2 + YZ)), pure Y^5: H = [1, 3, 6, 5, 4, 1], closed True, f_i in Ann(M) True, deg gcd K = 1, CI local length 37 (stable pair N = 10, 11; attempt 1) <= 37
   (1,3,6,5,4,1), x o G binary (G = X(Y^2 + YZ)), nonpure (3,4): attempt 1 used (local lengths of the attempts made: [36]); the triple:
      f_1 = x^5 - 5*x^4*y - 3*x^3*y^2 - x^2*y^3 + 5*x*y^4 - 2*y^5 - 4*x^4*z + x^3*y*z - 2*x^2*y^2*z + 2*x*y^3*z + y^4*z - 4*x^3*z^2 + 3*x*y^2*z^2 - 5*y^3*z^2 + 2*x^2*z^3 + 4*x*y*z^3 - 4*y^2*z^3 - 4*x*z^4 + 3*y*z^4 + 4*z^5 - 3*y^4 - 2*y^3*z + 2*y^2*z^2 + 30*y*z^3 - 1/5*z^4 + 4*x^3 + 2*x^2*y + x^2*z - 3*x*z^2
      f_2 = 4*x^5 - x^4*y + x^3*y^2 + 3*x^2*y^3 + 3*x*y^4 + y^5 + 3*x^4*z + 3*x^3*y*z + x^2*y^2*z + 5*x*y^3*z + 2*x^3*z^2 + 2*x^2*y*z^2 - 5*x*y^2*z^2 + 4*y^3*z^2 + 4*x^2*z^3 + 3*x*y*z^3 - 5*y^2*z^3 - 3*x*z^4 + 5*y*z^4 + 3*z^5 + 5*y^3*z - 5*y^2*z^2 + 1/2*z^4 + 4*x^3 + 4*x^2*y + x*y^2 - 3*x^2*z - 2*x*y*z - 5*x*z^2
      f_3 = 4*x^5 + x^4*y - 5*x^3*y^2 - 4*x^2*y^3 - x*y^4 - 2*y^5 - x^4*z - x^3*y*z - 5*x^2*y^2*z + 2*y^4*z - 2*x^3*z^2 - 4*x^2*y*z^2 + 4*x*y^2*z^2 + 3*y^3*z^2 + 5*x^2*z^3 + 4*x*y*z^3 - 4*y^2*z^3 + 3*x*z^4 - 4*y*z^4 - 2*z^5 - 4*y^4 + 4*y^2*z^2 + 40*y*z^3 - 2/5*z^4 - 2*x^3 - x^2*y - 4*x*y^2 - x^2*z + 8*x*y*z - 5*x*z^2
  ok   (1,3,6,5,4,1), x o G binary (G = X(Y^2 + YZ)), nonpure (3,4): H = [1, 3, 6, 5, 4, 1], closed True, f_i in Ann(M) True, deg gcd K = 0, CI local length 36 (stable pair N = 9, 10; attempt 1) <= 36
   (1,3,6,5,4,1), x o G binary (G = X(Y^2 + YZ)), nonpure (2,5): attempt 1 used (local lengths of the attempts made: [38]); the triple:
      f_1 = -2*x^5 + 5*x^4*y - 4*x^2*y^3 - 4*x*y^4 + 4*y^5 + 4*x^4*z + x^3*y*z + 3*x^2*y^2*z - 2*x*y^3*z - 4*y^4*z - x*y^2*z^2 - 3*y^3*z^2 - 2*x^2*z^3 - x*y*z^3 - 4*y^2*z^3 - x*z^4 + 5*y*z^4 + z^5 - 3*y^3*z + 5*y^2*z^2 + 3*y*z^3 + x^3 - 5*x^2*y + x*y^2 + 5*x^2*z - 2*x*y*z + x*z^2
      f_2 = 3*x^5 + x^4*y - 2*x^3*y^2 + 3*x^2*y^3 + x*y^4 - 3*y^5 + x^4*z + x^3*y*z + 4*x^2*y^2*z - x*y^3*z + y^4*z + 2*x^3*z^2 - 2*x^2*y*z^2 - 4*x*y^2*z^2 - 5*y^3*z^2 - x^2*z^3 + 4*x*y*z^3 - 3*y^2*z^3 - 4*y*z^4 - z^5 + 4*y^3*z + 5*y^2*z^2 + 5*y*z^3 - 3*x^3 - 2*x^2*y - 4*x*y^2 + 5*x^2*z + 8*x*y*z + 4*x*z^2
      f_3 = 4*x^5 - 4*x^4*y + 2*x^2*y^3 - 2*y^5 + 5*x^4*z - 4*x^3*y*z - 3*x^2*y^2*z - 2*x*y^3*z + 4*y^4*z - 5*x^3*z^2 + 3*x^2*y*z^2 + 2*x*y^2*z^2 + 2*y^3*z^2 - 4*x^2*z^3 + 5*x*y*z^3 - 4*y^2*z^3 + x*z^4 - 5*y*z^4 - 2*z^5 + y^3*z - 2*y^2*z^2 + 4*x^3 + 3*x^2*y + 4*x*y^2 + 4*x^2*z - 8*x*y*z - 3*x*z^2
  ok   (1,3,6,5,4,1), x o G binary (G = X(Y^2 + YZ)), nonpure (2,5): H = [1, 3, 6, 5, 4, 1], closed True, f_i in Ann(M) True, deg gcd K = 2, CI local length 38 (stable pair N = 10, 11; attempt 1) <= 38
   (1,3,6,5,4,1), x o G binary (G = X(Y^2 + YZ)), X-free Y^4 Z: attempt 1 used (local lengths of the attempts made: [38]); the triple:
      f_1 = -3*x^4*y + 4*x^3*y^2 - 2*x*y^4 + 5*y^5 + 3*x^4*z + 5*x^3*y*z - 4*x^2*y^2*z - 4*y^4*z - 4*x^3*z^2 - 4*x^2*y*z^2 - 2*y^3*z^2 - 2*x^2*z^3 - 3*x*y*z^3 + y^2*z^3 + 2*x*z^4 + 3*y*z^4 + 2*z^5 - 4*y^2*z^2 + 4*y*z^3 + 2*z^4 - 4*x^3 - 3*x^2*z + 4*x*z^2
      f_2 = 2*x^5 + x^4*y - 4*x^3*y^2 - 5*x^2*y^3 - 2*x*y^4 + 2*y^5 + 2*x^3*y*z - 4*x^2*y^2*z + 5*y^4*z - x^3*z^2 + 2*x^2*y*z^2 + x*y^2*z^2 - 4*y^3*z^2 - 3*x*y*z^3 + 4*y^2*z^3 - x*z^4 - 4*y*z^4 - z^5 - 3*y^2*z^2 + 5*y*z^3 - 5*z^4 + x^3 - 5*x^2*y + 4*x*y^2 - 4*x^2*z - 8*x*y*z
      f_3 = -2*x^5 - x^4*y - 4*x^3*y^2 - 4*x^2*y^3 - 3*x*y^4 + 2*y^5 + 5*x^4*z - 3*x^3*y*z + 3*x^2*y^2*z + 3*x*y^3*z - x^3*z^2 + 2*x^2*y*z^2 + x*y^2*z^2 - 3*y^3*z^2 + x^2*z^3 - 2*x*y*z^3 - 4*y^2*z^3 - 4*x*z^4 - 5*y*z^4 - 3*z^5 - 5*y^2*z^2 - 2*y*z^3 + z^4 + x^3 + 3*x*y^2 - 4*x^2*z - 6*x*y*z + x*z^2
  ok   (1,3,6,5,4,1), x o G binary (G = X(Y^2 + YZ)), X-free Y^4 Z: H = [1, 3, 6, 5, 4, 1], closed True, f_i in Ann(M) True, deg gcd K = 2, CI local length 38 (stable pair N = 10, 11; attempt 1) <= 38
   (1,3,6,5,3,2), x o G binary (G = X(Y^2 + YZ)), r = 3 generic: attempt 1 used (local lengths of the attempts made: [36]); the triple:
