Code for computations used in the proof of Theorems 2 and 3

[2]:

from sage.rings.invariants.invariant_theory import AlgebraicForm,transvectant

_.<b0, b1, b2, b3, b4, b5, c0, c1, mu1, mu2, t1, t2, x, y> = QQ[]


def quinticAsFourOneInvariantsRoots(lambda1, lambda2):

    q = x * (x-y) * (x - lambda1*y) * (x - lambda2*y)
    l = y


    p = q*l


    qDelta = p(b0, b1, b2, b3, b4, b5, c0, c1, mu1, mu2, t1, t2, x, y)
    Delta = qDelta.discriminant(x) # the discriminant


    q4 = p^2
    f4 = AlgebraicForm(2, 10, q4, x, y)
    T4 = transvectant(f4, f4, 10, scale='none')
    I4 = T4.polynomial() # the invariant I4


    q8 = p^4
    f8 = AlgebraicForm(2, 20, q8, x, y)
    T8 = transvectant(f8, f8, 20, scale='none')
    I8 = T8.polynomial() # the invariant I8


    q12 = p^6
    f12 = AlgebraicForm(2, 30, q12, x, y)
    T12 = transvectant(f12, f12, 30, scale='none')
    I12 = T12.polynomial() # the invariant I12


    f18_1 = AlgebraicForm(2, 25, p^5, x, y)
    f18_2 = AlgebraicForm(2, 30, p^6, x, y)
    T18   = transvectant(f18_1, f18_2, 10, scale='none')
    f18_3 = AlgebraicForm(2, 35, p^7, x, y)
    T18_1 = transvectant(f18_3, T18, 35, scale='none')
    I18   = T18_1.polynomial() # the invariant I18


    alpha = 1/(250822656000)
    beta  = 1/(299067455175152371993371049908040738485043200000000000000)

    H = beta*I12 -  396*(alpha*I4)^3 # the invariant H


    ### Notice that we are scaling the invariants to make their coefficients coprime integers.

    return [I4/41803776000,
            I8/8543177208700379867381760000000,
            I12/9062650156822799151314274239637598135910400000000000000,
            I18/150029545764234105222267552311394127852252417928072743057133909444111892480000000000000000000,
            Delta/1,
            22*H]




def FourOneInvariantsRoots(lambda1,lambda2):

    q = x * (x-y) * (x - lambda1*y) * (x - lambda2*y)
    l = y


    ######### j2
    fq = AlgebraicForm(2, 4, q, x, y)
    T4 = transvectant(fq, fq, 4, scale='none')
    j2 = T4.polynomial()


    ######### j3
    fq = AlgebraicForm(2, 4, q, x, y)

    Tinitial1 = transvectant(fq,fq,2,scale='none')
    Tinitial2 = transvectant(fq,Tinitial1,4,scale='none')
    j3 = Tinitial2.polynomial()


    ######### j5
    fq = AlgebraicForm(2, 4, q, x, y)
    gq = AlgebraicForm(2, 4, l^4, x, y)

    Tinitial1 = transvectant(gq,fq,4,scale='none')
    j5 = Tinitial1.polynomial()



    ######### j6
    fq = AlgebraicForm(2, 4, q, x, y)
    gq = AlgebraicForm(2, 4, l^4, x, y)

    Tinitial1 = transvectant(fq,fq,2,scale='none')
    Tinitial2 = transvectant(gq,Tinitial1,4,scale='none')

    j6 = Tinitial2.polynomial()


    ######### j9
    fq = AlgebraicForm(2, 4, q, x, y)
    gq = AlgebraicForm(2, 6, l^6, x, y)

    Tinitial1 = transvectant(fq, fq, 2, scale='none')
    Tinitial2 = transvectant(fq, Tinitial1, 1, scale='none')
    Tinitial3 = transvectant(Tinitial2, gq, 6, scale='none')

    j9 = Tinitial3.polynomial()


    return [j2/96,
            j3/1152,
            j5/576,
            j6/3456,
            j9/37324800]


def Reduction(form):

    return form(b0, b1, b2, b3, b4, b5, c0, c1, mu1, mu2, 0, 0, x, y)


def Factor(n):

    if n == 0:
        return 0
    else:
        return factor(n)

Marked trees of type I

This type is already determined using the invariant of the quintic $ g(x,z) = :nbsphinx-math:`ell`(x,z) q(x,z)$

Marked trees of type II

Type%20II.1.png 
[3]:

lambda1 = t1 * mu1
lambda2 = t1 * mu2

j2, j3, j5, j6, _ = FourOneInvariantsRoots(lambda1,lambda2)



print("j2:")
print(Factor(j2))
print("\n")

print("j3:")
print(Factor(j3))
print("\n")

print("j5:")
print(Factor(j5))
print("\n")

print("j6:")
print(Factor(j6))




j2_red = Reduction(j2/t1^2)
j3_red = Reduction(j3/t1^3)
j5_red = Reduction(j5)
j6_red = Reduction(j6)



print("\n")
print("\n")
print("\n")



print("j2_red:")
print(Factor(j2_red))
print("\n")

print("j3_red:")
print(Factor(j3_red))
print("\n")

print("j5_red:")
print(Factor(j5_red))
print("\n")

print("j6_red:")
print(Factor(j6_red))

j2:
t1^2 * (mu1^2*mu2^2*t1^2 - mu1^2*mu2*t1 - mu1*mu2^2*t1 + mu1^2 - mu1*mu2 + mu2^2)


j3:
(-1) * t1^3 * (mu1*mu2*t1 - 2*mu1 + mu2) * (mu1*mu2*t1 + mu1 - 2*mu2) * (2*mu1*mu2*t1 - mu1 - mu2)


j5:
1


j6:
(-1) * (3*mu1^2*t1^2 - 2*mu1*mu2*t1^2 + 3*mu2^2*t1^2 - 2*mu1*t1 - 2*mu2*t1 + 3)






j2_red:
mu1^2 - mu1*mu2 + mu2^2


j3_red:
(-1) * (mu1 - 2*mu2) * (mu1 + mu2) * (2*mu1 - mu2)


j5_red:
1


j6_red:
-1 * 3
TypeII.2.png 
[4]:

lambda1 = t1*mu1
lambda2 = mu2

j2, j3, j5, j6, _ = FourOneInvariantsRoots(lambda1,lambda2)


print("j2:")
print(Factor(j2))
print("\n")

print("j3:")
print(Factor(j3))
print("\n")

print("j5:")
print(Factor(j5))
print("\n")

print("j6:")
print(Factor(j6))



j2_red = Reduction(j2)
j3_red = Reduction(j3)
j5_red = Reduction(j5)
j6_red = Reduction(j6)

print("\n")
print("\n")
print("\n")



print("j2_red:")
print(Factor(j2_red))
print("\n")

print("j3_red:")
print(Factor(j3_red))
print("\n")

print("j5_red:")
print(Factor(j5_red))
print("\n")

print("j6_red:")
print(Factor(j6_red))



j2:
mu1^2*mu2^2*t1^2 - mu1^2*mu2*t1^2 - mu1*mu2^2*t1 + mu1^2*t1^2 - mu1*mu2*t1 + mu2^2


j3:
(-1) * (mu1*mu2*t1 - 2*mu1*t1 + mu2) * (mu1*mu2*t1 + mu1*t1 - 2*mu2) * (2*mu1*mu2*t1 - mu1*t1 - mu2)


j5:
1


j6:
(-1) * (3*mu1^2*t1^2 - 2*mu1*mu2*t1 + 3*mu2^2 - 2*mu1*t1 - 2*mu2 + 3)






j2_red:
mu2^2


j3_red:
(-2) * mu2^3


j5_red:
1


j6_red:
(-1) * (3*mu2^2 - 2*mu2 + 3)

