  [ok] (F8 orbit) R = X^3*Z + Z^4: QT(u=-3*x - 2*y + 2*z), triple in Ann, local length ds = 40, stabilization = 40, in [21,41]
      triple: [-x^7 - 3*x^6*y - x^5*y^2 - 3*x^4*y^3 + 2*x*y^6 + 2*y^7 - x^6*z + x^5*y*z - 3*x^4*y^2*z - 3*x^3*y^3*z + x^2*y^4*z - 2*x*y^5*z + 3*y^6*z + x^5*z^2 + x^4*y*z^2 + x^3*y^2*z^2 + 3*x^2*y^3*z^2 + 3*x*y^4*z^2 + y^5*z^2 - 2*x^4*z^3 - x^3*y*z^3 - 2*x^2*y^2*z^3 + 3*x*y^3*z^3 - 2*y^4*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 + 2*y^2*z^5 - 2*x*z^6 + y*z^6 + 2*z^7 + 3/20*x^6 - 2*x^5*y - x^4*y^2 + 2*x^3*y^3 - 3*x^2*y^4 + 3*x*y^5 + y^6 + x^5*z - x^4*y*z - 2*x^3*y^2*z - x^2*y^3*z + 3*x*y^4*z + 3*y^5*z + 3*x^4*z^2 + 2*x^3*y*z^2 + 2*x^2*y^2*z^2 - 2*x*y^3*z^2 + y^4*z^2 + x^2*y*z^3 - x*y^2*z^3 + 2*y^3*z^3 - 3*x^2*z^4 + x*y*z^4 + y^2*z^4 + 3*x*z^5 - 3*y*z^5 + z^6 - 13/360*x^5 - 13/108*x^4*y + 3*x^3*y^2 + 2*x^2*y^3 - 3*x*y^4 - y^5 + x^4*z - x^2*y^2*z - 3*x*y^3*z - 2*y^4*z + 3*x^3*z^2 - x*y^2*z^2 - 2*y^3*z^2 + 2*x^2*z^3 - x*y*z^3 - 2*x*z^4 + 3*y*z^4 - z^5 - x^2*y^2 - 2*y^4 + 4*x^3*z + 3*x^2*y*z + 3*x*y^2*z + 2*y^3*z - 3*y^2*z^2 + 2*x*z^3 - 3*y*z^3 - z^4 + x*y^2 - 2*y^3 - x*y*z - 3*y^2*z + y*z^2, -3*x^7 - 2*x^6*y + 3*x^3*y^4 - 2*x^2*y^5 + 3*x*y^6 - y^7 + x^5*y*z + 2*x^4*y^2*z + x^3*y^3*z + y^6*z + 3*x^5*z^2 + 2*x^4*y*z^2 + 3*x^3*y^2*z^2 - x^2*y^3*z^2 - y^5*z^2 - x^4*z^3 - 3*x^3*y*z^3 + x^2*y^2*z^3 - 2*x*y^3*z^3 - 2*y^4*z^3 + 3*x^3*z^4 + 2*x^2*y*z^4 - 3*x*y^2*z^4 + 2*y^3*z^4 - 2*x^2*z^5 - x*y*z^5 - y^2*z^5 + 2*x*z^6 - 3*y*z^6 - z^7 + 23/180*x^6 + x^5*y - x^4*y^2 + 3*x^2*y^4 + 2*y^6 + 3*x^5*z - 2*x^4*y*z - 2*x^3*y^2*z - x^2*y^3*z - 2*x*y^4*z - 2*y^5*z - 2*x^3*y*z^2 + 3*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 + 2*x*z^5 - 2*y*z^5 - z^6 + 23/360*x^5 + 23/108*x^4*y + x^3*y^2 - 3*x^2*y^3 - 2*x*y^4 + 2*y^5 + 3*x^3*y*z - 2*x^2*y^2*z - 2*x*y^3*z - 2*y^4*z - 2*x^2*y*z^2 - 2*x*y^2*z^2 + y^3*z^2 - x^2*z^3 - 3*x*y*z^3 + y^2*z^3 + 3*x*z^4 - 2*y*z^4 - z^5 - x^2*y^2 - 3*x*y^3 - 2*y^4 + 12*x^3*z + x^2*y*z + 2*x*y^2*z + 3*y^3*z + 3*x^2*z^2 - x*y*z^2 - 3*y^2*z^2 + 2*x*z^3 - 2*y*z^3 - 3*z^4 - x*y^2 + 3*x*y*z - y^2*z - 3*y*z^2, x^7 - x^6*y + 2*x^5*y^2 + 2*x^4*y^3 - 2*x^3*y^4 + 3*x^2*y^5 - x*y^6 - 3*y^7 + 2*x^6*z - 3*x^5*y*z - 3*x^4*y^2*z - 3*x^3*y^3*z + 2*x^2*y^4*z + x*y^5*z + 3*y^6*z + 2*x^5*z^2 - 2*x^4*y*z^2 + x^2*y^3*z^2 + x*y^4*z^2 + 2*y^5*z^2 - 2*x^4*z^3 - 3*x^3*y*z^3 - 3*x^2*y^2*z^3 + 2*x*y^3*z^3 + 3*y^4*z^3 - 2*x^2*y*z^4 - 3*x*y^2*z^4 - 3*y^3*z^4 + 3*x^2*z^5 - 2*x*y*z^5 + 3*x*z^6 - 3*z^7 + 7/180*x^6 - x^5*y - 3*x^4*y^2 - 3*x^3*y^3 + 2*x^2*y^4 - 2*x*y^5 + x^5*z + 3*x^3*y^2*z + x^2*y^3*z + x*y^4*z + 3*y^5*z - x^4*z^2 - x^3*y*z^2 + x*y^3*z^2 + y^4*z^2 + 2*x^3*z^3 - x^2*y*z^3 + 2*x*y^2*z^3 + 3*y^3*z^3 + 3*x^2*z^4 + 2*x*y*z^4 - y^2*z^4 - 3*y*z^5 + z^6 - 1/40*x^5 - 1/12*x^4*y - x^3*y^2 - x^2*y^3 + 2*x*y^4 - 2*y^5 - x^4*z - 2*x^2*y^2*z - x*y^3*z - x^3*z^2 - 3*x^2*y*z^2 + 3*x*y^2*z^2 - 3*y^3*z^2 + 3*x^2*z^3 + 2*x*y*z^3 - 3*y^2*z^3 + x*z^4 + 2*y*z^4 + 3*z^5 + x^2*y^2 - 2*x*y^3 + y^4 + 12*x^3*z - x^2*y*z - 2*y^3*z - x^2*z^2 - 2*y^2*z^2 - 2*x*z^3 + y*z^3 - 3*z^4 + 3*x*y^2 + 3*x*y*z - 2*y^2*z + y*z^2]