      f_1 = 3*x^5 - 4*x^4*y - 2*x^3*y^2 - 5*x^2*y^3 - x*y^4 - 5*y^5 - 5*x^4*z - 2*x^3*y*z + 2*x^2*y^2*z + 4*x*y^3*z - 5*y^4*z + 2*x^3*z^2 + 3*x^2*y*z^2 + 5*x*y^2*z^2 - 3*y^3*z^2 - 4*x^2*z^3 + 2*x*y*z^3 + y^2*z^3 + 3*x*z^4 + 3*y*z^4 + z^5 + 2*y^4 + y^3*z - 20*y*z^3 - 1/10*z^4 - 5*x^2*y - x*y^2 - 5*x^2*z + 2*x*y*z + 3*x*z^2
      f_2 = -2*x^5 - 5*x^4*y + x^3*y^2 - 5*x^2*y^3 - x*y^4 - 2*y^5 + x^4*z + x^3*y*z - 3*x^2*y^2*z - 2*x*y^3*z + y^4*z + x^3*z^2 + x*y^2*z^2 + y^3*z^2 - 3*x^2*z^3 - 4*x*y*z^3 - 2*y^2*z^3 - y*z^4 - 3*z^5 + 4*y^4 - 5*y^3*z - 40*y*z^3 + 1/2*z^4 + 4*x^3 + 5*x^2*y + 4*x*y^2 + 5*x^2*z - 8*x*y*z + 5*x*z^2
      f_3 = -3*x^5 + 2*x^4*y - 4*x^3*y^2 - 5*x^2*y^3 + 5*x*y^4 + 2*y^5 - 4*x^4*z + 5*x^3*y*z + x^2*y^2*z - 5*x*y^3*z - 3*y^4*z + x^3*z^2 - 3*x^2*y*z^2 + 2*x*y^2*z^2 + 2*y^3*z^2 - 2*x^2*z^3 - 2*y^2*z^3 + 4*x*z^4 - 4*y*z^4 + 5*y^4 - 4*y^3*z - 50*y*z^3 + 2/5*z^4 + 2*x^3 - 5*x*y^2 - x^2*z + 10*x*y*z + 5*x*z^2
  ok   (1,3,6,5,3,2), x o G binary (G = X(Y^2 + YZ)), r = 3 generic: H = [1, 3, 6, 5, 3, 2], closed True, f_i in Ann(M) True, deg gcd K = 0, CI local length 36 (stable pair N = 9, 10; attempt 1) <= 36
   (1,3,6,5,3,2), x o G binary (G = X(Y^2 + YZ)), r = 3 with a fifth power: attempt 1 used (local lengths of the attempts made: [37]); the triple:
      f_1 = -3*x^5 + 5*x^4*y - x^2*y^3 - x*y^4 - 2*x^3*y*z + 3*x^2*y^2*z - 4*x*y^3*z - y^4*z - 3*x^3*z^2 + 3*x^2*y*z^2 - 4*x*y^2*z^2 - 4*x^2*z^3 + 5*x*y*z^3 + 2*y^2*z^3 + x*z^4 + 3*y*z^4 + 4*z^5 - 4*y^3*z - y^2*z^2 + 1/10*z^4 + 4*x^3 - 5*x^2*y - 3*x*y^2 - 5*x^2*z + 6*x*y*z + x*z^2
      f_2 = -5*x^5 + x^4*y - 2*x^3*y^2 - 5*x^2*y^3 + x*y^4 - y^5 - 3*x^4*z + 2*x^3*y*z - 5*x^2*y^2*z - 2*x*y^3*z - 5*y^4*z + 5*x^3*z^2 - 5*x^2*y*z^2 + 3*x*y^2*z^2 + 3*y^3*z^2 - 2*x^2*z^3 + 4*x*y*z^3 - 2*y^2*z^3 - 2*x*z^4 - 5*y*z^4 + z^5 - 5*y^3*z + 3*y^2*z^2 - 3/10*z^4 + 2*x^3 - 2*x^2*y - 3*x*y^2 + 4*x^2*z + 6*x*y*z - x*z^2
      f_3 = 5*x^5 - 4*x^4*y + 4*x^3*y^2 + 4*x^2*y^3 - x*y^4 + 5*y^5 + 3*x^3*y*z + 4*x^2*y^2*z - 3*x*y^3*z + 2*y^4*z - x^3*z^2 + x^2*y*z^2 + 4*x*y^2*z^2 - 5*y^3*z^2 + x^2*z^3 + 5*x*y*z^3 - 2*y^2*z^3 - 4*x*z^4 + 3*y*z^4 + 4*z^5 + 2*y^3*z - 4*y^2*z^2 + 2/5*z^4 - 5*x^2*y - x*y^2 + 3*x^2*z + 2*x*y*z + 2*x*z^2
  ok   (1,3,6,5,3,2), x o G binary (G = X(Y^2 + YZ)), r = 3 with a fifth power: H = [1, 3, 6, 5, 3, 2], closed True, f_i in Ann(M) True, deg gcd K = 1, CI local length 37 (stable pair N = 10, 11; attempt 1) <= 37
   (1,3,6,5,3,2), x o G binary (G = X(Y^2 + YZ)), r = 2 secant: attempt 1 used (local lengths of the attempts made: [38]); the triple:
      f_1 = 4*x^5 + 4*x^4*y + 4*x^3*y^2 - 5*x^2*y^3 - 3*x*y^4 + 2*y^5 - 2*x^4*z + 2*x^3*y*z - 4*x^2*y^2*z - 3*y^4*z - 5*x^3*z^2 - 4*x^2*y*z^2 - 4*y^3*z^2 - x*y*z^3 - y^2*z^3 - x*z^4 - y*z^4 - 5*z^5 - 4*y^3*z + 2*y^2*z^2 - 5*y*z^3 + x^2*y + 2*x*y^2 - 3*x^2*z - 4*x*y*z
      f_2 = 2*x^5 - 3*x^4*y - 3*x^3*y^2 + 2*x*y^4 + 5*y^5 + 4*x^4*z + 4*x^3*y*z + 3*x^2*y^2*z - 5*x*y^3*z - 5*y^4*z - 5*x^3*z^2 - 4*x^2*y*z^2 + 4*x*y^2*z^2 - 5*y^3*z^2 - 5*x^2*z^3 + 2*x*y*z^3 + 4*y^2*z^3 + 5*x*z^4 + y*z^4 - 4*z^5 - 2*y^3*z - y^2*z^2 + 4*y*z^3 + 5*x^3 - x^2*y - 3*x*y^2 + x^2*z + 6*x*y*z + 5*x*z^2
      f_3 = -4*x^5 - 4*x^4*y + 4*x^3*y^2 - 4*x^2*y^3 + 3*x*y^4 - y^5 + 5*x^4*z - 4*x^3*y*z + 2*x^2*y^2*z + 3*x*y^3*z - 4*y^4*z - 5*x^3*z^2 - x^2*y*z^2 - 4*x*y^2*z^2 + y^3*z^2 + 5*x^2*z^3 + 5*x*y*z^3 - y^2*z^3 + 5*x*z^4 - y*z^4 + 2*z^5 + 4*y^3*z - 2*y^2*z^2 + 3*y*z^3 + x^3 + 3*x^2*y + 2*x*y^2 + 4*x^2*z - 4*x*y*z + 4*x*z^2
  ok   (1,3,6,5,3,2), x o G binary (G = X(Y^2 + YZ)), r = 2 secant: H = [1, 3, 6, 5, 3, 2], closed True, f_i in Ann(M) True, deg gcd K = 2, CI local length 38 (stable pair N = 10, 11; attempt 1) <= 38
   (1,3,6,5,3,2), x o G binary (G = X(Y^2 + YZ)), r = 2 tangent: attempt 1 used (local lengths of the attempts made: [38]); the triple:
      f_1 = 4*x^4*y + 2*x^3*y^2 + 2*x*y^4 - 3*y^5 - 2*x^4*z - 4*x^3*y*z - 4*x^2*y^2*z + 4*x*y^3*z - 2*y^4*z - 3*x^3*z^2 + 5*x^2*y*z^2 + 3*x*y^2*z^2 - 3*y^3*z^2 + x^2*z^3 + 5*x*y*z^3 + 2*y^2*z^3 + x*z^4 - 3*y*z^4 + 5*z^5 + 5*y^2*z^2 - 2*y*z^3 - z^4 + 5*x^3 + 2*x^2*y + x*y^2 + 2*x^2*z - 2*x*y*z - 3*x*z^2
      f_2 = 4*x^5 + 2*x^4*y + 3*x^3*y^2 - 2*x^2*y^3 + 3*x*y^4 - 4*y^5 + 3*x^4*z - 2*x^3*y*z + 5*x^2*y^2*z - x*y^3*z - 2*y^4*z + 3*x^3*z^2 - 3*x^2*y*z^2 - y^3*z^2 + x^2*z^3 - x*y*z^3 + y^2*z^3 + 4*x*z^4 + 3*z^5 - 3*y^2*z^2 + y*z^3 - z^4 + 3*x^3 + 2*x^2*y + 4*x^2*z - 5*x*z^2
      f_3 = -4*x^5 - 4*x^4*y - x^3*y^2 - 5*x^2*y^3 + 5*x*y^4 + 5*x^4*z + 4*x^3*y*z + 3*x^2*y^2*z - 4*y^4*z - 3*x^3*z^2 + x^2*y*z^2 - x*y^2*z^2 - y^3*z^2 - 3*x^2*z^3 - 4*x*y*z^3 - 4*y^2*z^3 + x*z^4 - y*z^4 - 4*y^2*z^2 - 4*y*z^3 - x^3 + 2*x^2*y + 5*x*y^2 + 5*x^2*z - 10*x*y*z - 4*x*z^2
  ok   (1,3,6,5,3,2), x o G binary (G = X(Y^2 + YZ)), r = 2 tangent: H = [1, 3, 6, 5, 3, 2], closed True, f_i in Ann(M) True, deg gcd K = 2, CI local length 38 (stable pair N = 10, 11; attempt 1) <= 38
   (1,3,6,5,3,2), x o G binary (G = X(Y^2 + YZ)), r = 3 gcd z^3: attempt 1 used (local lengths of the attempts made: [39]); the triple:
      f_1 = -5*x^5 - 4*x^4*y + x^3*y^2 + x^2*y^3 - 5*x*y^4 - 4*y^5 + 3*x^4*z - 4*x^3*y*z - 4*x^2*y^2*z + 4*x*y^3*z + 5*y^4*z + 2*x^3*z^2 - 2*x^2*y*z^2 + 3*x*y^2*z^2 + 3*y^3*z^2 + x^2*z^3 - x*y*z^3 + 5*x*z^4 - 4*z^5 - 4*y*z^3 - 3*z^4 + x^3 + x^2*y - 4*x*y^2 - 4*x^2*z + 8*x*y*z - x*z^2
      f_2 = 3*x^5 - 4*x^4*y + 5*x^3*y^2 + x^2*y^3 - x*y^4 - 2*y^5 - 4*x^4*z + 5*x^3*y*z + x^2*y^2*z + 3*x*y^3*z + 3*y^4*z + 3*x^3*z^2 - 4*x^2*y*z^2 + 2*x*y^2*z^2 - 3*y^3*z^2 - x^2*z^3 + x*y*z^3 + y^2*z^3 - 5*x*z^4 + 5*y*z^4 + 5*z^5 - 4*y*z^3 + 4*z^4 + 2*x^3 + x^2*y - 3*x*y^2 - 4*x^2*z + 6*x*y*z - 2*x*z^2
      f_3 = -5*x^5 + 5*x^4*y - x^3*y^2 + 4*x^2*y^3 - x*y^4 + 2*y^5 + 2*x^4*z + 2*x^3*y*z + 5*x^2*y^2*z - 3*x*y^3*z + y^4*z + 3*x^3*z^2 + 3*x^2*y*z^2 + 5*x*y^2*z^2 + 4*y^3*z^2 - x^2*z^3 - x*y*z^3 - 4*y^2*z^3 + 2*x*z^4 - 3*y*z^4 + 4*z^5 - 3*y*z^3 - z^4 - 3*x^3 - 2*x^2*y + 3*x*y^2 - 6*x*y*z - 4*x*z^2
  ok   (1,3,6,5,3,2), x o G binary (G = X(Y^2 + YZ)), r = 3 gcd z^3: H = [1, 3, 6, 5, 3, 2], closed True, f_i in Ann(M) True, deg gcd K = 3, CI local length 39 (stable pair N = 10, 11; attempt 1) <= 39
   (1,3,6,5,4,1), G = XYZ, pure Y^5: attempt 1 used (local lengths of the attempts made: [37]); the triple:
      f_1 = 2*x^5 + 5*x^4*y + 2*x^3*y^2 + 4*x^2*y^3 - 5*x*y^4 - 5*y^5 - 3*x^4*z - 4*x^3*y*z - x^2*y^2*z + 3*x*y^3*z - x^3*z^2 + 5*x^2*y*z^2 + 3*x*y^2*z^2 + 4*y^3*z^2 - 4*x*y*z^3 + 4*y^2*z^3 - 2*x*z^4 + 2*y*z^4 + 5*z^5 - 4*y^3*z + 2*y^2*z^2 - y*z^3 + 4*z^4 - 2*x^3 - 3*x^2*y - 3*x*y^2 - 2*x^2*z + 5*x*z^2
      f_2 = 3*x^5 + x^4*y + x^3*y^2 - x^2*y^3 - 5*x*y^4 + 5*y^5 - x^4*z + 5*x^2*y^2*z + 2*x*y^3*z - y^4*z - 3*x^3*z^2 - 2*x^2*y*z^2 - 3*x*y^2*z^2 + y^3*z^2 - 5*x^2*z^3 - x*y*z^3 + 4*y^2*z^3 + x*z^4 + 5*y*z^4 + 2*z^5 - 5*y^2*z^2 - y*z^3 - 3*z^4 + 2*x^3 + 3*x^2*y - x*y^2 + 4*x^2*z + 4*x*z^2
      f_3 = 5*x^5 - 5*x^4*y + 2*x^3*y^2 - 5*x^2*y^3 + 2*x*y^4 + 3*y^5 + 3*x^4*z + 2*x^3*y*z - 2*x^2*y^2*z - 4*x*y^3*z - 2*y^4*z + 3*x^2*y*z^2 - 2*x*y^2*z^2 + 5*y^3*z^2 - 5*x^2*z^3 - 3*x*y*z^3 + y^2*z^3 - 3*x*z^4 + 5*y*z^4 - 4*z^5 - 5*y^3*z - 3*y^2*z^2 + 4*y*z^3 + 2*z^4 + 3*x^2*y + 3*x^2*z + 4*x*z^2
  ok   (1,3,6,5,4,1), G = XYZ, pure Y^5: H = [1, 3, 6, 5, 4, 1], closed True, f_i in Ann(M) True, deg gcd K = 1, CI local length 37 (stable pair N = 10, 11; attempt 1) <= 37
   (1,3,6,5,4,1), G = XYZ, nonpure (3,4): attempt 1 used (local lengths of the attempts made: [36]); the triple:
      f_1 = -4*x^5 + 3*x^4*y + x^3*y^2 - 2*x^2*y^3 + 4*x*y^4 - y^5 + 4*x^3*y*z - 4*x^2*y^2*z + 3*x*y^3*z + 3*y^4*z - x^3*z^2 + 2*x*y^2*z^2 - 5*y^3*z^2 + 3*x^2*z^3 + 5*x*y*z^3 - 3*y^2*z^3 - 5*x*z^4 + y*z^4 - 5*z^5 + 5*y^4 - 2*y^3*z + 4*y^2*z^2 - 50*y*z^3 - 2/5*z^4 - 2*x^3 + 5*x^2*y - 5*x*y^2 - x^2*z - 2*x*z^2
