(A) F_3 member: H [1, 3, 6, 4, 3, 2, 1]; three elements of the ideal: local length 36 (standard basis), m^N certificate (N, length) = (9, 36)
    the triple: [-130061/2160*x^6 + 2*x^5*y + x^3*y^3 - 3*x*y^5 - y^6 - 3*x^5*z + 2*x^4*y*z - x^3*y^2*z - 2*x^2*y^3*z - x*y^4*z + 2*y^5*z + 2*x^4*z^2 - 3*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 - 3*x*y^2*z^3 - 2*y^3*z^3 + 3*x^2*z^4 + 3*x*y*z^4 - y^2*z^4 - 2*x*z^5 + 3*y*z^5 - 3*z^6 + 156173/2160*x^5 - 163967/720*x^4*y - x^3*y^2 - 2*x^2*y^3 - 3*x*y^4 - 3*y^5 + x^4*z + x^3*y*z + 2*x^2*y^2*z + x*y^3*z - 3*y^4*z + x^3*z^2 + 2*x^2*y*z^2 + x*y^2*z^2 + 3*y^3*z^2 + 2*x^2*z^3 - x*y*z^3 + 3*y^2*z^3 + 3*x*z^4 - 3*y*z^4 - 3*z^5 + 13/2*x^4 - 80*x^3*y + 155/2*x^2*y^2 - x*y^3 - x^3*z + x^2*y*z + 3*x*y^2*z - y^3*z + 3*x^2*z^2 + y^2*z^2 + 3*y*z^3 + z^4 - 2*x^2*z - x*y*z - 3*y^2*z + 3*x*z^2 + 2*y*z^2 - z^3, 209407/432*x^6 - 2*x^5*y + x^4*y^2 + x^3*y^3 - 3*x^2*y^4 + 3*x*y^5 - 3*y^6 + 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 + x^3*y*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 - 2*y^3*z^3 - 2*x^2*z^4 + 3*x*y*z^4 - 2*y^2*z^4 + 3*x*z^5 - 3*y*z^5 + 3*z^6 + 655/72*x^5 - 827/60*x^4*y - 2*x^3*y^2 + 3*x^2*y^3 + 3*x*y^4 - 2*y^5 + 2*x^4*z + x^3*y*z + x*y^3*z + 3*x^2*y*z^2 + 3*x*y^2*z^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 + 2*x^4 - 11*x^3*y - x^2*y^2 + 3*x*y^3 - y^4 + x^3*z - 3*x^2*y*z - x*y^2*z - 2*y^3*z - 3*x^2*z^2 - 3*x*y*z^2 - 2*y^2*z^2 - 3*x*z^3 - 2*y*z^3 + 3*z^4 + 3*x*y*z - y^2*z - x*z^2 + 2*y*z^2 - z^3, 175141/1080*x^6 + 2*x^4*y^2 + 3*x^3*y^3 + 2*x^2*y^4 - x*y^5 + y^6 + 2*x^5*z + 2*x^4*y*z + 3*x^3*y^2*z - 2*x^2*y^3*z + 2*y^5*z - x^4*z^2 - 3*x^3*y*z^2 + 2*x^2*y^2*z^2 - 3*y^4*z^2 + x^3*z^3 + 3*x^2*y*z^3 - 3*x*y^2*z^3 - 3*y^3*z^3 + 3*x^2*z^4 + 2*x*y*z^4 - x*z^5 - 3*y*z^5 - 13439/540*x^5 + 535/8*x^4*y - 3*x^3*y^2 + 3*x^2*y^3 + 2*x*y^4 - 3*y^5 + 2*x^4*z - 2*x^3*y*z + 2*x*y^3*z - 2*y^4*z - 2*x^3*z^2 - 3*x^2*y*z^2 + x*y^2*z^2 - 2*y^3*z^2 - x^2*z^3 - 2*x*y*z^3 + y^2*z^3 + y*z^4 - z^5 - 7/2*x^4 + 67*x^3*y - 99/2*x^2*y^2 - 3*x*y^3 + 2*y^4 - x^3*z + 2*x^2*y*z - 2*x*y^2*z + 3*y^3*z - 3*x^2*z^2 - x*y*z^2 + 3*x*z^3 - 3*y*z^3 + z^4 - 2*x^2*z - 2*x*y*z + 3*y^2*z + x*z^2 - 2*y*z^2 + 2*z^3]