  [ok] (F8 orbit) R = Z^4: QT(u=-2*x + 2*y - 2*z), triple in Ann, local length ds = 41, stabilization = 41, in [21,41]
      triple: [-x^6*y + 3*x^4*y^3 + 2*x^3*y^4 - 3*x^2*y^5 + x*y^6 - 2*y^7 + 2*x^6*z - 3*x^5*y*z + x^4*y^2*z - x^3*y^3*z + x^2*y^4*z + 3*x*y^5*z + 2*y^6*z - 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 - 2*y^5*z^2 + x^4*z^3 - 3*x^3*y*z^3 + 2*x^2*y^2*z^3 + 3*x*y^3*z^3 + 3*y^4*z^3 + x^3*z^4 + x^2*y*z^4 - 2*x*y^2*z^4 + 3*y^3*z^4 + x^2*z^5 - 3*x*y*z^5 + 3*y^2*z^5 + 3*x*z^6 + y*z^6 + 2*z^7 + 1/36*x^6 + 2*x^5*y + 3*x^4*y^2 + 2*x^2*y^4 + x*y^5 - 3*y^6 + x^5*z - 2*x^4*y*z - 3*x^3*y^2*z + 2*x^2*y^3*z + 2*x*y^4*z - 2*x^4*z^2 + 2*x^2*y^2*z^2 + x*y^3*z^2 + 3*y^4*z^2 + x^3*z^3 + 3*x^2*y*z^3 + 2*y^3*z^3 + 3*x*y*z^4 - 3*y^2*z^4 - y*z^5 - z^6 + 7/180*x^5 - 7/36*x^4*y + x^3*y^2 - 2*x^2*y^3 - 3*x*y^4 + y^5 - x^4*z - 3*x^3*y*z + 3*x^2*y^2*z - 2*x*y^3*z - 3*y^4*z + 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 - 3*x*y*z^3 - 3*y^2*z^3 + 3*x*z^4 + 2*y*z^4 - 3*z^5 + 2*x^2*y^2 + x*y^3 + y^4 + 3*x^3*z - 2*x^2*y*z + 3*y^3*z - 2*x^2*z^2 + 2*x*z^3 + 3*y*z^3 + 2*x*y^2 - y^3 + 2*x*y*z - y^2*z + 3*y*z^2, 2*x^7 + 3*x^6*y + 2*x^5*y^2 - x^4*y^3 - 2*x^3*y^4 + 2*x*y^6 + 3*y^7 - x^6*z + 2*x^4*y^2*z - 2*x^2*y^4*z - 2*x*y^5*z - 3*y^6*z + 2*x^5*z^2 - 3*x^3*y^2*z^2 - 2*x^2*y^3*z^2 + y^5*z^2 + 2*x^4*z^3 - 2*x^3*y*z^3 - x^2*y^2*z^3 - 3*x*y^3*z^3 - y^4*z^3 - x^3*z^4 - x*y^2*z^4 - 3*y^3*z^4 + 3*x^2*z^5 + 3*y^2*z^5 - 2*x*z^6 + 2*y*z^6 + 3*z^7 + 7/120*x^6 + 3*x^5*y + 3*x^4*y^2 + 2*x^3*y^3 - 2*x^2*y^4 - 2*x*y^5 + 3*y^6 - 2*x^5*z - 2*x^4*y*z + 3*x^3*y^2*z - x^2*y^3*z + 3*x*y^4*z - y^5*z + x^3*y*z^2 + x^2*y^2*z^2 + x^3*z^3 - x^2*y*z^3 + 2*x*y^2*z^3 - 3*y^3*z^3 - x^2*z^4 + 3*x*y*z^4 - y^2*z^4 - y*z^5 + 3*z^6 - 7/180*x^5 + 7/36*x^4*y - x^3*y^2 + x^2*y^3 - 3*x*y^4 + y^5 - 2*x^4*z + x*y^3*z + 3*y^4*z + 2*x^3*z^2 - 2*x*y^2*z^2 + y^3*z^2 - x^2*z^3 + 2*x*y*z^3 - 2*y^2*z^3 - 2*x*z^4 - 3*y*z^4 + 2*z^5 - x^2*y^2 - 2*x*y^3 - y^4 + 3*x^2*y*z - x*y^2*z - 2*y^3*z + 2*x^2*z^2 + 2*x*y*z^2 + 2*x*z^3 - 2*y*z^3 - 2*x*y^2 + 3*y^3 - 2*x*y*z - 3*y^2*z - 3*y*z^2, 2*x^6*y + x^5*y^2 + 3*x^4*y^3 - 2*x^3*y^4 - x^2*y^5 + 3*x*y^6 - 2*y^7 - 2*x^6*z + 3*x^5*y*z - 2*x^4*y^2*z - 2*x^2*y^4*z - 3*x^5*z^2 - x^4*y*z^2 - x^3*y^2*z^2 + 3*x^2*y^3*z^2 + x*y^4*z^2 + y^5*z^2 - 3*x^4*z^3 - 3*x^3*y*z^3 - x^2*y^2*z^3 + 3*x*y^3*z^3 + 2*y^4*z^3 + x^3*z^4 - x^2*y*z^4 - 3*x*y^2*z^4 + y^3*z^4 + 2*x^2*z^5 + x*y*z^5 + y^2*z^5 - 3*x*z^6 - y*z^6 + z^7 - 7/60*x^6 - x^4*y^2 + x^3*y^3 - 2*x^2*y^4 - 2*x*y^5 + 3*y^6 + 2*x^5*z - 2*x^4*y*z + x^3*y^2*z + 3*x^2*y^3*z - 3*y^5*z + x^4*z^2 - 3*x^3*y*z^2 - x^2*y^2*z^2 - 2*x*y^3*z^2 - 2*y^4*z^2 + 3*x^3*z^3 + 2*x^2*y*z^3 + x*y^2*z^3 + x^2*z^4 - 3*x*y*z^4 - 3*y^2*z^4 + y*z^5 - z^6 + 1/36*x^5 - 5/36*x^4*y - 3*x^3*y^2 + 3*x^2*y^3 - 2*x*y^4 - 2*y^5 + x^4*z + 3*x^3*y*z - 3*x^2*y^2*z - x*y^3*z - 2*y^4*z + 3*x^2*y*z^2 + 3*x*y^2*z^2 - 2*y^3*z^2 + x*y*z^3 + 3*x*z^4 - 2*y*z^4 + z^5 + x^2*y^2 - 3*y^4 + 3*x^3*z - 3*x^2*y*z - 2*x*y^2*z + y^3*z - x*y*z^2 + 2*y^2*z^2 - 2*x*z^3 - 3*y*z^3 + x*y^2 + y^2*z + 2*y*z^2]
  [ok] (F8 orbit) R = X^3*Z + X*Z^3: QT(u=2*x + 2*z), triple in Ann, local length ds = 40, stabilization = 40, in [21,41]