      f_2 = 2*x^5 + 2*x^4*y + 3*x^3*y^2 + 4*x^2*y^3 + 4*x*y^4 + 5*y^5 - x^4*z - 4*x^3*y*z + 4*x^2*y^2*z + 4*x*y^3*z + 2*y^4*z - 5*x^3*z^2 + 2*x^2*y*z^2 + 4*x*y^2*z^2 + 3*y^3*z^2 + 4*x^2*z^3 - 2*x*y*z^3 - 2*y^2*z^3 - 5*x*z^4 - 4*y*z^4 - 3*y^4 + 2*y^3*z + 2*y^2*z^2 + 30*y*z^3 - 1/5*z^4 + 2*x^3 - 5*x^2*y - 2*x*y^2 - 2*x^2*z - 4*x*z^2
      f_3 = -5*x^5 - 5*x^4*y - 5*x^3*y^2 + x^2*y^3 + x*y^4 + 2*y^5 + x^4*z - x^3*y*z + x^2*y^2*z - 2*x*y^3*z - 3*y^4*z - 2*x^3*z^2 - 3*x^2*y*z^2 - 2*x*y^2*z^2 + y^3*z^2 - 4*x^2*z^3 - 3*x*y*z^3 - 5*y^2*z^3 + x*z^4 - y*z^4 + 2*z^5 - 3*y^4 - 2*y^3*z + y^2*z^2 + 30*y*z^3 - 1/10*z^4 - 2*x^3 - 4*x^2*y + 3*x^2*z - 4*x*z^2
  ok   (1,3,6,5,4,1), G = XYZ, nonpure (3,4): H = [1, 3, 6, 5, 4, 1], closed True, f_i in Ann(M) True, deg gcd K = 0, CI local length 36 (stable pair N = 9, 10; attempt 1) <= 36
   (1,3,6,5,4,1), G = XYZ, nonpure (2,5): attempt 1 used (local lengths of the attempts made: [38]); the triple:
      f_1 = 2*x^5 - 2*x^4*y + 5*x^2*y^3 - 3*x*y^4 + y^5 - 2*x^4*z + 4*x^3*y*z - 5*x^2*y^2*z - x^3*z^2 - 2*x^2*y*z^2 + 5*x*y^2*z^2 - 3*y^3*z^2 + 3*x^2*z^3 - 4*x*y*z^3 + 5*y^2*z^3 + 4*x*z^4 - y*z^4 - 3*z^5 + 5*y^3*z + 5*y^2*z^2 + 3*y*z^3 - x^3 - 4*x^2*y + x*y^2 - 2*x^2*z + x*z^2
      f_2 = -4*x^4*y - 3*x^3*y^2 - 5*x^2*y^3 - x*y^4 + 3*y^5 + 2*x^4*z + 4*x^2*y^2*z + 4*x*y^3*z + y^4*z - 5*x^3*z^2 + 2*x*y^2*z^2 - 3*y^3*z^2 + 2*x^2*z^3 + y^2*z^3 + 4*x*z^4 + y*z^4 + 3*z^5 - 5*y^3*z + 3*y^2*z^2 + 2*y*z^3 - 5*x^3 + 3*x^2*y + x*y^2 - 3*x^2*z - 4*x*z^2
      f_3 = 3*x^5 + 2*x^4*y - 4*x^3*y^2 + 4*x^2*y^3 - 5*y^5 - 2*x^4*z + 5*x^3*y*z + x*y^3*z - y^4*z + 5*x^3*z^2 - 4*x^2*y*z^2 - 3*x*y^2*z^2 + 5*y^3*z^2 - 4*x^2*z^3 + 5*y^2*z^3 - 4*y*z^4 + 3*z^5 - 4*y^3*z + 3*y^2*z^2 - 4*y*z^3 + 2*x^3 - 5*x^2*y + 2*x*y^2 - 2*x^2*z - x*z^2
  ok   (1,3,6,5,4,1), G = XYZ, nonpure (2,5): H = [1, 3, 6, 5, 4, 1], closed True, f_i in Ann(M) True, deg gcd K = 2, CI local length 38 (stable pair N = 10, 11; attempt 1) <= 38
   (1,3,6,5,4,1), G = XYZ, X-free Y^4 Z: attempt 1 used (local lengths of the attempts made: [38]); the triple:
      f_1 = -x^5 - 3*x^4*y - x^3*y^2 + 4*x^2*y^3 - 2*x*y^4 - 4*y^5 + x^4*z + 4*x^3*y*z - 4*x^2*y^2*z - x*y^3*z + y^4*z + 2*x^3*z^2 + 5*x^2*y*z^2 - 4*x*y^2*z^2 - y^3*z^2 + 3*x^2*z^3 + 4*x*y*z^3 + y^2*z^3 - x*z^4 - 4*y*z^4 - 3*z^5 + 2*y^2*z^2 - 2*y*z^3 + 2*z^4 + 5*x^3 - 4*x^2*y + 2*x*y^2 - 5*x^2*z + 3*x*z^2
      f_2 = -4*x^5 + 2*x^4*y - 4*x^3*y^2 - 5*x^2*y^3 + 4*x*y^4 + 2*y^5 + 2*x^3*y*z + 5*x^2*y^2*z - 2*x*y^3*z + 3*y^4*z + 2*x^2*y*z^2 - x*y^2*z^2 + 3*y^3*z^2 - 5*x^2*z^3 - x*y*z^3 - x*z^4 + 3*y*z^4 + 3*z^5 - 2*y^2*z^2 + 5*z^4 + 5*x^2*y - 4*x*y^2 + x^2*z
      f_3 = x^5 + x^4*y - 3*x^3*y^2 + x*y^4 - 5*y^5 - 4*x^4*z - 4*x^3*y*z - 5*x^2*y^2*z + 4*x*y^3*z + 2*y^4*z + 2*x^3*z^2 - x^2*y*z^2 + 4*x*y^2*z^2 - 3*y^3*z^2 - 4*x*y*z^3 - y^2*z^3 - 3*x*z^4 - 5*y*z^4 + 4*z^5 + 5*y^2*z^2 - 2*y*z^3 + 4*z^4 - 4*x^3 + 3*x^2*y + 2*x*y^2 + 4*x^2*z + x*z^2
  ok   (1,3,6,5,4,1), G = XYZ, X-free Y^4 Z: H = [1, 3, 6, 5, 4, 1], closed True, f_i in Ann(M) True, deg gcd K = 2, CI local length 38 (stable pair N = 10, 11; attempt 1) <= 38
   (1,3,6,5,3,2), G = XYZ, r = 3 generic: attempt 1 used (local lengths of the attempts made: [36]); the triple:
      f_1 = -x^5 - x^4*y + 5*x^3*y^2 + x^2*y^3 + 2*y^5 + x^4*z - 2*x^3*y*z + 2*x^2*y^2*z + 3*x*y^3*z + 4*y^4*z - 5*x^3*z^2 - x^2*y*z^2 + 3*x*y^2*z^2 + 2*y^3*z^2 - 3*x^2*z^3 + 4*x*y*z^3 - 5*y^2*z^3 + x*z^4 - 3*y*z^4 - y^4 + 5*y^3*z + 10*y*z^3 - 1/2*z^4 - 4*x^3 + 4*x^2*y - 4*x*y^2 - 5*x^2*z - 4*x*z^2
      f_2 = -4*x^3*y^2 + 2*x^2*y^3 + 3*x*y^4 - 3*y^5 + 5*x^4*z + x^3*y*z - x^2*y^2*z + 2*x*y^3*z + y^4*z + x^3*z^2 - 3*x^2*y*z^2 - 5*y^3*z^2 - 3*x^2*z^3 - x*y*z^3 + 5*y^2*z^3 - 3*x*z^4 + y*z^4 + 2*y^4 - y^3*z - 20*y*z^3 + 1/10*z^4 + 2*x^3 - 4*x*y^2 + 4*x^2*z - 4*x*z^2
      f_3 = 2*x^5 - 4*x^4*y + x^3*y^2 - 2*x^2*y^3 + 2*x*y^4 - 4*y^5 - x^4*z - 5*x^3*y*z + 5*x^2*y^2*z + 4*x*y^3*z + 3*y^4*z - 2*x^3*z^2 + x^2*y*z^2 + 5*x*y^2*z^2 - y^3*z^2 - 2*x^2*z^3 + 3*x*y*z^3 - 4*y^2*z^3 - 5*x*z^4 + 4*y^4 + 3*y^3*z - 40*y*z^3 - 3/10*z^4 - x^3 - 2*x^2*y - 3*x*y^2 + 4*x^2*z + x*z^2
