Code for computations used in the proof of Theorems 1 and 3

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

Computing the invariants of a quintic \(f\) in terms of its coefficients

# importing essential packages
from sage.rings.invariants.invariant_theory import AlgebraicForm, transvectant


# Making the universal coordinate ring

_.<a0,a1,a2,a3,a4,a5, mu1,mu2, t1,t2, x,y> = QQ[]



# a universal quintic in terms of coefficients. Check that H reduces to the discriminant.



def Reduction(form):

    return form(a0,a1,a2,a3,a4,a5, mu1,mu2, 0,0, x,y)


def Factor(n):

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


def InvariantsFromCoefficients():

    p = a0*x^5 + a1*x^4*y + a2*x^3*y^2 + a3*x^2*y^3 + a4*x*y^4 + a5*y^5

    qDelta = p(a0,a1,a2,a3,a4,a5, mu1,mu2, t1,t2, x,1)
    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

    q18_1, q18_2, q18_3 = p^5, p^6, p^7
    f18_1 = AlgebraicForm(2, 25, q18_1, x, y)
    f18_2 = AlgebraicForm(2, 30, q18_2, x, y)
    T18 = transvectant(f18_1, f18_2, 10, scale='none')
    f18_3 = AlgebraicForm(2, 35, q18_3, 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,
            22*H]


I4,I8,I12,I18,Delta,H = InvariantsFromCoefficients()


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")

I4:
-2*a2^2*a3^2 + 6*a1*a3^3 + 6*a2^3*a4 - 19*a1*a2*a3*a4 - 15*a0*a3^2*a4 + 9*a1^2*a4^2 + 40*a0*a2*a4^2 - 15*a1*a2^2*a5 + 40*a1^2*a3*a5 + 25*a0*a2*a3*a5 - 250*a0*a1*a4*a5 + 625*a0^2*a5^2



I8:
14*a2^4*a3^4 - 84*a1*a2^2*a3^5 + 126*a1^2*a3^6 - 84*a2^5*a3^2*a4 + 518*a1*a2^3*a3^3*a4 - 798*a1^2*a2*a3^4*a4 + 210*a0*a2^2*a3^4*a4 - 630*a0*a1*a3^5*a4 + 126*a2^6*a4^2 - 798*a1*a2^4*a3*a4^2 + 1143*a1^2*a2^2*a3^2*a4^2 - 1212*a0*a2^3*a3^2*a4^2 + 356*a1^3*a3^3*a4^2 + 3774*a0*a1*a2*a3^3*a4^2 + 639*a0^2*a3^4*a4^2 + 356*a1^2*a2^3*a4^3 + 1768*a0*a2^4*a4^3 - 1098*a1^3*a2*a3*a4^3 - 5760*a0*a1*a2^2*a3*a4^3 - 978*a0*a1^2*a3^2*a4^3 - 3408*a0^2*a2*a3^2*a4^3 + 135*a1^4*a4^4 + 3312*a0*a1^2*a2*a4^4 + 4896*a0^2*a2^2*a4^4 - 1056*a0^2*a1*a3*a4^4 + 1408*a0^3*a4^5 + 210*a1*a2^4*a3^2*a5 - 1212*a1^2*a2^2*a3^3*a5 - 262*a0*a2^3*a3^3*a5 + 1768*a1^3*a3^4*a5 + 654*a0*a1*a2*a3^4*a5 + 594*a0^2*a3^5*a5 - 630*a1*a2^5*a4*a5 + 3774*a1^2*a2^3*a3*a4*a5 + 654*a0*a2^4*a3*a4*a5 - 5760*a1^3*a2*a3^2*a4*a5 + 3708*a0*a1*a2^2*a3^2*a4*a5 - 14568*a0*a1^2*a3^3*a4*a5 - 6090*a0^2*a2*a3^3*a4*a5 - 978*a1^3*a2^2*a4^2*a5 - 14568*a0*a1*a2^3*a4^2*a5 + 3312*a1^4*a3*a4^2*a5 + 41922*a0*a1^2*a2*a3*a4^2*a5 + 10080*a0^2*a2^2*a3*a4^2*a5 + 31860*a0^2*a1*a3^2*a4^2*a5 - 15948*a0*a1^3*a4^3*a5 - 69120*a0^2*a1*a2*a4^3*a5 - 8800*a0^3*a3*a4^3*a5 + 639*a1^2*a2^4*a5^2 + 594*a0*a2^5*a5^2 - 3408*a1^3*a2^2*a3*a5^2 - 6090*a0*a1*a2^3*a3*a5^2 + 4896*a1^4*a3^2*a5^2 + 10080*a0*a1^2*a2*a3^2*a5^2 - 2025*a0^2*a2^2*a3^2*a5^2 + 21300*a0^2*a1*a3^3*a5^2 - 1056*a1^4*a2*a4*a5^2 + 31860*a0*a1^2*a2^2*a4*a5^2 + 21300*a0^2*a2^3*a4*a5^2 - 69120*a0*a1^3*a3*a4*a5^2 - 138150*a0^2*a1*a2*a3*a4*a5^2 - 53250*a0^3*a3^2*a4*a5^2 + 257850*a0^2*a1^2*a4^2*a5^2 + 186000*a0^3*a2*a4^2*a5^2 + 1408*a1^5*a5^3 - 8800*a0*a1^3*a2*a5^3 - 53250*a0^2*a1*a2^2*a5^3 + 186000*a0^2*a1^2*a3*a5^3 + 88750*a0^3*a2*a3*a5^3 - 1107500*a0^3*a1*a4*a5^3 + 1384375*a0^4*a5^4