      triple: [-x^7 - x^6*y - 2*x^5*y^2 - x^4*y^3 - 3*x^3*y^4 - 2*x^2*y^5 - 3*x*y^6 - 3*y^7 - x^6*z - 2*x^5*y*z - 3*x^4*y^2*z - 3*x^3*y^3*z + 2*x^2*y^4*z + 3*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 + 3*y^5*z^2 + x^4*z^3 + x^3*y*z^3 + 3*x^2*y^2*z^3 + 2*x*y^3*z^3 + y^4*z^3 - 3*x^3*z^4 + 3*x^2*y*z^4 + x*y^2*z^4 - 3*y^3*z^4 + 2*x^2*z^5 + x*y*z^5 + 2*y^2*z^5 - 2*x*z^6 - 3*y*z^6 + 3*z^7 + 133/720*x^6 - 3*x^5*y + 3*x^4*y^2 - 2*x^3*y^3 - x^2*y^4 + 3*x*y^5 - 2*x^5*z - 2*x^4*y*z + x^2*y^3*z + 2*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 - 2*y^4*z^2 - x^3*z^3 + 2*x^2*y*z^3 + 3*x*y^2*z^3 - 3*y^3*z^3 + x^2*z^4 + 2*x*y*z^4 - 3*y^2*z^4 - x*z^5 + y*z^5 + 3*z^6 + 1/72*x^5 + 2*x^3*y^2 - 3*x^2*y^3 - 2*x*y^4 + 2*y^5 + 2*x^4*z - 3*x^3*y*z + 2*x^2*y^2*z + x*y^3*z + y^4*z + 3*x^2*y*z^2 + x*y^2*z^2 + y^3*z^2 + 3*x^2*z^3 + 2*y^2*z^3 + x*z^4 + 3*y*z^4 + 2*z^5 + 3*x^2*y^2 - x*y^3 + 3*y^4 + 2*x^3*z - x^2*y*z + 3*x*y^2*z + 3*y^3*z + 2*x^2*z^2 + 2*x*y*z^2 - 3*y^2*z^2 - 2*x*z^3 + 3*y*z^3 - 2*z^4 + 2*y^3 - x*y*z + 2*y^2*z + 3*y*z^2, 3*x^7 + x^5*y^2 - 2*x^3*y^4 + 2*x^2*y^5 - 3*x*y^6 + y^7 - x^6*z - x^5*y*z + x^4*y^2*z + 3*x^3*y^3*z + x^2*y^4*z - 2*x*y^5*z + y^6*z + 2*x^5*z^2 - x^4*y*z^2 + x^2*y^3*z^2 - x*y^4*z^2 + x^4*z^3 + x^3*y*z^3 - 3*x^2*y^2*z^3 - 2*y^4*z^3 - x^3*z^4 + 2*x^2*y*z^4 + 2*x*y^2*z^4 - 2*x^2*z^5 - x*y*z^5 - y^2*z^5 + 2*x*z^6 + 2*y*z^6 - 3*z^7 - 47/240*x^6 - 2*x^5*y + 3*x^4*y^2 + x^3*y^3 + 2*x^2*y^4 - 3*x*y^5 - 2*y^6 + 2*x^5*z + 2*x^4*y*z - x^3*y^2*z - 3*x^2*y^3*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 - y^4*z^2 + 2*x^3*z^3 - 2*x^2*y*z^3 + 2*x*y^2*z^3 + 2*y^3*z^3 - 3*x^2*z^4 - x*y*z^4 - 2*y^2*z^4 - 3*x*z^5 + 1/72*x^5 - x^3*y^2 - 3*x^2*y^3 + x*y^4 + x^4*z - 2*x^3*y*z - 2*x*y^3*z - y^4*z + x^3*z^2 - x^2*y*z^2 + x*y^2*z^2 + 2*y^3*z^2 + 3*x^2*z^3 + x*y*z^3 + 2*y^2*z^3 - 2*x*z^4 - 2*z^5 + 2*x^2*y^2 + 3*x*y^3 + y^4 - 2*x^3*z + 3*x*y^2*z - 3*y^3*z + 3*x^2*z^2 + x*y*z^2 - 3*y^2*z^2 + 2*x*z^3 + 2*z^4 + x*y^2 + 2*y^3 + x*y*z - 3*y^2*z - 3*y*z^2, -x^7 - 2*x^6*y - 3*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 - x^3*y^3*z + 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 + x^2*y^3*z^2 - x*y^4*z^2 + y^5*z^2 - 3*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 - 3*x^2*y*z^4 - 3*x*y^2*z^4 - 3*y^3*z^4 - 3*x^2*z^5 + 3*x*y*z^5 + 3*x*z^6 + y*z^6 + 3*z^7 - 23/45*x^6 + x^5*y + x^4*y^2 - x^3*y^3 + 3*x^2*y^4 + 3*x*y^5 - 2*y^6 + 2*x^5*z + 3*x^3*y^2*z + 3*x^2*y^3*z - x*y^4*z - 2*y^5*z + 3*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 + x^3*z^3 - x^2*y*z^3 + x*y^2*z^3 - y^3*z^3 - x^2*z^4 - 3*x*y*z^4 + 3*y^2*z^4 - 3*x*z^5 + 3*y*z^5 + 3*z^6 + 1/36*x^5 - 3*x^3*y^2 - 2*x^2*y^3 - 3*x*y^4 - y^5 + 3*x^4*z - x^2*y^2*z + x*y^3*z - 3*y^4*z + x^3*z^2 - 3*x^2*y*z^2 - 2*x*y^2*z^2 - 3*y^3*z^2 - x^2*z^3 - x*y*z^3 - 3*y^2*z^3 + x*z^4 + 2*z^5 - 3*x^2*y^2 - x*y^3 - 3*y^4 + 2*x^3*z + x*y^2*z - 2*y^3*z + 2*x^2*z^2 + 3*y^2*z^2 - 2*x*z^3 + 3*y*z^3 + z^4 + 3*y^3 - 2*x*y*z - 3*y^2*z - y*z^2]
  [ok] (F8 orbit) R = X*Z^3: QT(u=2*x + y - 2*z), triple in Ann, local length ds = 40, stabilization = 40, in [21,41]