(B) F_2 member: H [1, 3, 6, 4, 3, 2, 1]; three elements of the ideal: local length 38 (standard basis), m^N certificate (N, length) = (10, 38)
    the triple: [1409/720*x^6 + 2*x^5*y + 2*x^4*y^2 + 3*x^3*y^3 + x^2*y^4 - 3*x*y^5 + y^6 + x^4*y*z + 3*x^3*y^2*z - 3*x^2*y^3*z - x*y^4*z + 2*y^5*z - 2*x^4*z^2 - x^3*y*z^2 + x^2*y^2*z^2 + 3*x*y^3*z^2 + 2*y^4*z^2 - 2*x^3*z^3 - 2*x^2*y*z^3 + 2*x*y^2*z^3 - y^3*z^3 - 2*x^2*z^4 - x*y*z^4 - y^2*z^4 + x*z^5 + 3*y*z^5 + z^6 - 329/360*x^5 - 2*x^4*y + 3*x^2*y^3 - 2*x*y^4 - 133/72*y^5 + 2*x^4*z + 2*x^2*y^2*z + x*y^3*z + 2*y^4*z - x^3*z^2 + 2*x^2*y*z^2 + 2*x*y^2*z^2 - 2*y^3*z^2 + 2*x^2*z^3 - y^2*z^3 + 3*x*z^4 + 3*y*z^4 - 3*x^4 - x^3*y - 2*x^2*y^2 - 3*x*y^3 - y^4 + 3*x^2*y*z + y^3*z - 2*x^2*z^2 + 2*x*y*z^2 - y^2*z^2 + 3*x*z^3 - 2*y*z^3 - z^4 + 3*x^2*z - x*y*z - y^2*z + x*z^2 + 3*y*z^2 + z^3, -2153/720*x^6 - x^5*y + 2*x^4*y^2 + 3*x^3*y^3 + 3*x*y^5 + y^6 - 3*x^4*y*z - x^3*y^2*z - x^2*y^3*z + 2*x*y^4*z + 3*x^4*z^2 - x^2*y^2*z^2 - 3*x*y^3*z^2 + y^4*z^2 - 3*x^3*z^3 + x^2*y*z^3 + x*y^2*z^3 - 3*y^3*z^3 + 3*x^2*z^4 - x*y*z^4 - 3*y^2*z^4 - y*z^5 - 3*z^6 + 1109/360*x^5 + x^4*y - x^3*y^2 - 2*x^2*y^3 - 491/120*y^5 - 3*x^4*z + x^3*y*z - 2*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 + 3*x^2*z^3 - x*y*z^3 + 3*y^2*z^3 - 3*x*z^4 + 2*z^5 - x^3*y + 4*x^2*y^2 - 3*x*y^3 + y^4 - 3*x^3*z - 3*x^2*y*z - x*y^2*z - 2*y^3*z + 2*x^2*z^2 - x*y*z^2 + 2*y^2*z^2 - 2*x*z^3 + 2*y*z^3 + z^4 + 3*x*y*z + y^2*z + 2*y*z^2 - 2*z^3, -191/24*x^6 - 3*x^4*y^2 - 2*x^3*y^3 + x^2*y^4 + x*y^5 - y^6 + x^5*z + 3*x^4*y*z + 2*x^3*y^2*z + 2*x^2*y^3*z + x*y^4*z + y^5*z + 3*x^2*y^2*z^2 + 3*x*y^3*z^2 + 2*y^4*z^2 - 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 - 3*y^2*z^4 + 3*x*z^5 - y*z^5 + 2*z^6 - 7/90*x^5 - x^4*y - 3*x^3*y^2 - x^2*y^3 + x*y^4 + 176/45*y^5 + 2*x^3*y*z + 3*x^2*y^2*z - 3*x*y^3*z - 3*y^4*z + 3*x^3*z^2 - 3*x^2*y*z^2 + x*y^2*z^2 + 2*y^3*z^2 + 2*x^2*z^3 - 3*x*y*z^3 - 2*y^2*z^3 + y*z^4 + 3*z^5 + 3*x^4 + x^3*y + x^2*y^2 + 3*x*y^3 - 2*y^4 - 3*x^3*z + 2*x^2*y*z - x*y^2*z + 3*y^3*z + 3*x^2*z^2 + x*y*z^2 + 3*y^2*z^2 - x*z^3 + 2*y*z^3 - 2*z^4 - 3*x^2*z + 2*x*y*z - 2*y^2*z + 3*x*z^2 - y*z^2 - z^3]