Marked trees of type III

TypeIII.1.png 
[5]:

lambda1 = t1 * mu1
lambda2 = t1 * t2 * mu2


j2, j3, j5, j6, _ = FourOneInvariantsRoots(lambda1,lambda2)


print("j2:")
print(Factor(j2))
print("\n")

print("j3:")
print(Factor(j3))
print("\n")

print("j5:")
print(Factor(j5))
print("\n")

print("j6:")
print(Factor(j6))





j2_red = Reduction(j2/t1^2)
j3_red = Reduction(j3/t1^3)
j5_red = Reduction(j5)
j6_red = Reduction(j6)


print("\n")
print("\n")
print("\n")



print("j2_red:")
print(Factor(j2_red))
print("\n")

print("j3_red:")
print(Factor(j3_red))
print("\n")

print("j5_red:")
print(Factor(j5_red))
print("\n")

print("j6_red:")
print(Factor(j6_red))


print("\n \n")
print("########################################################")
print("\n \n")

I4,I8,I12,I18,Delta,H = quinticAsFourOneInvariantsRoots(lambda1,lambda2)

print("I4: ")
print(Factor(I4))
print("\n \n")

print("I8: ")
print(Factor(I8))
print("\n \n")

print("I12: ")
print(Factor(I12))
print("\n \n")


print("I18: ")
print(Factor(I18))
print("\n \n")

print("Delta: ")
print(Factor(Delta))
print("\n \n")


print("H: ")
print(Factor(H))
print("\n \n")


j2:
t1^2 * (mu1^2*mu2^2*t1^2*t2^2 - mu1*mu2^2*t1*t2^2 - mu1^2*mu2*t1*t2 + mu2^2*t2^2 - mu1*mu2*t2 + mu1^2)


j3:
(-1) * t1^3 * (mu1*mu2*t1*t2 - 2*mu2*t2 + mu1) * (mu1*mu2*t1*t2 + mu2*t2 - 2*mu1) * (2*mu1*mu2*t1*t2 - mu2*t2 - mu1)


j5:
1


j6:
(-1) * (3*mu2^2*t1^2*t2^2 - 2*mu1*mu2*t1^2*t2 + 3*mu1^2*t1^2 - 2*mu2*t1*t2 - 2*mu1*t1 + 3)






j2_red:
mu1^2


j3_red:
(-2) * mu1^3


j5_red:
1


j6_red:
-1 * 3



########################################################



I4:
(-1) * t1^2 * (2*mu1^2*mu2^4*t1^4*t2^4 - 2*mu1^3*mu2^3*t1^4*t2^3 + 2*mu1^4*mu2^2*t1^4*t2^2 - 2*mu1*mu2^4*t1^3*t2^4 - mu1^2*mu2^3*t1^3*t2^3 - mu1^3*mu2^2*t1^3*t2^2 + 2*mu2^4*t1^2*t2^4 - 2*mu1^4*mu2*t1^3*t2 - mu1*mu2^3*t1^2*t2^3 + 6*mu1^2*mu2^2*t1^2*t2^2 - mu1^3*mu2*t1^2*t2 - 2*mu2^3*t1*t2^3 + 2*mu1^4*t1^2 - mu1*mu2^2*t1*t2^2 - mu1^2*mu2*t1*t2 - 2*mu1^3*t1 + 2*mu2^2*t2^2 - 2*mu1*mu2*t2 + 2*mu1^2)



I8:
t1^4 * (14*mu1^4*mu2^8*t1^8*t2^8 - 28*mu1^5*mu2^7*t1^8*t2^7 + 42*mu1^6*mu2^6*t1^8*t2^6 - 28*mu1^3*mu2^8*t1^7*t2^8 - 28*mu1^7*mu2^5*t1^8*t2^5 + 14*mu1^4*mu2^7*t1^7*t2^7 + 14*mu1^8*mu2^4*t1^8*t2^4 - 28*mu1^5*mu2^6*t1^7*t2^6 + 42*mu1^2*mu2^8*t1^6*t2^8 - 28*mu1^6*mu2^5*t1^7*t2^5 - 28*mu1^3*mu2^7*t1^6*t2^7 + 14*mu1^7*mu2^4*t1^7*t2^4 + 149*mu1^4*mu2^6*t1^6*t2^6 - 28*mu1*mu2^8*t1^5*t2^8 - 28*mu1^8*mu2^3*t1^7*t2^3 - 88*mu1^5*mu2^5*t1^6*t2^5 - 28*mu1^2*mu2^7*t1^5*t2^7 + 149*mu1^6*mu2^4*t1^6*t2^4 - 88*mu1^3*mu2^6*t1^5*t2^6 + 14*mu2^8*t1^4*t2^8 - 28*mu1^7*mu2^3*t1^6*t2^3 - 66*mu1^4*mu2^5*t1^5*t2^5 + 14*mu1*mu2^7*t1^4*t2^7 + 42*mu1^8*mu2^2*t1^6*t2^2 - 66*mu1^5*mu2^4*t1^5*t2^4 + 149*mu1^2*mu2^6*t1^4*t2^6 - 88*mu1^6*mu2^3*t1^5*t2^3 - 66*mu1^3*mu2^5*t1^4*t2^5 - 28*mu2^7*t1^3*t2^7 - 28*mu1^7*mu2^2*t1^5*t2^2 + 282*mu1^4*mu2^4*t1^4*t2^4 - 28*mu1*mu2^6*t1^3*t2^6 - 28*mu1^8*mu2*t1^5*t2 - 66*mu1^5*mu2^3*t1^4*t2^3 - 88*mu1^2*mu2^5*t1^3*t2^5 + 149*mu1^6*mu2^2*t1^4*t2^2 - 66*mu1^3*mu2^4*t1^3*t2^4 + 42*mu2^6*t1^2*t2^6 + 14*mu1^7*mu2*t1^4*t2 - 66*mu1^4*mu2^3*t1^3*t2^3 - 28*mu1*mu2^5*t1^2*t2^5 + 14*mu1^8*t1^4 - 88*mu1^5*mu2^2*t1^3*t2^2 + 149*mu1^2*mu2^4*t1^2*t2^4 - 28*mu1^6*mu2*t1^3*t2 - 88*mu1^3*mu2^3*t1^2*t2^3 - 28*mu2^5*t1*t2^5 - 28*mu1^7*t1^3 + 149*mu1^4*mu2^2*t1^2*t2^2 + 14*mu1*mu2^4*t1*t2^4 - 28*mu1^5*mu2*t1^2*t2 - 28*mu1^2*mu2^3*t1*t2^3 + 42*mu1^6*t1^2 - 28*mu1^3*mu2^2*t1*t2^2 + 14*mu2^4*t2^4 + 14*mu1^4*mu2*t1*t2 - 28*mu1*mu2^3*t2^3 - 28*mu1^5*t1 + 42*mu1^2*mu2^2*t2^2 - 28*mu1^3*mu2*t2 + 14*mu1^4)