      triple: [-x^7 + 3*x^6*y - x^4*y^3 - 3*x^2*y^5 - 2*y^7 + 2*x^6*z - 2*x^5*y*z + 3*x^4*y^2*z + 2*x^3*y^3*z - 3*x^2*y^4*z - 2*x*y^5*z - 2*y^6*z - 2*x^5*z^2 - x^3*y^2*z^2 - 3*x^2*y^3*z^2 - 2*x*y^4*z^2 - 2*y^5*z^2 + x^3*y*z^3 - 3*x^2*y^2*z^3 + 3*y^4*z^3 - 3*x^3*z^4 + 2*x^2*y*z^4 - x*y^2*z^4 - y^3*z^4 + x^2*z^5 - 3*x*y*z^5 - y^2*z^5 - z^7 - 73/720*x^6 + 2*x^5*y + 2*x^4*y^2 + x^2*y^4 - 2*x*y^5 + 2*y^6 - 2*x^5*z + x^4*y*z + 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 + x^3*y*z^2 - 3*x^2*y^2*z^2 - 2*x*y^3*z^2 + 2*y^4*z^2 + 2*x^3*z^3 + 2*x^2*y*z^3 - 3*x*y^2*z^3 - 2*y^3*z^3 + 2*x^2*z^4 - 3*x*y*z^4 - 2*x*z^5 + 2*z^6 + 23/360*x^5 + 23/144*x^4*y - x^3*y^2 + 3*x^2*y^3 - x*y^4 - 2*y^5 - x^4*z - 2*x^3*y*z + 3*x^2*y^2*z - x*y^3*z + y^4*z - 2*x^2*y*z^2 - x*y^2*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 - y*z^4 + 2*z^5 + x^2*y^2 + 3*x*y^3 + y^4 + x^3*z - 3*x*y^2*z - 3*y^3*z - x^2*z^2 + 3*x*y*z^2 - y^2*z^2 + 3*z^4 + 2*x*y^2 + x*y*z - 3*y^2*z + 3*y*z^2, 2*x^7 + 2*x^6*y - 2*x^5*y^2 - 2*x^4*y^3 - 3*x^2*y^5 + 3*y^7 + 3*x^6*z - 2*x^5*y*z + 2*x^4*y^2*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 - 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 - 2*x*y^3*z^3 - y^4*z^3 - x^3*z^4 + x^2*y*z^4 + x*y^2*z^4 - y^3*z^4 - 3*x*y*z^5 - 2*y^2*z^5 + x*z^6 - 13/240*x^6 - 3*x^5*y - 2*x^4*y^2 + 3*x^3*y^3 - x^2*y^4 - 3*x*y^5 + y^6 - 2*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 - 3*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 - x^3*z^3 - 3*x^2*y*z^3 + 2*x*y^2*z^3 + 2*y^3*z^3 - 2*x^2*z^4 + x*y*z^4 - 2*y^2*z^4 - 3*y*z^5 + 3*z^6 - 13/120*x^5 - 13/48*x^4*y - 2*x^3*y^2 + 3*y^5 + 2*x^4*z - 3*x^3*y*z - 3*x^2*y^2*z + x*y^3*z + 3*y^4*z - x^3*z^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 - 2*x*z^4 + 3*y*z^4 + z^5 + 2*x*y^3 + 3*x^3*z - 2*x^2*y*z - 3*x*y^2*z - 2*y^3*z + x^2*z^2 + x*y*z^2 - 2*y^2*z^2 + 2*y*z^3 + z^4 - 3*x*y^2 + 2*y^3 - 3*x*y*z + y^2*z - 2*y*z^2, -2*x^7 + 3*x^6*y + 2*x^5*y^2 + x^4*y^3 - 3*x^3*y^4 - x^2*y^5 + 3*x*y^6 - y^7 - x^6*z - x^5*y*z + 2*x^4*y^2*z + 2*x^3*y^3*z - x*y^5*z + 2*x^5*z^2 + x^4*y*z^2 - x*y^4*z^2 - y^5*z^2 + x^4*z^3 - 2*x^3*y*z^3 - x*y^3*z^3 - 2*y^4*z^3 + 2*x^3*z^4 + 3*x^2*y*z^4 + x*y^2*z^4 - y^3*z^4 - x^2*z^5 + x*y*z^5 - y^2*z^5 - 2*y*z^6 - 3/40*x^6 - x^5*y - x^4*y^2 + 2*x^3*y^3 + 3*x^2*y^4 + 2*x*y^5 + 3*y^6 - 3*x^5*z - 3*x^4*y*z - x^2*y^3*z - 3*x*y^4*z + 3*y^5*z - 2*x^4*z^2 + x^2*y^2*z^2 - 3*x*y^3*z^2 + 3*y^4*z^2 - x^3*z^3 + 2*x^2*y*z^3 + y^3*z^3 + y^2*z^4 - 2*x*z^5 + y*z^5 - z^6 + 1/36*x^5 + 5/72*x^4*y + 2*x^3*y^2 - 3*x^2*y^3 - x*y^4 + y^5 - 3*x^3*y*z - 3*x^2*y^2*z - 3*x*y^3*z + x^2*y*z^2 + x*y^2*z^2 - 2*y^3*z^2 - 2*x^2*z^3 - x*y*z^3 - 3*y^2*z^3 + 3*x*z^4 + 2*y*z^4 - z^5 - 3*x^2*y^2 - 2*x*y^3 + y^4 - 2*x^3*z - x^2*y*z - x*y^2*z + 3*y^3*z - 3*x^2*z^2 + 2*x*y*z^2 + y^2*z^2 - 2*y*z^3 + x*y^2 - 3*y^2*z - 3*y*z^2]
  [ok] (F8 orbit) R = X^2*Z^2: QT(u=3*x), triple in Ann, local length ds = 40, stabilization = 40, in [21,41]