  ok   (1,3,6,5,3,2), G = XYZ, r = 3 generic: H = [1, 3, 6, 5, 3, 2], closed True, f_i in Ann(M) True, deg gcd K = 0, CI local length 36 (stable pair N = 9, 10; attempt 1) <= 36
   (1,3,6,5,3,2), G = XYZ, r = 3 with a fifth power: attempt 1 used (local lengths of the attempts made: [37]); the triple:
      f_1 = -5*x^5 - x^4*y + 3*x^3*y^2 - 2*x^2*y^3 - 2*y^5 + 2*x^4*z - 2*x^3*y*z + 2*x*y^3*z - 4*y^4*z - 5*x^3*z^2 - 5*x*y^2*z^2 - y^3*z^2 - 2*x^2*z^3 - 5*x*y*z^3 - 4*y^2*z^3 - 4*x*z^4 - 2*y*z^4 - 4*z^5 - 2*y^3*z + 2*y^2*z^2 - 1/5*z^4 + 4*x^3 - 5*x*y^2 - 3*x^2*z - 5*x*z^2
      f_2 = -3*x^5 - x^4*y - 4*x^3*y^2 - 2*x^2*y^3 + 2*x*y^4 + y^5 - x^4*z + x^3*y*z + 2*x^2*y^2*z - 2*x*y^3*z + 5*y^4*z - 3*x^3*z^2 - 5*x^2*y*z^2 - 5*x*y^2*z^2 + 4*y^3*z^2 - x^2*z^3 + 3*x*y*z^3 - 4*y^2*z^3 - 2*x*z^4 - y*z^4 + 2*z^5 - y^3*z + y^2*z^2 - 1/10*z^4 + 5*x^3 - 4*x^2*y + 4*x^2*z
      f_3 = -2*x^5 - 2*x^3*y^2 - 5*x^2*y^3 + 3*x*y^4 + 5*y^5 + 4*x^3*y*z - 5*x^2*y^2*z + x*y^3*z - 2*y^4*z + 2*x^3*z^2 + 4*x^2*y*z^2 + 4*y^3*z^2 - y^2*z^3 - x*z^4 + 2*y*z^4 - 3*z^5 - 4*y^3*z - 4*y^2*z^2 + 2/5*z^4 - 5*x^3 - x^2*y + 2*x*y^2 - 5*x^2*z - 3*x*z^2
  ok   (1,3,6,5,3,2), G = XYZ, r = 3 with a fifth power: H = [1, 3, 6, 5, 3, 2], closed True, f_i in Ann(M) True, deg gcd K = 1, CI local length 37 (stable pair N = 10, 11; attempt 1) <= 37
   (1,3,6,5,3,2), G = XYZ, r = 2 secant: attempt 1 used (local lengths of the attempts made: [38]); the triple:
      f_1 = x^5 + 5*x^4*y + 4*x^3*y^2 - 3*x^2*y^3 + 4*x*y^4 - 2*y^5 + 5*x^4*z - 2*x^3*y*z - 2*x^2*y^2*z + 4*x*y^3*z + 2*y^4*z - x^3*z^2 - 3*x^2*y*z^2 + 4*x*y^2*z^2 + 5*y^3*z^2 - 4*x^2*z^3 + 4*x*y*z^3 - y^2*z^3 + 2*x*z^4 - y*z^4 - 4*z^5 - 2*y^3*z - 2*y^2*z^2 - 2*y*z^3 - 2*x^3 - 5*x^2*y - x*y^2 + 4*x^2*z - 4*x*z^2
      f_2 = 5*x^5 + 2*x^3*y^2 - x^2*y^3 + 4*x*y^4 - y^5 + 4*x^4*z + 3*x^3*y*z - 5*x^2*y^2*z - 3*x*y^3*z + 2*y^4*z - 5*x^3*z^2 + 3*x^2*y*z^2 + 5*x*y^2*z^2 + 4*y^3*z^2 - 3*x^2*z^3 - x*y*z^3 + 4*y^2*z^3 + x*z^4 + 5*y*z^4 + 5*y^3*z - 4*y^2*z^2 + 5*y*z^3 + 4*x^3 - 3*x^2*y + 2*x*y^2 + 4*x^2*z + 3*x*z^2
      f_3 = 4*x^5 - 5*x^4*y - 3*x^3*y^2 - 5*x^2*y^3 - x*y^4 + 2*x^4*z - x^3*y*z - 2*x^2*y^2*z + x*y^3*z - y^4*z + 3*x^3*z^2 - x^2*y*z^2 - x*y^2*z^2 - 4*y^3*z^2 - 3*x^2*z^3 - 4*x*y*z^3 + 4*y^2*z^3 + 5*x*z^4 - 2*y*z^4 - 5*z^5 + 5*y^3*z - 2*y^2*z^2 - 4*y*z^3 - 3*x^3 - 2*x^2*y + 3*x*y^2 + 2*x^2*z + 3*x*z^2
  ok   (1,3,6,5,3,2), G = XYZ, r = 2 secant: H = [1, 3, 6, 5, 3, 2], closed True, f_i in Ann(M) True, deg gcd K = 2, CI local length 38 (stable pair N = 10, 11; attempt 1) <= 38
   (1,3,6,5,3,2), G = XYZ, r = 2 tangent: attempt 1 used (local lengths of the attempts made: [38]); the triple:
      f_1 = 2*x^5 + 3*x^4*y + x^3*y^2 - 5*x^2*y^3 - x*y^4 - 5*x^4*z + 4*x^3*y*z - 2*x^2*y^2*z + y^4*z + 4*x^3*z^2 + 3*x^2*y*z^2 + 3*x*y^2*z^2 + 4*y^3*z^2 + 5*x^2*z^3 + 5*x*y*z^3 + 4*y^2*z^3 - 3*x*z^4 + 5*y*z^4 + 5*z^5 - 3*y^2*z^2 + y*z^3 + 5*z^4 - x^3 - 4*x^2*y + x*y^2 + 2*x^2*z + 2*x*z^2
      f_2 = -x^5 - 2*x^4*y + 2*x^3*y^2 - 3*x^2*y^3 + 2*x*y^4 + x^4*z + 2*x^3*y*z + 2*x^2*y^2*z + 5*x*y^3*z - 2*y^4*z + 3*x^3*z^2 + 2*x^2*y*z^2 + 4*x*y^2*z^2 - 3*y^3*z^2 + 2*x^2*z^3 - 2*x*y*z^3 - 4*y^2*z^3 - 2*x*z^4 - 4*y*z^4 + 2*z^5 + 4*y^2*z^2 + 5*y*z^3 - 5*z^4 - 3*x^3 + x^2*y + 4*x*y^2 - 2*x^2*z + 4*x*z^2
      f_3 = 2*x^5 + 4*x^4*y - 3*x^3*y^2 - 4*x^2*y^3 - 4*x*y^4 + 5*y^5 + 5*x^4*z - 4*x^3*y*z + x^2*y^2*z - 5*x*y^3*z + 3*y^4*z - 5*x^3*z^2 + 2*x^2*y*z^2 + 5*y^3*z^2 - 3*x^2*z^3 + x*y*z^3 + 3*x*z^4 - 2*y*z^4 - 3*z^5 + 5*y^2*z^2 - 3*y*z^3 - 5*z^4 + 5*x^3 + 2*x^2*y + 5*x*y^2 + 3*x^2*z - 5*x*z^2
  ok   (1,3,6,5,3,2), G = XYZ, r = 2 tangent: H = [1, 3, 6, 5, 3, 2], closed True, f_i in Ann(M) True, deg gcd K = 2, CI local length 38 (stable pair N = 10, 11; attempt 1) <= 38
   (1,3,6,5,3,2), G = XYZ, r = 3 gcd z^3: attempt 1 used (local lengths of the attempts made: [39]); the triple:
      f_1 = 4*x^5 + 2*x^3*y^2 - 2*x^2*y^3 - x*y^4 + 5*y^5 - 5*x^4*z - 5*x^3*y*z + 3*x^2*y^2*z - 2*x*y^3*z - 4*y^4*z - 2*x^3*z^2 - x^2*y*z^2 + 5*x*y^2*z^2 + 4*y^3*z^2 - x^2*z^3 - 2*x*y*z^3 + 2*y^2*z^3 + 2*x*z^4 + 2*y*z^4 - 2*z^5 - 5*z^4 + 2*x^3 + 3*x^2*y + 5*x*y^2 + 2*x^2*z + 3*x*z^2
      f_2 = -4*x^4*y - 2*x^3*y^2 - 2*x^2*y^3 - 4*x*y^4 - 4*y^5 - 3*x^4*z + 5*x*y^3*z + 3*y^4*z - x^3*z^2 - 2*x^2*y*z^2 + 3*x*y^2*z^2 + 3*y^3*z^2 - x^2*z^3 + 2*x*y*z^3 + 5*y^2*z^3 - x*z^4 - y*z^4 - 5*z^5 + y*z^3 - 5*z^4 + x^2*y - 2*x*y^2 - x^2*z
      f_3 = 3*x^5 - 2*x^4*y + x^2*y^3 - 5*x*y^4 - 4*y^5 - 3*x^3*y*z + 5*x^2*y^2*z - 4*x*y^3*z - 4*y^4*z - 5*x^3*z^2 + x^2*y*z^2 - x*y^2*z^2 - 5*y^3*z^2 + x^2*z^3 + 3*x*y*z^3 + x*z^4 - y*z^4 + 3*z^5 - y*z^3 - 3*z^4 - 3*x^3 + 5*x^2*y - 4*x*y^2 - 4*x^2*z + 2*x*z^2
  ok   (1,3,6,5,3,2), G = XYZ, r = 3 gcd z^3: H = [1, 3, 6, 5, 3, 2], closed True, f_i in Ann(M) True, deg gcd K = 3, CI local length 39 (stable pair N = 10, 11; attempt 1) <= 39
   (1,3,6,5,4,1), G = X^2 Y + X Z^2, pure Y^5: attempt 1 used (local lengths of the attempts made: [37]); the triple:
      f_1 = 2*x^5 + 2*x^4*y + 2*x^3*y^2 + 3*x^2*y^3 - 4*x*y^4 + 3*y^5 - x^4*z - 4*x^3*y*z - 4*x^2*y^2*z - 4*x*y^3*z + 4*y^4*z - x^3*z^2 - 2*x^2*y*z^2 + 4*x*y^2*z^2 + y^3*z^2 + x^2*z^3 - 2*x*y*z^3 + 3*y^2*z^3 + 5*x*z^4 - 2*y*z^4 + 5*z^5 + y^3*z + y^2*z^2 + 3*y*z^3 - 5*z^4 + 3*x^2*y - x*y^2 - x^2*z - 4*x*y*z - 3*x*z^2
      f_2 = -5*x^5 - 4*x^4*y + x^3*y^2 + 5*x^2*y^3 - 2*x*y^4 - 2*y^5 + 4*x^4*z + x^3*y*z + 2*x^2*y^2*z + 2*y^4*z - 5*x^3*z^2 + 2*x*y^2*z^2 + 2*y^3*z^2 + 3*x^2*z^3 - x*y*z^3 - y^2*z^3 - 2*x*z^4 - 4*y*z^4 - z^5 - 2*y^2*z^2 - 3*y*z^3 - 5*z^4 + 5*x^3 + 5*x^2*y + 3*x*y^2 + 2*x^2*z + 3*x*y*z - 5*x*z^2
      f_3 = -5*x^5 + 5*x^4*y - 5*x^3*y^2 + 3*x^2*y^3 + y^5 - 3*x^4*z - 2*x^3*y*z + 4*x^2*y^2*z - 2*x*y^3*z - 3*y^4*z + x^3*z^2 - 3*x^2*y*z^2 + 5*x*y^2*z^2 - 2*y^3*z^2 - 2*x^2*z^3 - 4*x*y*z^3 + 2*y^2*z^3 + 3*x*z^4 + 5*y*z^4 + 2*z^5 + y^3*z - 5*y^2*z^2 + 5*y*z^3 + 2*z^4 - 4*x^3 + 4*x^2*y + 3*x*y^2 + 4*x^2*z - x*y*z - 4*x*z^2
  ok   (1,3,6,5,4,1), G = X^2 Y + X Z^2, pure Y^5: H = [1, 3, 6, 5, 4, 1], closed True, f_i in Ann(M) True, deg gcd K = 1, CI local length 37 (stable pair N = 10, 11; attempt 1) <= 37
   (1,3,6,5,4,1), G = X^2 Y + X Z^2, nonpure (3,4): attempt 1 used (local lengths of the attempts made: [36]); the triple:
      f_1 = 5*x^4*y + 4*x^3*y^2 + 3*x^2*y^3 - 2*x^3*y*z - x^2*y^2*z - x*y^3*z - 4*x^3*z^2 + 5*x^2*y*z^2 + 5*x*y^2*z^2 + 5*x^2*z^3 + 3*x*y*z^3 - 3*y^2*z^3 - 5*x*z^4 - 2*y*z^4 + z^5 - y^4 - 4*y^3*z + 2*y^2*z^2 + 10*y*z^3 - 1/5*z^4 - 3*x^3 - 4*x*y^2 + 2*x^2*z + 2*x*y*z
      f_2 = 4*x^5 - 3*x^4*y - 2*x^2*y^3 + 2*x*y^4 - 5*x^4*z - 5*x^3*y*z + x^2*y^2*z + 5*x*y^3*z - 2*y^4*z - 4*x^3*z^2 + 3*x^2*y*z^2 - 4*x*y^2*z^2 + y^3*z^2 + 4*x^2*z^3 + x*y*z^3 - 4*y^2*z^3 - 2*x*z^4 + 4*y*z^4 - 5*z^5 + y^4 + y^3*z - 2*y^2*z^2 - 10*y*z^3 + 1/5*z^4 + 3*x^3 + x^2*y + 4*x*y^2 - x*y*z - x*z^2
      f_3 = -x^5 + 2*x^4*y - 3*x^3*y^2 + 2*x^2*y^3 + 5*x*y^4 - y^5 - 2*x^3*y*z + 2*x*y^3*z - 3*y^4*z - 5*x^3*z^2 - 2*x^2*y*z^2 + 4*y^3*z^2 - 4*x^2*z^3 - x*y*z^3 - 4*x*z^4 + 3*y*z^4 + 2*z^5 + y^4 - y^2*z^2 - 10*y*z^3 + 1/10*z^4 + x^3 - 2*x^2*z - 3*x*y*z
  ok   (1,3,6,5,4,1), G = X^2 Y + X Z^2, nonpure (3,4): H = [1, 3, 6, 5, 4, 1], closed True, f_i in Ann(M) True, deg gcd K = 0, CI local length 36 (stable pair N = 9, 10; attempt 1) <= 36
   (1,3,6,5,4,1), G = X^2 Y + X Z^2, nonpure (2,5): attempt 1 used (local lengths of the attempts made: [38]); the triple:
      f_1 = 4*x^5 - 2*x^4*y + x^3*y^2 - 5*x^2*y^3 - x*y^4 - 4*x^4*z - 3*x^3*y*z + 5*x^2*y^2*z - 5*x*y^3*z + y^4*z + 3*x^3*z^2 + 4*x^2*y*z^2 - 5*x*y^2*z^2 + 5*y^3*z^2 + 5*x*y*z^3 + 5*y^2*z^3 + 3*x*z^4 - 4*y*z^4 + 3*z^5 + 4*y^3*z + 4*y^2*z^2 - 5*y*z^3 - 2*x^3 + x^2*y - x*y^2 + 2*x^2*z + 5*x*y*z - x*z^2