I12:
(-1) * t1^6 * (484*mu1^6*mu2^12*t1^12*t2^12 - 1452*mu1^7*mu2^11*t1^12*t2^11 + 2937*mu1^8*mu2^10*t1^12*t2^10 - 1452*mu1^5*mu2^12*t1^11*t2^12 - 3454*mu1^9*mu2^9*t1^12*t2^9 + 2178*mu1^6*mu2^11*t1^11*t2^11 + 2937*mu1^10*mu2^8*t1^12*t2^8 - 3762*mu1^7*mu2^10*t1^11*t2^10 + 2937*mu1^4*mu2^12*t1^10*t2^12 - 1452*mu1^11*mu2^7*t1^12*t2^7 + 858*mu1^8*mu2^9*t1^11*t2^9 - 3762*mu1^5*mu2^11*t1^10*t2^11 + 484*mu1^12*mu2^6*t1^12*t2^6 + 858*mu1^9*mu2^8*t1^11*t2^8 + 13014*mu1^6*mu2^10*t1^10*t2^10 - 3454*mu1^3*mu2^12*t1^9*t2^12 - 3762*mu1^10*mu2^7*t1^11*t2^7 - 13170*mu1^7*mu2^9*t1^10*t2^9 + 858*mu1^4*mu2^11*t1^9*t2^11 + 2178*mu1^11*mu2^6*t1^11*t2^6 + 20871*mu1^8*mu2^8*t1^10*t2^8 - 13170*mu1^5*mu2^10*t1^9*t2^10 + 2937*mu1^2*mu2^12*t1^8*t2^12 - 1452*mu1^12*mu2^5*t1^11*t2^5 - 13170*mu1^9*mu2^7*t1^10*t2^7 + 946*mu1^6*mu2^9*t1^9*t2^9 + 858*mu1^3*mu2^11*t1^8*t2^11 + 13014*mu1^10*mu2^6*t1^10*t2^6 - 11448*mu1^7*mu2^8*t1^9*t2^8 + 20871*mu1^4*mu2^10*t1^8*t2^10 - 1452*mu1*mu2^12*t1^7*t2^12 - 3762*mu1^11*mu2^5*t1^10*t2^5 - 11448*mu1^8*mu2^7*t1^9*t2^7 - 11448*mu1^5*mu2^9*t1^8*t2^9 - 3762*mu1^2*mu2^11*t1^7*t2^11 + 2937*mu1^12*mu2^4*t1^10*t2^4 + 946*mu1^9*mu2^6*t1^9*t2^6 + 51696*mu1^6*mu2^8*t1^8*t2^8 - 13170*mu1^3*mu2^10*t1^7*t2^10 + 484*mu2^12*t1^6*t2^12 - 13170*mu1^10*mu2^5*t1^9*t2^5 - 25812*mu1^7*mu2^7*t1^8*t2^7 - 11448*mu1^4*mu2^9*t1^7*t2^9 + 2178*mu1*mu2^11*t1^6*t2^11 + 858*mu1^11*mu2^4*t1^9*t2^4 + 51696*mu1^8*mu2^6*t1^8*t2^6 - 25812*mu1^5*mu2^8*t1^7*t2^8 + 13014*mu1^2*mu2^10*t1^6*t2^10 - 3454*mu1^12*mu2^3*t1^9*t2^3 - 11448*mu1^9*mu2^5*t1^8*t2^5 - 21510*mu1^6*mu2^7*t1^7*t2^7 + 946*mu1^3*mu2^9*t1^6*t2^9 - 1452*mu2^11*t1^5*t2^11 + 20871*mu1^10*mu2^4*t1^8*t2^4 - 21510*mu1^7*mu2^6*t1^7*t2^6 + 51696*mu1^4*mu2^8*t1^6*t2^8 - 3762*mu1*mu2^10*t1^5*t2^10 + 858*mu1^11*mu2^3*t1^8*t2^3 - 25812*mu1^8*mu2^5*t1^7*t2^5 - 21510*mu1^5*mu2^7*t1^6*t2^7 - 13170*mu1^2*mu2^9*t1^5*t2^9 + 2937*mu1^12*mu2^2*t1^8*t2^2 - 11448*mu1^9*mu2^4*t1^7*t2^4 + 81966*mu1^6*mu2^6*t1^6*t2^6 - 11448*mu1^3*mu2^8*t1^5*t2^8 + 2937*mu2^10*t1^4*t2^10 - 13170*mu1^10*mu2^3*t1^7*t2^3 - 21510*mu1^7*mu2^5*t1^6*t2^5 - 25812*mu1^4*mu2^7*t1^5*t2^7 + 858*mu1*mu2^9*t1^4*t2^9 - 3762*mu1^11*mu2^2*t1^7*t2^2 + 51696*mu1^8*mu2^4*t1^6*t2^4 - 21510*mu1^5*mu2^6*t1^5*t2^6 + 20871*mu1^2*mu2^8*t1^4*t2^8 - 1452*mu1^12*mu2*t1^7*t2 + 946*mu1^9*mu2^3*t1^6*t2^3 - 21510*mu1^6*mu2^5*t1^5*t2^5 - 11448*mu1^3*mu2^7*t1^4*t2^7 - 3454*mu2^9*t1^3*t2^9 + 13014*mu1^10*mu2^2*t1^6*t2^2 - 25812*mu1^7*mu2^4*t1^5*t2^4 + 51696*mu1^4*mu2^6*t1^4*t2^6 + 858*mu1*mu2^8*t1^3*t2^8 + 2178*mu1^11*mu2*t1^6*t2 - 11448*mu1^8*mu2^3*t1^5*t2^3 - 25812*mu1^5*mu2^5*t1^4*t2^5 - 13170*mu1^2*mu2^7*t1^3*t2^7 + 484*mu1^12*t1^6 - 13170*mu1^9*mu2^2*t1^5*t2^2 + 51696*mu1^6*mu2^4*t1^4*t2^4 + 946*mu1^3*mu2^6*t1^3*t2^6 + 2937*mu2^8*t1^2*t2^8 - 3762*mu1^10*mu2*t1^5*t2 - 11448*mu1^7*mu2^3*t1^4*t2^3 - 11448*mu1^4*mu2^5*t1^3*t2^5 - 3762*mu1*mu2^7*t1^2*t2^7 - 1452*mu1^11*t1^5 + 20871*mu1^8*mu2^2*t1^4*t2^2 - 11448*mu1^5*mu2^4*t1^3*t2^4 + 13014*mu1^2*mu2^6*t1^2*t2^6 + 858*mu1^9*mu2*t1^4*t2 + 946*mu1^6*mu2^3*t1^3*t2^3 - 13170*mu1^3*mu2^5*t1^2*t2^5 - 1452*mu2^7*t1*t2^7 + 2937*mu1^10*t1^4 - 13170*mu1^7*mu2^2*t1^3*t2^2 + 20871*mu1^4*mu2^4*t1^2*t2^4 + 2178*mu1*mu2^6*t1*t2^6 + 858*mu1^8*mu2*t1^3*t2 - 13170*mu1^5*mu2^3*t1^2*t2^3 - 3762*mu1^2*mu2^5*t1*t2^5 - 3454*mu1^9*t1^3 + 13014*mu1^6*mu2^2*t1^2*t2^2 + 858*mu1^3*mu2^4*t1*t2^4 + 484*mu2^6*t2^6 - 3762*mu1^7*mu2*t1^2*t2 + 858*mu1^4*mu2^3*t1*t2^3 - 1452*mu1*mu2^5*t2^5 + 2937*mu1^8*t1^2 - 3762*mu1^5*mu2^2*t1*t2^2 + 2937*mu1^2*mu2^4*t2^4 + 2178*mu1^6*mu2*t1*t2 - 3454*mu1^3*mu2^3*t2^3 - 1452*mu1^7*t1 + 2937*mu1^4*mu2^2*t2^2 - 1452*mu1^5*mu2*t2 + 484*mu1^6)