      triple: [x^7 + 2*x^6*y + 3*x^5*y^2 + 3*x^3*y^4 + 3*x^2*y^5 + 2*x*y^6 + 2*y^7 + 3*x^6*z - x^4*y^2*z + x^3*y^3*z + 3*x^2*y^4*z - 3*x*y^5*z - 2*y^6*z - x^5*z^2 - 3*x^3*y^2*z^2 - 2*x*y^4*z^2 - 2*y^5*z^2 - 2*x^4*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 + 3*x^2*y*z^4 + x*y^2*z^4 - 3*y^3*z^4 - 2*x^2*z^5 + 3*x*y*z^5 + 3*y^2*z^5 - 2*x*z^6 - y*z^6 - 53/360*x^6 + 2*x^5*y - x^4*y^2 + x^2*y^4 - x*y^5 - 2*y^6 + x^5*z - x^4*y*z - 2*x^3*y^2*z - 3*x^2*y^3*z + 2*x*y^4*z + x^4*z^2 - 3*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 - 2*x^2*y*z^3 - 2*y^3*z^3 - 2*x^2*z^4 + x*y*z^4 + 3*y^2*z^4 - 3*x*z^5 - 2*y*z^5 + 2*z^6 + 7/360*x^5 - 3*x^3*y^2 + 2*x^2*y^3 + 3*x*y^4 - 2*y^5 + 2*x^4*z + 2*x^3*y*z + x^2*y^2*z + 3*x*y^3*z - x^3*z^2 - 2*x^2*y*z^2 + 3*x*y^2*z^2 + x^2*z^3 + 2*x*y*z^3 + 3*y^2*z^3 + 2*x*z^4 - z^5 + 2*x^2*y^2 + 2*x*y^3 - 2*x^3*z + 3*x^2*y*z + 2*x*y^2*z + 3*x*y*z^2 + y^2*z^2 + 2*x*z^3 + 2*y*z^3 - z^4 - 2*x*y^2 + 2*y^3 - x*y*z + 3*y^2*z - 3*y*z^2, -3*x^7 - 3*x^6*y + x^5*y^2 + 3*x^3*y^4 - 3*x^2*y^5 + 3*x*y^6 - y^7 - 2*x^6*z + x^5*y*z + x^4*y^2*z - x^3*y^3*z - 2*x^2*y^4*z - x*y^5*z - 3*y^6*z + 2*x^4*y*z^2 - x^3*y^2*z^2 + 3*x^2*y^3*z^2 + 3*y^5*z^2 - 2*x^4*z^3 - 2*x^3*y*z^3 - x^2*y^2*z^3 + 2*x*y^3*z^3 - 3*y^4*z^3 + x^3*z^4 - 2*x*y^2*z^4 - x^2*z^5 - 2*x*y*z^5 + 3*y^2*z^5 + 2*x*z^6 - 2*y*z^6 + 3*z^7 + 83/360*x^6 + 3*x^5*y - x^4*y^2 + x^2*y^4 - 2*x*y^5 + y^6 - 3*x^4*y*z - 2*x^2*y^3*z + x*y^4*z - y^5*z - 3*x^4*z^2 - x^3*y*z^2 + 3*x^2*y^2*z^2 - x*y^3*z^2 + 2*y^4*z^2 + 3*x^2*y*z^3 - 2*x*y^2*z^3 + 2*y^3*z^3 + x*y*z^4 - 2*y^2*z^4 - 2*x*z^5 + y*z^5 - z^6 - 1/72*x^5 + x^3*y^2 + 2*x^2*y^3 + 2*y^5 - 2*x^3*y*z - x^2*y^2*z + 3*x*y^3*z + 3*y^4*z + x^3*z^2 + 2*x^2*y*z^2 - x*y^2*z^2 + y^3*z^2 + 2*x^2*z^3 - x*z^4 + 3*y*z^4 + 3*z^5 + x^2*y^2 - 2*x*y^3 + x^3*z - x^2*y*z + x*y^2*z - 2*y^2*z^2 - 3*x*z^3 - y*z^3 + 3*z^4 + x*y^2 + 2*y^3 + x*y*z + 3*y^2*z, -x^7 + 3*x^6*y + 3*x^5*y^2 - 3*x^4*y^3 - 2*x^3*y^4 + x^2*y^5 - x*y^6 + y^7 + 2*x^6*z + x^5*y*z - x^4*y^2*z - 2*x^3*y^3*z - 3*x^2*y^4*z + 2*x*y^5*z + x^5*z^2 - 3*x^2*y^3*z^2 + 3*x*y^4*z^2 - 3*y^5*z^2 + x^4*z^3 + 3*x^2*y^2*z^3 + 3*y^4*z^3 - 3*x^3*z^4 - x^2*y*z^4 - x*y^2*z^4 + 3*y^3*z^4 - 2*x^2*z^5 + 3*x*y*z^5 - 2*y^2*z^5 - x*z^6 + 3*y*z^6 - 2*z^7 - 1/12*x^6 + 2*x^5*y - 2*x^4*y^2 + x^3*y^3 - x^2*y^4 - 3*x*y^5 - 3*y^6 + x^5*z - 2*x^4*y*z - 3*x^3*y^2*z - 3*x*y^4*z + y^5*z - 3*x^3*y*z^2 - x*y^3*z^2 + y^4*z^2 + x^3*z^3 + 3*x*y^2*z^3 - 3*y^3*z^3 - x^2*z^4 + 2*x*y*z^4 + 2*y^2*z^4 + 2*y*z^5 + 1/72*x^5 + 3*x^3*y^2 - 3*y^5 + 2*x^4*z - 3*x^3*y*z - 2*x^2*y^2*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 - 2*z^5 - 3*x*y^3 - 2*y^4 + 3*x^3*z + 3*y^3*z + 3*x*y*z^2 - 3*y^2*z^2 - 3*x*z^3 + 3*y*z^3 + 2*x*y^2 + 3*y^3 - 3*x*y*z - 3*y^2*z + y*z^2]
certified values: {'X^3*Z + Z^4': 40, 'Z^4': 41, 'X^3*Z + X*Z^3': 40, 'X*Z^3': 40, 'X^2*Z^2': 40}
  [ok] five orbits certified in [21, 41]
  [info] (F8 orbit) R = X^3Z: graded tests give 43 here (not certified; free algebras by Lemma FQ in l21_1365321_r1fq.sage)
  [info] (F8 family) m = 17/5: best S6.2/QT value found 40
  [info] (F8 family) m = -2: best S6.2/QT value found 40
  [ok] (F4 orbit) R = X^2*Y^2 + Y^4: S6.2, triple in Ann, local length ds = 41, stabilization = 41, in [21,41]
