PASS (A) secant: B6 space = [X^6, X^5*Y, X^5*Z, Y^6]  [0.4s]
PASS (A) tangent: B6 space = [X^6, X^5*Y, X^5*Z, X^4*Y^2]  [0.5s]
PASS (B) secant: x^3 o F / 336 = X^5 modulo T<=2 (all tails symbolic)  [0.5s]
PASS (B) secant: x^5 y and x^5 z kill F, x o F, x^2 o F (all tails symbolic)  [0.5s]
PASS (B) tangent: x^3 o F / 336 = X^5 modulo T<=2 (all tails symbolic)  [0.5s]
PASS (B) tangent: x^5 y and x^5 z kill F, x o F, x^2 o F (all tails symbolic)  [0.5s]
(C) degree >= 4 conditions (degree, monomial, coefficient):
     (4, (0, 0, 4), 5*c0)
     (4, (0, 1, 3), 4*c1)
     (4, (0, 2, 2), 3*c3)
     (4, (0, 3, 1), 2*c6)
     (4, (0, 4, 0), c10)
     (4, (1, 0, 3), 4*c2)
     (4, (1, 1, 2), 3*c4)
     (4, (1, 2, 1), 2*c7)
     (4, (1, 3, 0), c11)
     (4, (2, 0, 2), 3*c5)
     (4, (2, 1, 1), 2*c8)
     (4, (2, 2, 0), c12)
     (4, (3, 0, 1), 2*c9)
(C) nonzero cubic remainders modulo E0 = <X^3, X^2Y>, generator (degree, pivot, derivative):
     (8, (8, 0, 0), 'z') : -3/(4*d)*c13*c16*X*Y^2 - 1/(2*d)*c13*c15*Y^3 - 3/(8*d)*c13^2*X^2*Z - 1/(2*d)*c12*c13*X*Y*Z - 3/(8*d)*c11*c13*Y^2*Z - 1/(4*d)*c8*c13*X*Z^2 - 1/(4*d)*c7*c13*Y*Z^2 - 1/(8*d)*c4*c13*Z^3 + h7*X*Y^2 + h6*Y^3 + 2*h5*X^2*Z + 2*h4*X*Y*Z + 2*h3*Y^2*Z + 3*h2*X*Z^2 + 3*h1*Y*Z^2 + 4*h0*Z^3
     (7, (7, 0, 0), 'z') : 1/4*c12*X*Y^2 + 1/8*c11*Y^3 + 3/4*c9*X^2*Z + 1/2*c8*X*Y*Z + 1/4*c7*Y^2*Z + 3/4*c5*X*Z^2 + 3/8*c4*Y*Z^2 + 1/2*c2*Z^3
     (5, (4, 1, 0), 'y') : 6/d*c15*X*Y^2 + 10/d*c14*Y^3 + 1/d*c12*X^2*Z + 3/d*c11*X*Y*Z + 6/d*c10*Y^2*Z + 1/d*c7*X*Z^2 + 3/d*c6*Y*Z^2 + 1/d*c3*Z^3
     (5, (4, 1, 0), 'z') : 3/(2*d)*c11*X*Y^2 + 2/d*c10*Y^3 + 1/d*c8*X^2*Z + 2/d*c7*X*Y*Z + 3/d*c6*Y^2*Z + 3/(2*d)*c4*X*Z^2 + 3/d*c3*Y*Z^2 + 2/d*c1*Z^3
PASS (C) the conditions in degree >= 4 say exactly: z o C5 in <X^4, X^3Y> (all tails symbolic)  [0.7s]
PASS (D) (U_0) member (G = Z^3): closure of F, D_3 has H = [1, 3, 6, 3, 2, 2, 1, 1, 1]  [1.0s]
     cubic base scheme: length 3  primary components: [Ideal (z, y^2) of Multivariate Polynomial Ring in x, y, z over Rational Field, Ideal (y, x) of Multivariate Polynomial Ring in x, y, z over Rational Field]
PASS (D) (U_0) member (G = Z^3): the cubic base scheme has length 3  [1.4s]
PASS (D) (U_1) member (C5 = XY^4 + Y^5): closure of F, D_3 has H = [1, 3, 6, 3, 2, 2, 1, 1, 1]  [1.8s]
     cubic base scheme: length 3  primary components: [Ideal (z, y^2) of Multivariate Polynomial Ring in x, y, z over Rational Field, Ideal (z, 5*x - y) of Multivariate Polynomial Ring in x, y, z over Rational Field]
PASS (D) (U_1) member (C5 = XY^4 + Y^5): the cubic base scheme has length 3  [2.1s]
PASS (D) (U_1) member with c_5 = 0 (C5 = XY^4): closure of F, D_3 has H = [1, 3, 6, 3, 2, 2, 1, 1, 1]  [2.5s]
     cubic base scheme: length 3  primary components: [Ideal (z, y^3) of Multivariate Polynomial Ring in x, y, z over Rational Field]
PASS (D) (U_1) member with c_5 = 0 (C5 = XY^4): the cubic base scheme has length 3  [2.8s]
PASS (E) (U_0) member: H of the closure of <Phi, X^3, X^2Y, XZ^2> is (1,3,6,3,2,2,1,1,1)  [3.2s]
PASS (E) (U_0) member: the three generators [y^3, x^2*z, -1/840*x^5 + x*y^2 + z^3] annihilate M  [3.2s]
PASS (E) (U_0) member: local length 33 (dim S/(J + m^N) = 33 for N = 11 and N = 12)  [3.2s]
PASS (E) (U_1) member with c_5 = 0: H of the closure of <Phi, X^3, X^2Y, XY^2> is (1,3,6,3,2,2,1,1,1)  [3.6s]
PASS (E) (U_1) member with c_5 = 0: the three generators [-1/2*x^3*y + y^3, x^2*z, -1/840*x^6 + x^2*y^2 + z^3] annihilate M  [3.6s]
PASS (E) (U_1) member with c_5 = 0: local length 34 (dim S/(J + m^N) = 34 for N = 11 and N = 12)  [3.6s]
parts run: all; total time 3.6 s
ALL CHECKS PASSED