I18:
(-1) * t1^9 * (mu2*t1*t2 - mu1*t1 - 1) * (mu2*t1*t2 - mu1*t1 + 1) * (mu2*t1*t2 + mu1*t1 - 1) * (-mu1^2*t1 + mu2*t2) * (mu1^2*t1 + mu2*t2 - 2*mu1) * (mu1*mu2*t1*t2 - mu2*t2 - mu1) * (mu1*mu2*t1*t2 - mu2*t2 + mu1) * (mu1*mu2*t1*t2 + mu2*t2 - mu1) * (2*mu1*mu2*t1*t2 - mu1^2*t1 - mu2*t2) * (mu2^2*t1*t2^2 - mu1) * (mu2^2*t1*t2^2 - 2*mu2*t2 + mu1) * (mu2^2*t1*t2^2 - 2*mu1*mu2*t1*t2 + mu1) * (mu1*mu2*t1^2*t2 - 1) * (mu1*mu2*t1^2*t2 - 2*mu1*t1 + 1) * (mu1*mu2*t1^2*t2 - 2*mu2*t1*t2 + 1)



Delta:
t2^2 * mu2^2 * mu1^2 * t1^6 * y^18 * (mu2*t2 - mu1)^2 * (mu1*t1 - 1)^2 * (mu2*t1*t2 - 1)^2



H:
(-1) * t1^6 * (22*mu1^8*mu2^10*t1^12*t2^10 - 44*mu1^9*mu2^9*t1^12*t2^9 + 22*mu1^10*mu2^8*t1^12*t2^8 - 88*mu1^7*mu2^10*t1^11*t2^10 + 22*mu1^4*mu2^12*t1^10*t2^12 + 88*mu1^8*mu2^9*t1^11*t2^9 - 88*mu1^5*mu2^11*t1^10*t2^11 + 88*mu1^9*mu2^8*t1^11*t2^8 + 690*mu1^6*mu2^10*t1^10*t2^10 - 44*mu1^3*mu2^12*t1^9*t2^12 - 88*mu1^10*mu2^7*t1^11*t2^7 - 1278*mu1^7*mu2^9*t1^10*t2^9 + 88*mu1^4*mu2^11*t1^9*t2^11 + 1330*mu1^8*mu2^8*t1^10*t2^8 - 1278*mu1^5*mu2^10*t1^9*t2^10 + 22*mu1^2*mu2^12*t1^8*t2^12 - 1278*mu1^9*mu2^7*t1^10*t2^7 + 1639*mu1^6*mu2^9*t1^9*t2^9 + 88*mu1^3*mu2^11*t1^8*t2^11 + 690*mu1^10*mu2^6*t1^10*t2^6 - 493*mu1^7*mu2^8*t1^9*t2^8 + 1330*mu1^4*mu2^10*t1^8*t2^10 - 88*mu1^11*mu2^5*t1^10*t2^5 - 493*mu1^8*mu2^7*t1^9*t2^7 - 493*mu1^5*mu2^9*t1^8*t2^9 - 88*mu1^2*mu2^11*t1^7*t2^11 + 22*mu1^12*mu2^4*t1^10*t2^4 + 1639*mu1^9*mu2^6*t1^9*t2^6 - 868*mu1^6*mu2^8*t1^8*t2^8 - 1278*mu1^3*mu2^10*t1^7*t2^10 - 1278*mu1^10*mu2^5*t1^9*t2^5 + 458*mu1^7*mu2^7*t1^8*t2^7 - 493*mu1^4*mu2^9*t1^7*t2^9 + 88*mu1^11*mu2^4*t1^9*t2^4 - 868*mu1^8*mu2^6*t1^8*t2^6 + 458*mu1^5*mu2^8*t1^7*t2^8 + 690*mu1^2*mu2^10*t1^6*t2^10 - 44*mu1^12*mu2^3*t1^9*t2^3 - 493*mu1^9*mu2^5*t1^8*t2^5 + 785*mu1^6*mu2^7*t1^7*t2^7 + 1639*mu1^3*mu2^9*t1^6*t2^9 + 1330*mu1^10*mu2^4*t1^8*t2^4 + 785*mu1^7*mu2^6*t1^7*t2^6 - 868*mu1^4*mu2^8*t1^6*t2^8 - 88*mu1*mu2^10*t1^5*t2^10 + 88*mu1^11*mu2^3*t1^8*t2^3 + 458*mu1^8*mu2^5*t1^7*t2^5 + 785*mu1^5*mu2^7*t1^6*t2^7 - 1278*mu1^2*mu2^9*t1^5*t2^9 + 22*mu1^12*mu2^2*t1^8*t2^2 - 493*mu1^9*mu2^4*t1^7*t2^4 - 2952*mu1^6*mu2^6*t1^6*t2^6 - 493*mu1^3*mu2^8*t1^5*t2^8 + 22*mu2^10*t1^4*t2^10 - 1278*mu1^10*mu2^3*t1^7*t2^3 + 785*mu1^7*mu2^5*t1^6*t2^5 + 458*mu1^4*mu2^7*t1^5*t2^7 + 88*mu1*mu2^9*t1^4*t2^9 - 88*mu1^11*mu2^2*t1^7*t2^2 - 868*mu1^8*mu2^4*t1^6*t2^4 + 785*mu1^5*mu2^6*t1^5*t2^6 + 1330*mu1^2*mu2^8*t1^4*t2^8 + 1639*mu1^9*mu2^3*t1^6*t2^3 + 785*mu1^6*mu2^5*t1^5*t2^5 - 493*mu1^3*mu2^7*t1^4*t2^7 - 44*mu2^9*t1^3*t2^9 + 690*mu1^10*mu2^2*t1^6*t2^2 + 458*mu1^7*mu2^4*t1^5*t2^4 - 868*mu1^4*mu2^6*t1^4*t2^6 + 88*mu1*mu2^8*t1^3*t2^8 - 493*mu1^8*mu2^3*t1^5*t2^3 + 458*mu1^5*mu2^5*t1^4*t2^5 - 1278*mu1^2*mu2^7*t1^3*t2^7 - 1278*mu1^9*mu2^2*t1^5*t2^2 - 868*mu1^6*mu2^4*t1^4*t2^4 + 1639*mu1^3*mu2^6*t1^3*t2^6 + 22*mu2^8*t1^2*t2^8 - 88*mu1^10*mu2*t1^5*t2 - 493*mu1^7*mu2^3*t1^4*t2^3 - 493*mu1^4*mu2^5*t1^3*t2^5 - 88*mu1*mu2^7*t1^2*t2^7 + 1330*mu1^8*mu2^2*t1^4*t2^2 - 493*mu1^5*mu2^4*t1^3*t2^4 + 690*mu1^2*mu2^6*t1^2*t2^6 + 88*mu1^9*mu2*t1^4*t2 + 1639*mu1^6*mu2^3*t1^3*t2^3 - 1278*mu1^3*mu2^5*t1^2*t2^5 + 22*mu1^10*t1^4 - 1278*mu1^7*mu2^2*t1^3*t2^2 + 1330*mu1^4*mu2^4*t1^2*t2^4 + 88*mu1^8*mu2*t1^3*t2 - 1278*mu1^5*mu2^3*t1^2*t2^3 - 88*mu1^2*mu2^5*t1*t2^5 - 44*mu1^9*t1^3 + 690*mu1^6*mu2^2*t1^2*t2^2 + 88*mu1^3*mu2^4*t1*t2^4 - 88*mu1^7*mu2*t1^2*t2 + 88*mu1^4*mu2^3*t1*t2^3 + 22*mu1^8*t1^2 - 88*mu1^5*mu2^2*t1*t2^2 + 22*mu1^2*mu2^4*t2^4 - 44*mu1^3*mu2^3*t2^3 + 22*mu1^4*mu2^2*t2^2)
TypeIII.2.png 
[6]:

lambda1 =     t1 * mu1
lambda2 = 1 + t2 * mu2


j2, j3, j5, j6, _ = FourOneInvariantsRoots(lambda1,lambda2)


print("j2:")
print(Factor(j2))
print("\n")

print("j3:")
print(Factor(j3))
print("\n")

print("j5:")
print(Factor(j5))
print("\n")

print("j6:")
print(Factor(j6))





j2_red = Reduction(j2)
j3_red = Reduction(j3)
j5_red = Reduction(j5)
j6_red = Reduction(j6)

print("\n")
print("\n")
print("\n")



print("j2_red:")
print(Factor(j2_red))
print("\n")

print("j3_red:")
print(Factor(j3_red))
print("\n")

print("j5_red:")
print(Factor(j5_red))
print("\n")

print("j6_red:")
print(Factor(j6_red))




print("\n \n")
print("########################################################")
print("\n \n")


I4,I8,I12,I18,Delta,H = quinticAsFourOneInvariantsRoots(lambda1,lambda2)

print("I4: ")
print(Factor(I4))
print("\n \n")

print("I8: ")
print(Factor(I8))
print("\n \n")

print("I12: ")
print(Factor(I12))
print("\n \n")


print("I18: ")
print(Factor(I18))
print("\n \n")

print("Delta: ")
print(Factor(Delta))
print("\n \n")


print("H: ")
print(Factor(H))
print("\n \n")


j2:
mu1^2*mu2^2*t1^2*t2^2 + mu1^2*mu2*t1^2*t2 - mu1*mu2^2*t1*t2^2 + mu1^2*t1^2 - 3*mu1*mu2*t1*t2 + mu2^2*t2^2 - 2*mu1*t1 + 2*mu2*t2 + 1


j3:
(-1) * (mu1*mu2*t1*t2 - mu1*t1 + mu2*t2 + 1) * (mu1*mu2*t1*t2 + 2*mu1*t1 - 2*mu2*t2 - 2) * (2*mu1*mu2*t1*t2 + mu1*t1 - mu2*t2 - 1)


j5:
1


j6:
(-1) * (3*mu1^2*t1^2 - 2*mu1*mu2*t1*t2 + 3*mu2^2*t2^2 - 4*mu1*t1 + 4*mu2*t2 + 4)






j2_red:
1


j3_red:
-1 * 2


j5_red:
1


j6_red:
-1 * 2^2



########################################################



I4:
(-1) * (2*mu1^4*mu2^2*t1^4*t2^2 - 2*mu1^3*mu2^3*t1^3*t2^3 + 2*mu1^2*mu2^4*t1^2*t2^4 + 2*mu1^4*mu2*t1^4*t2 - 7*mu1^3*mu2^2*t1^3*t2^2 + 7*mu1^2*mu2^3*t1^2*t2^3 - 2*mu1*mu2^4*t1*t2^4 + 2*mu1^4*t1^4 - 9*mu1^3*mu2*t1^3*t2 + 15*mu1^2*mu2^2*t1^2*t2^2 - 9*mu1*mu2^3*t1*t2^3 + 2*mu2^4*t2^4 - 6*mu1^3*t1^3 + 16*mu1^2*mu2*t1^2*t2 - 16*mu1*mu2^2*t1*t2^2 + 6*mu2^3*t2^3 + 8*mu1^2*t1^2 - 15*mu1*mu2*t1*t2 + 8*mu2^2*t2^2 - 6*mu1*t1 + 6*mu2*t2 + 2)