      triple: [-2*x^7 + 3*x^6*y - x^5*y^2 + x^4*y^3 - 3*x^3*y^4 - 2*x*y^6 - 3*y^7 - 3*x^6*z + 2*x^5*y*z - 2*x^4*y^2*z + x^3*y^3*z - x^2*y^4*z + 3*x*y^5*z + 2*x^5*z^2 - 3*x^4*y*z^2 + 3*x^3*y^2*z^2 - x*y^4*z^2 - 2*y^5*z^2 + 3*x^4*z^3 - 2*x^3*y*z^3 + 3*x^2*y^2*z^3 - 3*x*y^3*z^3 - 2*y^4*z^3 + 2*x^3*z^4 + 2*x^2*y*z^4 + x*y^2*z^4 + 3*y^3*z^4 - 3*x^2*z^5 - 3*x*y*z^5 + 3*y^2*z^5 + y*z^6 - 2*z^7 + 13/240*x^6 + 2*x^5*y - 2*x^4*y^2 - x^3*y^3 - x^2*y^4 - 3*x*y^5 + 3*y^6 + 2*x^3*y^2*z + 3*x^2*y^3*z - x*y^4*z - 2*y^5*z - 3*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 - 3*x^3*z^3 - x^2*y*z^3 - 2*x*y^2*z^3 + 2*y^3*z^3 + 3*x^2*z^4 + 3*x*y*z^4 - 2*x*z^5 - 3*y*z^5 - 3*z^6 - 2*x^3*y^2 - 3*x^2*y^3 + x*y^4 + 3*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 - x^3*z^2 + x^2*y*z^2 - y^3*z^2 + 3*x^2*z^3 - x*y*z^3 - x*z^4 - 3*y*z^4 - z^5 + 18*x^2*y^2 + 2*x*y^3 - 3*y^4 + 3*x^3*z - 3*x*y^2*z - 2*y^3*z + x^2*z^2 - 3*x*y*z^2 - y^2*z^2 - 3*x*z^3 - 2*y*z^3 + x*y*z - 3*x*z^2 - y*z^2 + 2*z^3, 3*x^7 - x^6*y - x^5*y^2 - 3*x^3*y^4 + x^2*y^5 + 2*x*y^6 - 2*x^6*z - x^5*y*z + x^4*y^2*z - 2*x^3*y^3*z - x^2*y^4*z + 2*x*y^5*z - y^6*z + x^5*z^2 - 2*x^4*y*z^2 - 3*x^3*y^2*z^2 + 3*x^2*y^3*z^2 + 2*x*y^4*z^2 - x^4*z^3 + 2*x^3*y*z^3 + 3*x*y^3*z^3 + 2*x^3*z^4 - x^2*y*z^4 + 3*x*y^2*z^4 + 3*y^3*z^4 - 3*x^2*z^5 - 2*x*y*z^5 + y^2*z^5 + x*z^6 + 2*y*z^6 + 2*z^7 + 1/16*x^6 + 3*x^5*y + x^4*y^2 + 2*x^3*y^3 + x^2*y^4 - 3*y^6 + 2*x^5*z - 2*x^4*y*z - x^3*y^2*z - 2*x^2*y^3*z - 2*x*y^4*z + y^5*z + 2*x^4*z^2 + 2*x^3*y*z^2 + 3*x*y^3*z^2 + x^3*z^3 + 3*x^2*y*z^3 - x*y^2*z^3 - y^3*z^3 - 3*x*y*z^4 + y^2*z^4 + 2*x*z^5 + 3*y*z^5 - 3*z^6 + x^3*y^2 + 3*x^2*y^3 - 3*x*y^4 - 2*y^5 + 2*x^4*z + 2*x^3*y*z - x^2*y^2*z + 2*x*y^3*z + 2*x^2*y*z^2 + 3*x*y^2*z^2 + 2*y^3*z^2 - 2*x^2*z^3 - x*y*z^3 + 3*x*z^4 - y*z^4 + 3*z^5 - 12*x^2*y^2 + x*y^3 + 2*y^4 + 2*x^3*z - x^2*y*z - x*y^2*z - 2*y^3*z - 2*x^2*z^2 - 2*x*y*z^2 + y^2*z^2 - 3*x*z^3 - y*z^3 + 2*x^2*z + x*y*z - 2*x*z^2 - y*z^2 + 2*z^3, -2*x^7 + 3*x^6*y + 3*x^5*y^2 + 3*x^4*y^3 + 3*x^3*y^4 - 3*x^2*y^5 + 2*x*y^6 + 3*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 - y^6*z - 3*x^4*y*z^2 + 2*x^3*y^2*z^2 + 2*x^2*y^3*z^2 - 2*x*y^4*z^2 - 3*y^5*z^2 - 3*x^4*z^3 - 3*x^2*y^2*z^3 - x*y^3*z^3 - 2*x*y^2*z^4 + 2*y^3*z^4 + 2*x^2*z^5 - x*y*z^5 + 3*y^2*z^5 + x*z^6 - 3*y*z^6 + 2*z^7 - 13/720*x^6 - x^5*y + 2*x^4*y^2 - 2*x^2*y^4 - y^6 + 3*x^5*z - 3*x^4*y*z + 2*x^3*y^2*z + 3*x^2*y^3*z + 2*x*y^4*z + y^5*z + 3*x^4*z^2 + x^3*y*z^2 + 3*x^2*y^2*z^2 + 3*x*y^3*z^2 - 2*y^4*z^2 - 2*x^3*z^3 + x^2*y*z^3 + 3*x*y^2*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 + y^5 + 2*x^4*z + 3*x^3*y*z - 2*x^2*y^2*z + 2*x*y^3*z - 2*y^4*z + x^3*z^2 + 3*x*y^2*z^2 - 3*y^3*z^2 + x^2*z^3 - x*y*z^3 - 2*y^2*z^3 - 2*x*z^4 - 2*z^5 + 18*x^2*y^2 + x*y^3 - 3*y^4 + x^3*z - x*y^2*z - 2*x^2*z^2 - 2*x*y*z^2 + 2*y^2*z^2 + 3*x*z^3 + y*z^3 - 2*z^4 - 3*x^2*z - x*y*z + 2*x*z^2 + 3*y*z^2 + 2*z^3]
  [ok] (F4 orbit) R = X*Y^3: S6.2, triple in Ann, local length ds = 41, stabilization = 41, in [21,41]