      f_2 = 4*x^5 + x^4*y - 3*x^3*y^2 - x^2*y^3 - y^5 - 3*x^4*z + 4*x^3*y*z + 4*x^2*y^2*z - x*y^3*z + 5*y^4*z + 2*x^3*z^2 + x^2*y*z^2 + y^3*z^2 + 2*x^2*z^3 + 2*y^2*z^3 - 5*x*z^4 + 4*y*z^4 + 3*z^5 - y^2*z^2 - 2*y*z^3 - 2*x^3 + 5*x^2*y + 4*x*y^2 - 3*x^2*z + 5*x*y*z - 5*x*z^2
      f_3 = 4*x^5 + x^4*y + 4*x^3*y^2 - 3*x^2*y^3 + 2*x*y^4 + 4*y^5 + 5*x^4*z - 4*x^3*y*z + 2*x^2*y^2*z + 4*x*y^3*z - y^4*z + x^2*y*z^2 - 4*y^3*z^2 - 5*x^2*z^3 + 3*y^2*z^3 + 5*x*z^4 + 5*y*z^4 + 3*z^5 - 4*y^3*z + 4*y^2*z^2 - 4*y*z^3 + 5*x^3 - 2*x^2*y - 4*x*y^2 - x^2*z - 4*x*y*z + 2*x*z^2
  ok   (1,3,6,5,4,1), G = X^2 Y + X Z^2, nonpure (2,5): H = [1, 3, 6, 5, 4, 1], closed True, f_i in Ann(M) True, deg gcd K = 2, CI local length 38 (stable pair N = 10, 11; attempt 1) <= 38
   (1,3,6,5,4,1), G = X^2 Y + X Z^2, X-free Y^4 Z: attempt 1 used (local lengths of the attempts made: [38]); the triple:
      f_1 = 4*x^5 - 2*x^4*y + x^3*y^2 + 2*x^2*y^3 + 2*x*y^4 + 4*y^5 - 5*x^4*z - 4*x^3*y*z - 4*x^2*y^2*z - 4*x*y^3*z + 3*y^4*z - 3*x^3*z^2 - 2*x^2*y*z^2 - x*y^2*z^2 - 2*y^3*z^2 + 5*x^2*z^3 - 5*x*y*z^3 + 4*y^2*z^3 - 4*x*z^4 + 4*y*z^4 + 5*z^5 - y^2*z^2 - 3*y*z^3 + 5*z^4 - 4*x^3 + 2*x^2*y - 4*x*y^2 - 3*x^2*z - 2*x*z^2
      f_2 = -2*x^5 - 3*x^4*y + 2*x^3*y^2 + 5*x^2*y^3 + 5*x*y^4 - y^5 + 3*x^4*z + 4*x^3*y*z + 4*x^2*y^2*z - 3*x*y^3*z - 3*y^4*z + 2*x^2*y*z^2 + 5*x*y^2*z^2 + 5*y^3*z^2 - 5*x^2*z^3 + x*y*z^3 - 5*y^2*z^3 + 5*x*z^4 + y*z^4 - 5*z^5 + 2*y^2*z^2 - y*z^3 - 4*z^4 - 2*x^3 + 4*x^2*y + 3*x*y^2 + x^2*z + 4*x*y*z - 4*x*z^2
      f_3 = 2*x^5 + x^4*y + 5*x^2*y^3 + 3*x*y^4 + 2*y^5 - 2*x^4*z + x^2*y^2*z - 5*x*y^3*z - 3*y^4*z + 2*x^3*z^2 - 4*x^2*y*z^2 - 2*x*y^2*z^2 - 5*x^2*z^3 + 5*x*y*z^3 + 3*x*z^4 + 5*y*z^4 + 5*z^5 + 2*y^2*z^2 + 5*y*z^3 + 2*z^4 + x^3 - 3*x^2*y + 4*x*y^2 + 5*x*y*z + 3*x*z^2
  ok   (1,3,6,5,4,1), G = X^2 Y + X Z^2, X-free Y^4 Z: H = [1, 3, 6, 5, 4, 1], closed True, f_i in Ann(M) True, deg gcd K = 2, CI local length 38 (stable pair N = 10, 11; attempt 1) <= 38
   (1,3,6,5,3,2), G = X^2 Y + X Z^2, r = 3 generic: attempt 1 used (local lengths of the attempts made: [36]); the triple:
      f_1 = -4*x^5 + 5*x^4*y - 4*x^3*y^2 - 3*x^2*y^3 + x*y^4 - 4*x^4*z + 5*x^3*y*z - 4*x^2*y^2*z - 2*y^4*z + 3*x^3*z^2 - 2*x^2*y*z^2 + x*y^2*z^2 - 2*y^3*z^2 + 2*x^2*z^3 - 2*x*y*z^3 - 4*y^2*z^3 + 4*x*z^4 + 5*y*z^4 + 5*z^5 - 4*y^4 - 4*y^3*z + 40*y*z^3 + 2/5*z^4 + x^3 - 4*x^2*y - x*y^2 + x*y*z + 4*x*z^2
      f_2 = 4*x^5 + 4*x^3*y^2 - 5*x^2*y^3 - 2*x*y^4 - y^5 - 3*x^4*z + 4*x^3*y*z - 2*x^2*y^2*z - 5*x*y^3*z + 2*y^4*z - 3*x^3*z^2 + 2*x^2*y*z^2 - 5*x*y^2*z^2 + 2*y^3*z^2 + 5*x^2*z^3 + 5*x*y*z^3 + 2*y^2*z^3 - 4*y*z^4 + 2*z^5 + y^4 - 2*y^3*z - 10*y*z^3 + 1/5*z^4 - x^3 - x^2*y + x*y^2 - 5*x^2*z + 4*x*y*z + x*z^2
      f_3 = -5*x^5 + 4*x^4*y + 5*x^3*y^2 - 5*x*y^4 - 5*y^5 - 4*x^4*z - 5*x^3*y*z - 4*x^2*y^2*z + 3*x*y^3*z - 2*y^4*z + 4*x^3*z^2 - 4*x^2*y*z^2 - 2*x*y^2*z^2 + y^3*z^2 + 2*x^2*z^3 + 4*x*y*z^3 + 2*x*z^4 - 4*y*z^4 - 5*z^5 + 4*y^4 + 3*y^3*z - 40*y*z^3 - 3/10*z^4 + 3*x^3 - 4*x^2*y + 5*x^2*z - 3*x*y*z + 4*x*z^2
  ok   (1,3,6,5,3,2), G = X^2 Y + X Z^2, r = 3 generic: H = [1, 3, 6, 5, 3, 2], closed True, f_i in Ann(M) True, deg gcd K = 0, CI local length 36 (stable pair N = 9, 10; attempt 1) <= 36
   (1,3,6,5,3,2), G = X^2 Y + X Z^2, r = 3 with a fifth power: attempt 1 used (local lengths of the attempts made: [37]); the triple:
      f_1 = 4*x^5 + 2*x^4*y - 2*x^3*y^2 - 3*x^2*y^3 + x*y^4 - y^5 + 4*x^4*z - 2*x^3*y*z - 2*x^2*y^2*z + 5*x*y^3*z + 4*y^4*z - 2*x^3*z^2 - 4*x^2*y*z^2 + 2*x*y^2*z^2 + 2*y^3*z^2 + 4*x^2*z^3 - 5*x*y*z^3 - 2*y^2*z^3 + 5*x*z^4 - 5*y*z^4 - 2*z^5 - 2*y^3*z - 4*x^3 - 2*x^2*y + 2*x^2*z - x*y*z + 2*x*z^2
      f_2 = -3*x^5 - 4*x^3*y^2 - 2*x^2*y^3 + 5*x*y^4 + 2*y^5 + 3*x^4*z - 2*y^4*z - x^3*z^2 + 5*x^2*y*z^2 - 3*x*y^2*z^2 + 3*x^2*z^3 + x*y*z^3 - 5*y^2*z^3 - 3*x*z^4 - 3*y*z^4 + 5*z^5 + y^3*z - y^2*z^2 + 1/10*z^4 - 5*x^3 + 3*x^2*y - 3*x*y^2 + x^2*z - 3*x*y*z - 3*x*z^2
      f_3 = -4*x^5 - 4*x^4*y - 3*x^3*y^2 - x^2*y^3 + 5*x*y^4 + 3*y^5 + 3*x^4*z - x^3*y*z - 5*x^2*y^2*z + 2*x*y^3*z - 3*y^4*z + 2*x^2*y*z^2 - 2*x*y^2*z^2 - y^3*z^2 - 2*x^2*z^3 + 2*x*y*z^3 + 5*y^2*z^3 - 3*x*z^4 + 5*z^5 - 4*y^3*z + 2*y^2*z^2 - 1/5*z^4 + 3*x^3 + 3*x^2*y - x*y^2 - x^2*z - 3*x*y*z - 3*x*z^2