I8:
14*mu1^8*mu2^4*t1^8*t2^4 - 28*mu1^7*mu2^5*t1^7*t2^5 + 42*mu1^6*mu2^6*t1^6*t2^6 - 28*mu1^5*mu2^7*t1^5*t2^7 + 14*mu1^4*mu2^8*t1^4*t2^8 + 28*mu1^8*mu2^3*t1^8*t2^3 - 126*mu1^7*mu2^4*t1^7*t2^4 + 224*mu1^6*mu2^5*t1^6*t2^5 - 224*mu1^5*mu2^6*t1^5*t2^6 + 126*mu1^4*mu2^7*t1^4*t2^7 - 28*mu1^3*mu2^8*t1^3*t2^8 + 42*mu1^8*mu2^2*t1^8*t2^2 - 252*mu1^7*mu2^3*t1^7*t2^3 + 639*mu1^6*mu2^4*t1^6*t2^4 - 844*mu1^5*mu2^5*t1^5*t2^5 + 639*mu1^4*mu2^6*t1^4*t2^6 - 252*mu1^3*mu2^7*t1^3*t2^7 + 42*mu1^2*mu2^8*t1^2*t2^8 + 28*mu1^8*mu2*t1^8*t2 - 308*mu1^7*mu2^2*t1^7*t2^2 + 1068*mu1^6*mu2^3*t1^6*t2^3 - 1906*mu1^5*mu2^4*t1^5*t2^4 + 1906*mu1^4*mu2^5*t1^4*t2^5 - 1068*mu1^3*mu2^6*t1^3*t2^6 + 308*mu1^2*mu2^7*t1^2*t2^7 - 28*mu1*mu2^8*t1*t2^8 + 14*mu1^8*t1^8 - 210*mu1^7*mu2*t1^7*t2 + 1129*mu1^6*mu2^2*t1^6*t2^2 - 2750*mu1^5*mu2^3*t1^5*t2^3 + 3657*mu1^4*mu2^4*t1^4*t2^4 - 2750*mu1^3*mu2^5*t1^3*t2^5 + 1129*mu1^2*mu2^6*t1^2*t2^6 - 210*mu1*mu2^7*t1*t2^7 + 14*mu2^8*t2^8 - 84*mu1^7*t1^7 + 714*mu1^6*mu2*t1^6*t2 - 2570*mu1^5*mu2^2*t1^5*t2^2 + 4656*mu1^4*mu2^3*t1^4*t2^3 - 4656*mu1^3*mu2^4*t1^3*t2^4 + 2570*mu1^2*mu2^5*t1^2*t2^5 - 714*mu1*mu2^6*t1*t2^6 + 84*mu2^7*t2^7 + 238*mu1^6*t1^6 - 1470*mu1^5*mu2*t1^5*t2 + 3904*mu1^4*mu2^2*t1^4*t2^2 - 5320*mu1^3*mu2^3*t1^3*t2^3 + 3904*mu1^2*mu2^4*t1^2*t2^4 - 1470*mu1*mu2^5*t1*t2^5 + 238*mu2^6*t2^6 - 420*mu1^5*t1^5 + 2016*mu1^4*mu2*t1^4*t2 - 4040*mu1^3*mu2^2*t1^3*t2^2 + 4040*mu1^2*mu2^3*t1^2*t2^3 - 2016*mu1*mu2^4*t1*t2^4 + 420*mu2^5*t2^5 + 504*mu1^4*t1^4 - 1890*mu1^3*mu2*t1^3*t2 + 2795*mu1^2*mu2^2*t1^2*t2^2 - 1890*mu1*mu2^3*t1*t2^3 + 504*mu2^4*t2^4 - 420*mu1^3*t1^3 + 1190*mu1^2*mu2*t1^2*t2 - 1190*mu1*mu2^2*t1*t2^2 + 420*mu2^3*t2^3 + 238*mu1^2*t1^2 - 462*mu1*mu2*t1*t2 + 238*mu2^2*t2^2 - 84*mu1*t1 + 84*mu2*t2 + 14