(C) F_1 member: H [1, 3, 6, 4, 3, 2, 1]; three elements of the ideal: local length 38 (standard basis), m^N certificate (N, length) = (11, 38)
    D_4 = <X^4, Q_x, Q_y>: True
    the triple: [-19/90*x^6 + x^5*y - 3*x^4*y^2 - x*y^5 + y^6 - 2*x^5*z + 2*x^4*y*z - 3*x^3*y^2*z + 2*x^2*y^3*z + 3*y^5*z + 3*x^4*z^2 - x^3*y*z^2 + 3*x*y^3*z^2 - 2*y^4*z^2 - 2*x^3*z^3 + x^2*y*z^3 + 2*x*y^2*z^3 - 2*y^3*z^3 - x^2*z^4 - 3*x*y*z^4 - 3*y^2*z^4 + 2*x*z^5 + 3*z^6 - 3*x^4*y + 3*x^3*y^2 - 109/12*x^2*y^3 + 2*x*y^4 + y^5 - x^4*z + 2*x^3*y*z + 2*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 + 2*x*y*z^3 - 3*y^2*z^3 + 3*x*z^4 - 3*y*z^4 - 2*z^5 + 3*x^3*y + 32*x^2*y^2 + 4/3*x*y^3 - 3*y^4 + 3*x^3*z - 3*x*y^2*z + 3*y^3*z + 3*x*y*z^2 + 2*x*z^3 - 2*z^4 - 3*x^2*z + x*y*z + 3*y^2*z + y*z^2 - 3*z^3, 73/1080*x^6 + 3*x^5*y - 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 - 3*x^2*y^3*z + 2*x*y^4*z - 3*y^5*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 - 3*x^3*z^3 - x^2*y*z^3 - 2*x*y^2*z^3 + y^3*z^3 - x^2*z^4 + 3*y^2*z^4 + x*z^5 + 3*y*z^5 - z^6 - x^4*y - 3*x^3*y^2 - 343/12*x^2*y^3 + 2*x*y^4 + 3*y^5 - x^4*z - 2*x^3*y*z - 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 + 2*x*y^2*z^2 + 2*y^3*z^2 + 3*x^2*z^3 + 3*x*y*z^3 - y^2*z^3 - 3*x*z^4 - 3*z^5 - 3*x^3*y - 121/6*x^2*y^2 + 2*y^4 + 3*x^3*z - 2*x^2*y*z - 2*x*y^2*z + 2*y^3*z - 3*x^2*z^2 + 2*y^2*z^2 - 3*x*z^3 + 2*y*z^3 + 3*z^4 + x^2*z - 3*x*y*z - 2*x*z^2 - 3*y*z^2 - 3*z^3, -47/720*x^6 - 3*x^5*y + x^4*y^2 - x^3*y^3 - x^2*y^4 - x*y^5 - 2*y^6 - 3*x^5*z - x^4*y*z - x^2*y^3*z + 2*x*y^4*z + y^5*z + 2*x^4*z^2 - 3*x^3*y*z^2 - x^2*y^2*z^2 + x*y^3*z^2 - 2*y^4*z^2 + 2*x^3*z^3 - 2*x^2*y*z^3 - x*y^2*z^3 - x^2*z^4 + 3*x*y*z^4 - y^2*z^4 + 2*x*z^5 - y*z^5 - 3*z^6 + 3*x^3*y^2 + 139/6*x^2*y^3 - 2*x*y^4 - 2*y^5 + x^3*y*z - x^2*y^2*z + 2*x*y^3*z + y^4*z + 2*x^3*z^2 - 3*x^2*y*z^2 + 3*x*y^2*z^2 - 2*x^2*z^3 + 3*x*y*z^3 - 3*y^2*z^3 + 3*x*z^4 - 3*y*z^4 - 3*z^5 - x^3*y + 23/2*x^2*y^2 + 1/2*x*y^3 - y^4 - x^3*z - 3*x^2*y*z + 3*x*y^2*z + 3*y^3*z + 2*x^2*z^2 + x*y*z^2 + y^2*z^2 + x*z^3 - y*z^3 + 2*z^4 - 3*x*y*z + 3*y^2*z - 2*x*z^2 + 2*y*z^2]

ALL CHECKS PASSED