      triple: [-3*x^7 + 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 - y^7 - x^6*z - 2*x^5*y*z + 2*x^4*y^2*z + x^3*y^3*z - x^2*y^4*z - 3*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 - x^2*y^3*z^2 - x*y^4*z^2 + 3*y^5*z^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 - 2*x*y^2*z^4 - 3*y^3*z^4 + x^2*z^5 - 3*x*y*z^5 - 2*y^2*z^5 + 2*x*z^6 + y*z^6 + z^7 - 1/45*x^6 - x^5*y - x^4*y^2 - x^3*y^3 + 3*x^2*y^4 + 2*y^6 - 3*x^5*z - 2*x^3*y^2*z + 3*x*y^4*z - y^5*z + 2*x^3*y*z^2 + 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 - x*y^2*z^3 - 3*y^3*z^3 - 2*x^2*z^4 - y^2*z^4 - 2*x*z^5 + y*z^5 + 2*z^6 + x^3*y^2 - 3*x^2*y^3 + 3*x*y^4 + 2*y^5 - 2*x^4*z - 3*x^3*y*z - 3*x^2*y^2*z + 3*y^4*z + 2*x^3*z^2 + x^2*y*z^2 + 3*x*y^2*z^2 + 2*y^3*z^2 - x^2*z^3 - 3*x*y*z^3 - 2*y^2*z^3 + 2*x*z^4 - 3*z^5 + 2*x^2*y^2 - y^4 - 3*x^2*y*z - 2*x*y^2*z - 3*y^3*z + x^2*z^2 + 3*x*y*z^2 + y^2*z^2 - x*z^3 - y*z^3 - 2*z^4 + x^2*z + 2*x*y*z - 2*x*z^2 - 2*y*z^2, x^7 - 2*x^6*y - 3*x^5*y^2 - 2*x^4*y^3 + 3*x^3*y^4 - 2*y^7 + x^6*z - x^5*y*z + x^4*y^2*z + 2*x^2*y^4*z - 3*y^6*z - 2*x^5*z^2 + 3*x^4*y*z^2 - 3*x^3*y^2*z^2 - 2*x^2*y^3*z^2 + x*y^4*z^2 - y^5*z^2 - 3*x^4*z^3 - x^3*y*z^3 - 2*x^2*y^2*z^3 - 3*x*y^3*z^3 - 2*y^4*z^3 - 2*x^2*y*z^4 + x*y^2*z^4 + y^3*z^4 - 3*x^2*z^5 + 2*x*y*z^5 + 3*y^2*z^5 - 3*x*z^6 - 2*y*z^6 + 3*z^7 - 1/48*x^6 - x^5*y - 3*x^4*y^2 + 2*x^3*y^3 - x^2*y^4 - 2*x*y^5 - 2*y^6 + 3*x^5*z + 2*x^4*y*z - x^3*y^2*z + x^2*y^3*z - x*y^4*z + x^3*y*z^2 + x^2*y^2*z^2 + 3*x*y^3*z^2 + x^2*y*z^3 - x*y^2*z^3 + 2*y^3*z^3 - x^2*z^4 + 2*x*y*z^4 + 3*y^2*z^4 - y*z^5 + z^6 - x^3*y^2 - 2*x^2*y^3 - 3*x*y^4 - y^5 + x^4*z + 2*x^3*y*z - 2*x^2*y^2*z - 3*x*y^3*z - x^3*z^2 + 2*y^3*z^2 + 3*x^2*z^3 - 2*x*y*z^3 - 2*y*z^4 - 3*z^5 - 3*x^2*y^2 - 2*y^4 - 3*x^3*z - x^2*y*z + 2*x*y^2*z + 3*x^2*z^2 - 3*x*y*z^2 + y^2*z^2 - 2*x*z^3 + y*z^3 + 3*z^4 + 3*x^2*z + x*y*z + 3*x*z^2 - 3*y*z^2 - 2*z^3, 3*x^7 + x^5*y^2 + 2*x^4*y^3 - 3*x^2*y^5 + 3*x*y^6 - 3*y^7 + 3*x^6*z + x^5*y*z + 2*x^4*y^2*z + x^3*y^3*z + 2*x^2*y^4*z - 2*x*y^5*z - 2*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 + 3*x*y^4*z^2 + y^5*z^2 - 3*x^4*z^3 - 2*x^3*y*z^3 + 2*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 + 3*x*y^2*z^4 - 2*y^3*z^4 + 3*x^2*z^5 + x*y*z^5 + y^2*z^5 + x*z^6 - y*z^6 - 37/720*x^6 + 2*x^3*y^3 - 2*x^2*y^4 + 3*y^6 + x^4*y*z - 2*x^3*y^2*z + x^2*y^3*z + 2*x*y^4*z - y^5*z + 2*x^4*z^2 + x^3*y*z^2 - 3*x^2*y^2*z^2 + 2*x*y^3*z^2 - y^4*z^2 + 2*x^3*z^3 - 3*x^2*y*z^3 - 2*x*y^2*z^3 + 3*y^3*z^3 - 2*x^2*z^4 + x*y*z^4 + 3*y^2*z^4 - 2*x*z^5 + 3*y*z^5 + 3*z^6 - x^3*y^2 - 3*x^2*y^3 - 2*x*y^4 + x^4*z + x^3*y*z + 3*x^2*y^2*z - x*y^3*z - y^4*z + 3*x^3*z^2 + 2*x^2*y*z^2 - 2*x*y^2*z^2 - 2*y^3*z^2 + 2*x*y*z^3 - 3*y^2*z^3 + x*z^4 - y*z^4 - 3*z^5 + 2*x^2*y^2 - y^4 - 2*x^3*z - 2*x^2*y*z + 3*x*y^2*z - 2*y^3*z + x^2*z^2 - 3*x*y*z^2 - y*z^3 - 2*z^4 + x^2*z - 3*x*y*z - 2*x*z^2 + 3*z^3]
  [ok] (F4 orbit) R = X^2*Y^2 + X*Y^3: S6.2, triple in Ann, local length ds = 41, stabilization = 41, in [21,41]
      triple: [-2*x^7 - 3*x^6*y - x^5*y^2 + 3*x^3*y^4 - 2*x^2*y^5 + 3*x*y^6 + y^7 - 3*x^6*z + 2*x^5*y*z + 3*x^4*y^2*z + 2*x^2*y^4*z + 2*x*y^5*z - 3*x^5*z^2 - 2*x^4*y*z^2 + 2*x^2*y^3*z^2 - 3*x*y^4*z^2 - y^5*z^2 + 3*x^4*z^3 + 2*x^3*y*z^3 + 3*x^2*y^2*z^3 + 2*x*y^3*z^3 + 2*y^4*z^3 + 2*x^3*z^4 + x^2*y*z^4 - 3*x*y^2*z^4 - 3*y^3*z^4 - 3*x^2*z^5 + 3*y^2*z^5 + 3*x*z^6 + 3*y*z^6 - 2*z^7 - 1/48*x^6 + 2*x^5*y - x^4*y^2 - 3*x^3*y^3 + 3*x^2*y^4 - x*y^5 - y^6 - 2*x^5*z + 2*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^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 + 3*x^2*y*z^3 + 2*y^3*z^3 - 2*x^2*z^4 - 3*x*y*z^4 - y^2*z^4 + x*z^5 - 3*y*z^5 + 2*z^6 + 2*x^3*y^2 - x^2*y^3 - 3*x*y^4 - 3*y^5 + 3*x^4*z + 2*x^3*y*z + x^2*y^2*z - 3*y^4*z + x^3*z^2 + 2*x^2*y*z^2 - 2*x*y^2*z^2 + 2*y^3*z^2 - 3*x*y*z^3 + 3*y^2*z^3 - 2*x*z^4 + 3*y*z^4 + 3/2*x^2*y^2 - x*y^3 + 3*y^4 + x^3*z + 2*x^2*y*z + 2*x*y^2*z + 3*x^2*z^2 - 3*x*y*z^2 - y^2*z^2 - 3*x*z^3 - 3*y*z^3 - z^4 - x^2*z - x*y*z + 2*x*z^2 - y*z^2 + z^3, -x^7 - 2*x^6*y - 2*x^5*y^2 + 3*x^2*y^5 + 3*x*y^6 + 2*y^7 + 2*x^6*z + 3*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 + 3*y^6*z + x^5*z^2 - x^4*y*z^2 + 3*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 - 2*x^2*y^2*z^3 + x*y^3*z^3 - y^4*z^3 + 3*x^3*z^4 + 2*x^2*z^5 - 3*x*y*z^5 - 3*y^2*z^5 + 3*x*z^6 + 2*y*z^6 + 2*z^7 - 7/360*x^6 + 2*x^5*y - x^4*y^2 - x*y^5 + y^6 - 2*x^5*z - x^3*y^2*z - x^2*y^3*z - 3*x*y^4*z - 3*x^3*y*z^2 + x^2*y^2*z^2 - 3*x*y^3*z^2 + 3*y^4*z^2 + 3*x^3*z^3 + x^2*y*z^3 - 3*x^2*z^4 + 3*x*y*z^4 - y^2*z^4 - 2*x*z^5 - y*z^5 - 3*z^6 + x^3*y^2 + 2*x^2*y^3 + 2*x*y^4 - 3*y^5 - 2*x^3*y*z + 2*x^2*y^2*z - 3*x*y^3*z - 2*y^4*z - 3*x^3*z^2 + 3*x*y^2*z^2 + y^3*z^2 - 3*x^2*z^3 - x*y*z^3 - y^2*z^3 - 2*x*z^4 - y*z^4 - 2*z^5 - 3/2*x^2*y^2 + x*y^3 + 2*y^4 - 2*x^3*z + 2*x^2*y*z - x*y^2*z + 2*y^3*z - 2*x^2*z^2 - 2*x*y*z^2 - 2*y^2*z^2 - 3*y*z^3 + 2*z^4 - 3*x^2*z + 2*x*y*z - 2*x*z^2 + 3*y*z^2 - z^3, 3*x^7 + x^6*y - x^5*y^2 - 2*x^4*y^3 - 2*x^3*y^4 + x^2*y^5 + 2*x*y^6 - 3*y^7 - 3*x^6*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 - 3*x^4*y*z^2 + 2*x^3*y^2*z^2 - x^2*y^3*z^2 + 3*x*y^4*z^2 - 2*y^5*z^2 - 3*x^4*z^3 - 3*x^2*y^2*z^3 + 3*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 + 2*x*y*z^5 + y^2*z^5 + 3*x*z^6 + y*z^6 - 2*z^7 - 41/720*x^6 - 3*x^5*y + x^4*y^2 + 3*x^3*y^3 - 2*x^2*y^4 + 3*y^6 + 2*x^5*z - 3*x^4*y*z - x^3*y^2*z + x^2*y^3*z - x*y^4*z - 3*y^5*z + x^4*z^2 + x^3*y*z^2 + 2*x^2*y^2*z^2 - 3*x*y^3*z^2 + 3*x^3*z^3 + 2*x^2*y*z^3 - 2*x*y^2*z^3 + y^3*z^3 + 2*x*y*z^4 - 2*y^2*z^4 - 3*x*z^5 - 3*y*z^5 + z^6 + x^2*y^3 - 2*x*y^4 - 3*y^5 - 3*x^3*y*z - 2*x^2*y^2*z + 2*x*y^3*z - 2*y^4*z + x^2*y*z^2 - 3*x*y^2*z^2 + 2*y^3*z^2 + x^2*z^3 - 2*x*y*z^3 - 3*y^2*z^3 - 2*y*z^4 + z^5 + 9/2*x^2*y^2 - 3*x*y^3 + 3*y^4 + 2*x^3*z + 3*x^2*y*z - 2*x*y^2*z - 3*y^3*z + 3*x^2*z^2 - 3*x*y*z^2 + y^2*z^2 - x*z^3 + 3*y*z^3 - 2*z^4 + 3*x^2*z - 3*x*y*z - x*z^2 - 3*y*z^2 + 3*z^3]
certified values (F4 orbits): {'X^2*Y^2 + Y^4': 41, 'X*Y^3': 41, 'X^2*Y^2 + X*Y^3': 41}
  [info] (F4 orbit) R = Y^4, X^2Y^2: not certified by graded tests (44, 42 in an exploratory run); these two q = 1 orbits are
         covered by the non-free family of l21_1365321_p1t (R = Y^4, X^2Y^2, every D_3 in N) and, for free algebras, by Lemma FQ
         (l21_1365321_r1fq (F4); l21_1365321_F1F8_f1fr (U9))
  [ok] (S) F4 configuration, R = Y^4, non-free: M spanned by T_{<=2}, C[X,Y]_3 + C Y^2Z and the derivatives of Phi = X^6 + Y^5/5 + XY^2Z, Psi = X^4Y + Y^3Z/3 is closed with H = (1,3,6,5,3,2,1), D_4 = <X^4, X^3Y, Y^4>, D_5 = <X^5, X^4Y>, D_3 in N = m o M + T_{<=2}, and y o Phi = Y^4 + 2XYZ (the extra quartic is not free)
10 checks, 0 failed
ALL CHECKS PASSED