I12:
(-1) * (484*mu1^12*mu2^6*t1^12*t2^6 - 1452*mu1^11*mu2^7*t1^11*t2^7 + 2937*mu1^10*mu2^8*t1^10*t2^8 - 3454*mu1^9*mu2^9*t1^9*t2^9 + 2937*mu1^8*mu2^10*t1^8*t2^10 - 1452*mu1^7*mu2^11*t1^7*t2^11 + 484*mu1^6*mu2^12*t1^6*t2^12 + 1452*mu1^12*mu2^5*t1^12*t2^5 - 7986*mu1^11*mu2^6*t1^11*t2^6 + 19734*mu1^10*mu2^7*t1^10*t2^7 - 30228*mu1^9*mu2^8*t1^9*t2^8 + 30228*mu1^8*mu2^9*t1^8*t2^9 - 19734*mu1^7*mu2^10*t1^7*t2^10 + 7986*mu1^6*mu2^11*t1^6*t2^11 - 1452*mu1^5*mu2^12*t1^5*t2^12 + 2937*mu1^12*mu2^4*t1^12*t2^4 - 21186*mu1^11*mu2^5*t1^11*t2^5 + 68916*mu1^10*mu2^6*t1^10*t2^6 - 130650*mu1^9*mu2^7*t1^9*t2^7 + 160758*mu1^8*mu2^8*t1^8*t2^8 - 130650*mu1^7*mu2^9*t1^7*t2^9 + 68916*mu1^6*mu2^10*t1^6*t2^10 - 21186*mu1^5*mu2^11*t1^5*t2^11 + 2937*mu1^4*mu2^12*t1^4*t2^12 + 3454*mu1^12*mu2^3*t1^12*t2^3 - 36102*mu1^11*mu2^4*t1^11*t2^4 + 150384*mu1^10*mu2^5*t1^10*t2^5 - 357356*mu1^9*mu2^6*t1^9*t2^6 + 538848*mu1^8*mu2^7*t1^8*t2^7 - 538848*mu1^7*mu2^8*t1^7*t2^8 + 357356*mu1^6*mu2^9*t1^6*t2^9 - 150384*mu1^5*mu2^10*t1^5*t2^10 + 36102*mu1^4*mu2^11*t1^4*t2^11 - 3454*mu1^3*mu2^12*t1^3*t2^12 + 2937*mu1^12*mu2^2*t1^12*t2^2 - 40590*mu1^11*mu2^3*t1^11*t2^3 + 224151*mu1^10*mu2^4*t1^10*t2^4 - 669498*mu1^9*mu2^5*t1^9*t2^5 + 1244790*mu1^8*mu2^6*t1^8*t2^6 - 1522116*mu1^7*mu2^7*t1^7*t2^7 + 1244790*mu1^6*mu2^8*t1^6*t2^8 - 669498*mu1^5*mu2^9*t1^5*t2^9 + 224151*mu1^4*mu2^10*t1^4*t2^10 - 40590*mu1^3*mu2^11*t1^3*t2^11 + 2937*mu1^2*mu2^12*t1^2*t2^12 + 1452*mu1^12*mu2*t1^12*t2 - 31482*mu1^11*mu2^2*t1^11*t2^2 + 231696*mu1^10*mu2^3*t1^10*t2^3 - 890592*mu1^9*mu2^4*t1^9*t2^4 + 2060964*mu1^8*mu2^5*t1^8*t2^5 - 3089862*mu1^7*mu2^6*t1^7*t2^6 + 3089862*mu1^6*mu2^7*t1^6*t2^7 - 2060964*mu1^5*mu2^8*t1^5*t2^8 + 890592*mu1^4*mu2^9*t1^4*t2^9 - 231696*mu1^3*mu2^10*t1^3*t2^10 + 31482*mu1^2*mu2^11*t1^2*t2^11 - 1452*mu1*mu2^12*t1*t2^12 + 484*mu1^12*t1^12 - 15246*mu1^11*mu2*t1^11*t2 + 165474*mu1^10*mu2^2*t1^10*t2^2 - 843444*mu1^9*mu2^3*t1^9*t2^3 + 2483244*mu1^8*mu2^4*t1^8*t2^4 - 4611978*mu1^7*mu2^5*t1^7*t2^5 + 5644740*mu1^6*mu2^6*t1^6*t2^6 - 4611978*mu1^5*mu2^7*t1^5*t2^7 + 2483244*mu1^4*mu2^8*t1^4*t2^8 - 843444*mu1^3*mu2^9*t1^3*t2^9 + 165474*mu1^2*mu2^10*t1^2*t2^10 - 15246*mu1*mu2^11*t1*t2^11 + 484*mu2^12*t2^12 - 4356*mu1^11*t1^11 + 75636*mu1^10*mu2*t1^10*t2 - 556200*mu1^9*mu2^2*t1^9*t2^2 + 2163744*mu1^8*mu2^3*t1^8*t2^3 - 5089392*mu1^7*mu2^4*t1^7*t2^4 + 7701840*mu1^6*mu2^5*t1^6*t2^5 - 7701840*mu1^5*mu2^6*t1^5*t2^6 + 5089392*mu1^4*mu2^7*t1^4*t2^7 - 2163744*mu1^3*mu2^8*t1^3*t2^8 + 556200*mu1^2*mu2^9*t1^2*t2^9 - 75636*mu1*mu2^10*t1*t2^10 + 4356*mu2^11*t2^11 + 18909*mu1^10*t1^10 - 236412*mu1^9*mu2*t1^9*t2 + 1321056*mu1^8*mu2^2*t1^8*t2^2 - 4101804*mu1^7*mu2^3*t1^7*t2^3 + 7849962*mu1^6*mu2^4*t1^6*t2^4 - 9701370*mu1^5*mu2^5*t1^5*t2^5 + 7849962*mu1^4*mu2^6*t1^4*t2^6 - 4101804*mu1^3*mu2^7*t1^3*t2^7 + 1321056*mu1^2*mu2^8*t1^2*t2^8 - 236412*mu1*mu2^9*t1*t2^9 + 18909*mu2^10*t2^10 - 52536*mu1^9*t1^9 + 520080*mu1^8*mu2*t1^8*t2 - 2326002*mu1^7*mu2^2*t1^7*t2^2 + 5881070*mu1^6*mu2^3*t1^6*t2^3 - 9191196*mu1^5*mu2^4*t1^5*t2^4 + 9191196*mu1^4*mu2^5*t1^4*t2^5 - 5881070*mu1^3*mu2^6*t1^3*t2^6 + 2326002*mu1^2*mu2^7*t1^2*t2^7 - 520080*mu1*mu2^8*t1*t2^8 + 52536*mu2^9*t2^9 + 104016*mu1^8*t1^8 - 848694*mu1^7*mu2*t1^7*t2 + 3107616*mu1^6*mu2^2*t1^6*t2^2 - 6427806*mu1^5*mu2^3*t1^5*t2^3 + 8131788*mu1^4*mu2^4*t1^4*t2^4 - 6427806*mu1^3*mu2^5*t1^3*t2^5 + 3107616*mu1^2*mu2^6*t1^2*t2^6 - 848694*mu1*mu2^7*t1*t2^7 + 104016*mu2^8*t2^8 - 154308*mu1^7*t1^7 + 1053492*mu1^6*mu2*t1^6*t2 - 3174696*mu1^5*mu2^2*t1^5*t2^2 + 5325936*mu1^4*mu2^3*t1^4*t2^3 - 5325936*mu1^3*mu2^4*t1^3*t2^4 + 3174696*mu1^2*mu2^5*t1^2*t2^5 - 1053492*mu1*mu2^6*t1*t2^6 + 154308*mu2^7*t2^7 + 175582*mu1^6*t1^6 - 1003002*mu1^5*mu2*t1^5*t2 + 2465232*mu1^4*mu2^2*t1^4*t2^2 - 3273816*mu1^3*mu2^3*t1^3*t2^3 + 2465232*mu1^2*mu2^4*t1^2*t2^4 - 1003002*mu1*mu2^5*t1*t2^5 + 175582*mu2^6*t2^6 - 154308*mu1^5*t1^5 + 728112*mu1^4*mu2*t1^4*t2 - 1423044*mu1^3*mu2^2*t1^3*t2^2 + 1423044*mu1^2*mu2^3*t1^2*t2^3 - 728112*mu1*mu2^4*t1*t2^4 + 154308*mu2^5*t2^5 + 104016*mu1^4*t1^4 - 394020*mu1^3*mu2*t1^3*t2 + 581472*mu1^2*mu2^2*t1^2*t2^2 - 394020*mu1*mu2^3*t1*t2^3 + 104016*mu2^4*t2^4 - 52536*mu1^3*t1^3 + 151272*mu1^2*mu2*t1^2*t2 - 151272*mu1*mu2^2*t1*t2^2 + 52536*mu2^3*t2^3 + 18909*mu1^2*t1^2 - 37026*mu1*mu2*t1*t2 + 18909*mu2^2*t2^2 - 4356*mu1*t1 + 4356*mu2*t2 + 484)



I18:
(-1) * (-mu1*t1 + mu2*t2) * (-mu1*t1 + mu2*t2 + 2) * (mu1*t1 + mu2*t2) * (mu2^2*t2^2 - mu1*t1 + 2*mu2*t2 + 1) * (mu2^2*t2^2 + mu1*t1 - 1) * (-2*mu1*mu2*t1*t2 + mu2^2*t2^2 - mu1*t1 + 2*mu2*t2 + 1) * (mu1*mu2*t1*t2 - mu2*t2 - 1) * (mu1*mu2*t1*t2 + mu2*t2 + 1) * (mu1*mu2*t1*t2 - mu1*t1 + 1) * (mu1*mu2*t1*t2 + mu1*t1 - 1) * (mu1*mu2*t1*t2 + mu1*t1 - 2*mu2*t2 - 1) * (mu1*mu2*t1*t2 + 2*mu1*t1 - mu2*t2 - 1) * (-mu1^2*t1^2 + mu2*t2 + 1) * (-mu1^2*t1^2 + 2*mu1*mu2*t1*t2 + 2*mu1*t1 - mu2*t2 - 1) * (mu1^2*t1^2 - 2*mu1*t1 + mu2*t2 + 1)



Delta:
t2^2 * t1^2 * mu2^2 * mu1^2 * y^18 * (mu2*t2 + 1)^2 * (-mu1*t1 + mu2*t2 + 1)^2 * (mu1*t1 - 1)^2