  ok   (1,3,6,5,3,2), G = X^2 Y + X Z^2, r = 3 with a fifth power: H = [1, 3, 6, 5, 3, 2], closed True, f_i in Ann(M) True, deg gcd K = 1, CI local length 37 (stable pair N = 10, 11; attempt 1) <= 37
   (1,3,6,5,3,2), G = X^2 Y + X Z^2, r = 2 secant: attempt 1 used (local lengths of the attempts made: [38]); the triple:
      f_1 = 2*x^5 + 4*x^4*y + 3*x^3*y^2 - 4*x^2*y^3 + x*y^4 + 5*y^5 + 5*x^3*y*z + 3*x*y^3*z + 4*y^4*z + 4*x^3*z^2 + 2*x^2*y*z^2 - 4*x*y^2*z^2 + 2*y^3*z^2 + 2*x^2*z^3 + 4*x*y*z^3 - 2*y^2*z^3 - x*z^4 + y*z^4 - z^5 - 4*y^3*z + 4*y^2*z^2 - 5*y*z^3 - x^3 - 2*x*y^2 + x^2*z
      f_2 = -2*x^5 - 4*x^4*y - 3*x^3*y^2 - 4*x^2*y^3 + 5*y^5 - x^4*z - x^3*y*z - 2*x^2*y^2*z + 4*x*y^3*z + 4*y^4*z - 3*x^3*z^2 - 4*x^2*y*z^2 - 4*x*y^2*z^2 - y^3*z^2 - 3*x^2*z^3 - 3*x*y*z^3 + 4*y^2*z^3 + 3*x*z^4 - y*z^4 + 5*z^5 + 3*y^2*z^2 - y*z^3 - 5*x^3 - 4*x^2*y + x*y*z + 4*x*z^2
      f_3 = 5*x^5 - 4*x^4*y - 2*x^3*y^2 - 2*x^2*y^3 + x*y^4 + y^5 - 3*x^4*z + 4*x^3*y*z + 3*x^2*y^2*z - 3*x*y^3*z + 4*y^4*z - 2*x^2*y*z^2 + 2*x*y^2*z^2 - 3*y^3*z^2 - x^2*z^3 - y^2*z^3 + x*z^4 - 4*z^5 + 2*y^3*z + y^2*z^2 - 2*y*z^3 + x^3 - 4*x^2*y - x*y^2 + x^2*z + x*y*z + 4*x*z^2
  ok   (1,3,6,5,3,2), G = X^2 Y + X Z^2, r = 2 secant: H = [1, 3, 6, 5, 3, 2], closed True, f_i in Ann(M) True, deg gcd K = 2, CI local length 38 (stable pair N = 10, 11; attempt 1) <= 38
   (1,3,6,5,3,2), G = X^2 Y + X Z^2, r = 2 tangent: attempt 1 used (local lengths of the attempts made: [38]); the triple:
      f_1 = -4*x^5 + 3*x^4*y - x^3*y^2 + 3*x^2*y^3 - 2*x*y^4 + 5*y^5 + 3*x^4*z - 3*x^3*y*z + 5*x^2*y^2*z - 5*x*y^3*z - 5*x^3*z^2 + 2*x^2*y*z^2 + 2*x*y^2*z^2 + 3*x^2*z^3 + 2*x*y*z^3 - 3*y^2*z^3 - 4*x*z^4 - 2*y*z^4 + 5*z^5 + 4*y^2*z^2 + 4*y*z^3 + z^4 - 4*x^3 + 5*x^2*y + 5*x*y^2 + 5*x^2*z + 4*x*y*z - 5*x*z^2
      f_2 = 5*x^5 + 3*x^2*y^3 + 5*x*y^4 - 3*y^5 + 4*x^4*z + 3*x^3*y*z + 5*x^2*y^2*z - 2*x*y^3*z - y^4*z - 3*x^3*z^2 - x^2*y*z^2 - 3*x*y^2*z^2 + y^3*z^2 - 2*x^2*z^3 - 3*x*y*z^3 + 2*y^2*z^3 + 5*x*z^4 + 5*y*z^4 + 2*z^5 + 2*y^2*z^2 - 4*y*z^3 + z^4 + 2*x^3 + 3*x^2*y - x*y^2 - 5*x^2*z - 3*x*y*z - 3*x*z^2
      f_3 = 4*x^5 - 2*x^4*y - x^3*y^2 + 4*x^2*y^3 - 5*x*y^4 - 3*y^5 - 5*x^4*z - 2*x^3*y*z + 4*x^2*y^2*z - 3*x*y^3*z - 4*y^4*z + x^3*z^2 + 4*x^2*y*z^2 + 5*x*y^2*z^2 + 2*y^3*z^2 + 5*x^2*z^3 + 2*x*y*z^3 - x*z^4 + 5*y*z^4 - 3*z^5 - 3*y^2*z^2 - 3*y*z^3 - 2*x^3 - 3*x^2*y + 3*x*y^2 + x^2*z - 5*x*y*z + 3*x*z^2
  ok   (1,3,6,5,3,2), G = X^2 Y + X Z^2, r = 2 tangent: H = [1, 3, 6, 5, 3, 2], closed True, f_i in Ann(M) True, deg gcd K = 2, CI local length 38 (stable pair N = 10, 11; attempt 1) <= 38
   (1,3,6,5,3,2), G = X^2 Y + X Z^2, r = 3 gcd z^3: attempt 1 used (local lengths of the attempts made: [39]); the triple:
      f_1 = 2*x^4*y - 4*x^3*y^2 - x^2*y^3 - 4*x*y^4 + 2*y^5 + 2*x^4*z + 4*x^3*y*z - 4*x^2*y^2*z - 4*x*y^3*z - 2*y^4*z - 3*x^3*z^2 - 3*x^2*y*z^2 + 5*x*y^2*z^2 - 2*y^3*z^2 - 4*x^2*z^3 - x*y*z^3 + 5*y^2*z^3 + x*z^4 + 2*y*z^4 + z^5 + 5*y*z^3 + 5*z^4 - 4*x^3 + 3*x^2*y + 2*x*y^2 - 2*x^2*z - 2*x*y*z - 3*x*z^2
      f_2 = 2*x^5 - 5*x^4*y - 5*x^3*y^2 + 2*x*y^4 + 2*y^5 - 3*x^4*z + 5*x^3*y*z + 4*x^2*y^2*z + 5*x*y^3*z - 4*y^4*z - x^3*z^2 + 4*x^2*y*z^2 + 3*x*y^2*z^2 + 3*y^3*z^2 - 4*x^2*z^3 + x*y*z^3 + 3*y^2*z^3 - 2*x*z^4 + 4*y*z^4 + 5*z^5 - 2*z^4 + 2*x^3 - 4*x^2*y + 2*x*y^2 + x^2*z - x*y*z + 4*x*z^2
      f_3 = -2*x^3*y^2 + 3*x^2*y^3 + 5*x*y^4 - 3*y^5 + 4*x^4*z - 2*x^3*y*z + 5*x^2*y^2*z + 3*x*y^3*z + 3*y^4*z + 2*x^3*z^2 - 4*x^2*y*z^2 + x*y^2*z^2 - y^3*z^2 + 3*x*y*z^3 - x*z^4 - y*z^4 - 2*y*z^3 - 3*z^4 + 2*x^3 - 2*x^2*y + x*y^2 + x*y*z + 2*x*z^2
  ok   (1,3,6,5,3,2), G = X^2 Y + X Z^2, r = 3 gcd z^3: H = [1, 3, 6, 5, 3, 2], closed True, f_i in Ann(M) True, deg gcd K = 3, CI local length 39 (stable pair N = 10, 11; attempt 1) <= 39
total time 399.0 s
ALL CHECKS PASSED