H:
(-1) * (22*mu1^10*mu2^8*t1^10*t2^8 - 44*mu1^9*mu2^9*t1^9*t2^9 + 22*mu1^8*mu2^10*t1^8*t2^10 + 88*mu1^10*mu2^7*t1^10*t2^7 - 308*mu1^9*mu2^8*t1^9*t2^8 + 308*mu1^8*mu2^9*t1^8*t2^9 - 88*mu1^7*mu2^10*t1^7*t2^10 + 22*mu1^12*mu2^4*t1^12*t2^4 - 88*mu1^11*mu2^5*t1^11*t2^5 + 690*mu1^10*mu2^6*t1^10*t2^6 - 2158*mu1^9*mu2^7*t1^9*t2^7 + 3112*mu1^8*mu2^8*t1^8*t2^8 - 2158*mu1^7*mu2^9*t1^7*t2^9 + 690*mu1^6*mu2^10*t1^6*t2^10 - 88*mu1^5*mu2^11*t1^5*t2^11 + 22*mu1^4*mu2^12*t1^4*t2^12 + 44*mu1^12*mu2^3*t1^12*t2^3 - 352*mu1^11*mu2^4*t1^11*t2^4 + 2246*mu1^10*mu2^5*t1^10*t2^5 - 8539*mu1^9*mu2^6*t1^9*t2^6 + 15955*mu1^8*mu2^7*t1^8*t2^7 - 15955*mu1^7*mu2^8*t1^7*t2^8 + 8539*mu1^6*mu2^9*t1^6*t2^9 - 2246*mu1^5*mu2^10*t1^5*t2^10 + 352*mu1^4*mu2^11*t1^4*t2^11 - 44*mu1^3*mu2^12*t1^3*t2^12 + 22*mu1^12*mu2^2*t1^12*t2^2 - 440*mu1^11*mu2^3*t1^11*t2^3 + 3750*mu1^10*mu2^4*t1^10*t2^4 - 18113*mu1^9*mu2^5*t1^9*t2^5 + 44933*mu1^8*mu2^6*t1^8*t2^6 - 60054*mu1^7*mu2^7*t1^7*t2^7 + 44933*mu1^6*mu2^8*t1^6*t2^8 - 18113*mu1^5*mu2^9*t1^5*t2^9 + 3750*mu1^4*mu2^10*t1^4*t2^10 - 440*mu1^3*mu2^11*t1^3*t2^11 + 22*mu1^2*mu2^12*t1^2*t2^12 - 176*mu1^11*mu2^2*t1^11*t2^2 + 3214*mu1^10*mu2^3*t1^10*t2^3 - 22487*mu1^9*mu2^4*t1^9*t2^4 + 76009*mu1^8*mu2^5*t1^8*t2^5 - 135645*mu1^7*mu2^6*t1^7*t2^6 + 135645*mu1^6*mu2^7*t1^6*t2^7 - 76009*mu1^5*mu2^8*t1^5*t2^8 + 22487*mu1^4*mu2^9*t1^4*t2^9 - 3214*mu1^3*mu2^10*t1^3*t2^10 + 176*mu1^2*mu2^11*t1^2*t2^11 + 1174*mu1^10*mu2^2*t1^10*t2^2 - 15981*mu1^9*mu2^3*t1^9*t2^3 + 79955*mu1^8*mu2^4*t1^8*t2^4 - 195699*mu1^7*mu2^5*t1^7*t2^5 + 260815*mu1^6*mu2^6*t1^6*t2^6 - 195699*mu1^5*mu2^7*t1^5*t2^7 + 79955*mu1^4*mu2^8*t1^4*t2^8 - 15981*mu1^3*mu2^9*t1^3*t2^9 + 1174*mu1^2*mu2^10*t1^2*t2^10 + 88*mu1^10*mu2*t1^10*t2 - 5622*mu1^9*mu2^2*t1^9*t2^2 + 50512*mu1^8*mu2^3*t1^8*t2^3 - 181830*mu1^7*mu2^4*t1^7*t2^4 + 331344*mu1^6*mu2^5*t1^6*t2^5 - 331344*mu1^5*mu2^6*t1^5*t2^6 + 181830*mu1^4*mu2^7*t1^4*t2^7 - 50512*mu1^3*mu2^8*t1^3*t2^8 + 5622*mu1^2*mu2^9*t1^2*t2^9 - 88*mu1*mu2^10*t1*t2^10 + 22*mu1^10*t1^10 - 792*mu1^9*mu2*t1^9*t2 + 17248*mu1^8*mu2^2*t1^8*t2^2 - 104601*mu1^7*mu2^3*t1^7*t2^3 + 276906*mu1^6*mu2^4*t1^6*t2^4 - 377529*mu1^5*mu2^5*t1^5*t2^5 + 276906*mu1^4*mu2^6*t1^4*t2^6 - 104601*mu1^3*mu2^7*t1^3*t2^7 + 17248*mu1^2*mu2^8*t1^2*t2^8 - 792*mu1*mu2^9*t1*t2^9 + 22*mu2^10*t2^10 - 176*mu1^9*t1^9 + 3080*mu1^8*mu2*t1^8*t2 - 34538*mu1^7*mu2^2*t1^7*t2^2 + 146320*mu1^6*mu2^3*t1^6*t2^3 - 286431*mu1^5*mu2^4*t1^5*t2^4 + 286431*mu1^4*mu2^5*t1^4*t2^5 - 146320*mu1^3*mu2^6*t1^3*t2^6 + 34538*mu1^2*mu2^7*t1^2*t2^7 - 3080*mu1*mu2^8*t1*t2^8 + 176*mu2^9*t2^9 + 616*mu1^8*t1^8 - 6776*mu1^7*mu2*t1^7*t2 + 46204*mu1^6*mu2^2*t1^6*t2^2 - 139139*mu1^5*mu2^3*t1^5*t2^3 + 198227*mu1^4*mu2^4*t1^4*t2^4 - 139139*mu1^3*mu2^5*t1^3*t2^5 + 46204*mu1^2*mu2^6*t1^2*t2^6 - 6776*mu1*mu2^7*t1*t2^7 + 616*mu2^8*t2^8 - 1232*mu1^7*t1^7 + 9240*mu1^6*mu2*t1^6*t2 - 41314*mu1^5*mu2^2*t1^5*t2^2 + 88088*mu1^4*mu2^3*t1^4*t2^3 - 88088*mu1^3*mu2^4*t1^3*t2^4 + 41314*mu1^2*mu2^5*t1^2*t2^5 - 9240*mu1*mu2^6*t1*t2^6 + 1232*mu2^7*t2^7 + 1540*mu1^6*t1^6 - 8008*mu1^5*mu2*t1^5*t2 + 24024*mu1^4*mu2^2*t1^4*t2^2 - 35399*mu1^3*mu2^3*t1^3*t2^3 + 24024*mu1^2*mu2^4*t1^2*t2^4 - 8008*mu1*mu2^5*t1*t2^5 + 1540*mu2^6*t2^6 - 1232*mu1^5*t1^5 + 4312*mu1^4*mu2*t1^4*t2 - 8526*mu1^3*mu2^2*t1^3*t2^2 + 8526*mu1^2*mu2^3*t1^2*t2^3 - 4312*mu1*mu2^4*t1*t2^4 + 1232*mu2^5*t2^5 + 616*mu1^4*t1^4 - 1320*mu1^3*mu2*t1^3*t2 + 1658*mu1^2*mu2^2*t1^2*t2^2 - 1320*mu1*mu2^3*t1*t2^3 + 616*mu2^4*t2^4 - 176*mu1^3*t1^3 + 176*mu1^2*mu2*t1^2*t2 - 176*mu1*mu2^2*t1*t2^2 + 176*mu2^3*t2^3 + 22*mu1^2*t1^2 + 22*mu2^2*t2^2)