I12:
-484*a2^6*a3^6 + 4356*a1*a2^4*a3^7 - 13101*a1^2*a2^2*a3^8 + 132*a0*a2^3*a3^8 + 13200*a1^3*a3^9 - 594*a0*a1*a2*a3^9 + 891*a0^2*a3^10 + 4356*a2^7*a3^4*a4 - 39930*a1*a2^5*a3^5*a4 + 122364*a1^2*a2^3*a3^6*a4 - 12474*a0*a2^4*a3^6*a4 - 125796*a1^3*a2*a3^7*a4 + 72732*a0*a1*a2^2*a3^7*a4 - 97812*a0*a1^2*a3^8*a4 - 11880*a0^2*a2*a3^8*a4 - 13101*a2^8*a3^2*a4^2 + 122364*a1*a2^6*a3^3*a4^2 - 376740*a1^2*a2^4*a3^4*a4^2 + 103008*a0*a2^5*a3^4*a4^2 + 367836*a1^3*a2^2*a3^5*a4^2 - 622872*a0*a1*a2^3*a3^5*a4^2 + 52260*a1^4*a3^6*a4^2 + 897192*a0*a1^2*a2*a3^6*a4^2 - 9708*a0^2*a2^2*a3^6*a4^2 + 219204*a0^2*a1*a3^7*a4^2 + 13200*a2^9*a4^3 - 125796*a1*a2^7*a3*a4^3 + 367836*a1^2*a2^5*a3^2*a4^3 - 297996*a0*a2^6*a3^2*a4^3 - 224600*a1^3*a2^3*a3^3*a4^3 + 1857900*a0*a1*a2^4*a3^3*a4^3 - 331140*a1^4*a2*a3^4*a4^3 - 2686200*a0*a1^2*a2^2*a3^4*a4^3 + 453720*a0^2*a2^3*a3^4*a4^3 - 260820*a0*a1^3*a3^5*a4^3 - 1925244*a0^2*a1*a2*a3^5*a4^3 - 120660*a0^3*a3^6*a4^3 + 52260*a1^2*a2^6*a4^4 + 294144*a0*a2^7*a4^4 - 331140*a1^3*a2^4*a3*a4^4 - 1896528*a0*a1*a2^5*a3*a4^4 + 542100*a1^4*a2^2*a3^2*a4^4 + 2332080*a0*a1^2*a2^3*a3^2*a4^4 - 1587600*a0^2*a2^4*a3^2*a4^4 + 34584*a1^5*a3^3*a4^4 + 1699800*a0*a1^3*a2*a3^3*a4^4 + 5895840*a0^2*a1*a2^2*a3^3*a4^4 + 58500*a0^2*a1^2*a3^4*a4^4 + 965280*a0^3*a2*a3^4*a4^4 + 34584*a1^4*a2^3*a4^5 + 1047888*a0*a1^2*a2^4*a4^5 + 1697280*a0^2*a2^5*a4^5 - 156924*a1^5*a2*a3*a4^5 - 3078816*a0*a1^3*a2^2*a3*a4^5 - 6379200*a0^2*a1*a2^3*a3*a4^5 + 55764*a0*a1^4*a3^2*a4^5 - 1185408*a0^2*a1^2*a2*a3^2*a4^5 - 2865216*a0^3*a2^2*a3^2*a4^5 + 873408*a0^3*a1*a3^3*a4^5 + 7452*a1^6*a4^6 + 538272*a0*a1^4*a2*a4^6 + 4004544*a0^2*a1^2*a2^2*a4^6 + 3166208*a0^3*a2^3*a4^6 - 1030464*a0^2*a1^3*a3*a4^6 - 2787072*a0^3*a1*a2*a3*a4^6 - 529920*a0^4*a3^2*a4^6 + 1373952*a0^3*a1^2*a4^7 + 1413120*a0^4*a2*a4^7 + 132*a2^8*a3^3*a5 - 12474*a1*a2^6*a3^4*a5 + 103008*a1^2*a2^4*a3^5*a5 + 12150*a0*a2^5*a3^5*a5 - 297996*a1^3*a2^2*a3^6*a5 - 59940*a0*a1*a2^3*a3^6*a5 + 294144*a1^4*a3^7*a5 + 63804*a0*a1^2*a2*a3^7*a5 - 39708*a0^2*a2^2*a3^7*a5 + 148824*a0^2*a1*a3^8*a5 - 594*a2^9*a3*a4*a5 + 72732*a1*a2^7*a3^2*a4*a5 - 622872*a1^2*a2^5*a3^3*a4*a5 - 59940*a0*a2^6*a3^3*a4*a5 + 1857900*a1^3*a2^3*a3^4*a4*a5 - 97080*a0*a1*a2^4*a3^4*a4*a5 - 1896528*a1^4*a2*a3^5*a4*a5 + 1924668*a0*a1^2*a2^2*a3^5*a4*a5 + 566784*a0^2*a2^3*a3^5*a4*a5 - 3069048*a0*a1^3*a3^6*a4*a5 - 1946076*a0^2*a1*a2*a3^6*a4*a5 - 372060*a0^3*a3^7*a4*a5 - 97812*a1*a2^8*a4^2*a5 + 897192*a1^2*a2^6*a3*a4^2*a5 + 63804*a0*a2^7*a3*a4^2*a5 - 2686200*a1^3*a2^4*a3^2*a4^2*a5 + 1924668*a0*a1*a2^5*a3^2*a4^2*a5 + 2332080*a1^4*a2^2*a3^3*a4^2*a5 - 12779640*a0*a1^2*a2^3*a3^3*a4^2*a5 - 2033340*a0^2*a2^4*a3^3*a4^2*a5 + 1047888*a1^5*a3^4*a4^2*a5 + 18426300*a0*a1^3*a2*a3^4*a4^2*a5 + 3129300*a0^2*a1*a2^2*a3^4*a4^2*a5 + 11145276*a0^2*a1^2*a3^5*a4^2*a5 + 4075860*a0^3*a2*a3^5*a4^2*a5 - 260820*a1^3*a2^5*a4^3*a5 - 3069048*a0*a1*a2^6*a4^3*a5 + 1699800*a1^4*a2^3*a3*a4^3*a5 + 18426300*a0*a1^2*a2^4*a3*a4^3*a5 + 2035440*a0^2*a2^5*a3*a4^3*a5 - 3078816*a1^5*a2*a3^2*a4^3*a5 - 23337000*a0*a1^3*a2^2*a3^2*a4^3*a5 + 12810240*a0^2*a1*a2^3*a3^2*a4^3*a5 - 8221440*a0*a1^4*a3^3*a4^3*a5 - 61511640*a0^2*a1^2*a2*a3^3*a4^3*a5 - 12410400*a0^3*a2^2*a3^3*a4^3*a5 - 13180200*a0^3*a1*a3^4*a4^3*a5 + 55764*a1^5*a2^2*a4^4*a5 - 8221440*a0*a1^3*a2^3*a4^4*a5 - 25286880*a0^2*a1*a2^4*a4^4*a5 + 538272*a1^6*a3*a4^4*a5 + 21759780*a0*a1^4*a2*a3*a4^4*a5 + 81790560*a0^2*a1^2*a2^2*a3*a4^4*a5 + 11092800*a0^3*a2^3*a3*a4^4*a5 + 15991920*a0^2*a1^3*a3^2*a4^4*a5 + 60360000*a0^3*a1*a2*a3^2*a4^4*a5 + 1992000*a0^4*a3^3*a4^4*a5 - 2376648*a0*a1^5*a4^5*a5 - 39710400*a0^2*a1^3*a2*a4^5*a5 - 69678720*a0^3*a1*a2^2*a4^5*a5 + 7369920*a0^3*a1^2*a3*a4^5*a5 - 2668800*a0^4*a2*a3*a4^5*a5 - 19392000*a0^4*a1*a4^6*a5 + 891*a2^10*a5^2 - 11880*a1*a2^8*a3*a5^2 - 9708*a1^2*a2^6*a3^2*a5^2 - 39708*a0*a2^7*a3^2*a5^2 + 453720*a1^3*a2^4*a3^3*a5^2 + 566784*a0*a1*a2^5*a3^3*a5^2 - 1587600*a1^4*a2^2*a3^4*a5^2 - 2033340*a0*a1^2*a2^3*a3^4*a5^2 + 58500*a0^2*a2^4*a3^4*a5^2 + 1697280*a1^5*a3^5*a5^2 + 2035440*a0*a1^3*a2*a3^5*a5^2 - 1767960*a0^2*a1*a2^2*a3^5*a5^2 + 4619460*a0^2*a1^2*a3^6*a5^2 + 620100*a0^3*a2*a3^6*a5^2 + 219204*a1^2*a2^7*a4*a5^2 + 148824*a0*a2^8*a4*a5^2 - 1925244*a1^3*a2^5*a3*a4*a5^2 - 1946076*a0*a1*a2^6*a3*a4*a5^2 + 5895840*a1^4*a2^3*a3^2*a4*a5^2 + 3129300*a0*a1^2*a2^4*a3^2*a4*a5^2 - 1767960*a0^2*a2^5*a3^2*a4*a5^2 - 6379200*a1^5*a2*a3^3*a4*a5^2 + 12810240*a0*a1^3*a2^2*a3^3*a4*a5^2 + 22958400*a0^2*a1*a2^3*a3^3*a4*a5^2 - 25286880*a0*a1^4*a3^4*a4*a5^2 - 49769100*a0^2*a1^2*a2*a3^4*a4*a5^2 - 1161000*a0^3*a2^2*a3^4*a4*a5^2 - 24337500*a0^3*a1*a3^5*a4*a5^2 + 58500*a1^4*a2^4*a4^2*a5^2 + 11145276*a0*a1^2*a2^5*a4^2*a5^2 + 4619460*a0^2*a2^6*a4^2*a5^2 - 1185408*a1^5*a2^2*a3*a4^2*a5^2 - 61511640*a0*a1^3*a2^3*a3*a4^2*a5^2 - 49769100*a0^2*a1*a2^4*a3*a4^2*a5^2 + 4004544*a1^6*a3^2*a4^2*a5^2 + 81790560*a0*a1^4*a2*a3^2*a4^2*a5^2 + 47806200*a0^2*a1^2*a2^2*a3^2*a4^2*a5^2 - 20550000*a0^3*a2^3*a3^2*a4^2*a5^2 + 141270000*a0^2*a1^3*a3^3*a4^2*a5^2 + 194841000*a0^3*a1*a2*a3^3*a4^2*a5^2 + 24982500*a0^4*a3^4*a4^2*a5^2 - 1030464*a1^6*a2*a4^3*a5^2 + 15991920*a0*a1^4*a2^2*a4^3*a5^2 + 141270000*a0^2*a1^2*a2^3*a4^3*a5^2 + 46578000*a0^3*a2^4*a4^3*a5^2 - 39710400*a0*a1^5*a3*a4^3*a5^2 - 394183800*a0^2*a1^3*a2*a3*a4^3*a5^2 - 308196000*a0^3*a1*a2^2*a3*a4^3*a5^2 - 277509000*a0^3*a1^2*a3^2*a4^3*a5^2 - 146520000*a0^4*a2*a3^2*a4^3*a5^2 + 109128900*a0^2*a1^4*a4^4*a5^2 + 565488000*a0^3*a1^2*a2*a4^4*a5^2 + 178200000*a0^4*a2^2*a4^4*a5^2 + 84840000*a0^4*a1*a3*a4^4*a5^2 + 48480000*a0^5*a4^5*a5^2 - 120660*a1^3*a2^6*a5^3 - 372060*a0*a1*a2^7*a5^3 + 965280*a1^4*a2^4*a3*a5^3 + 4075860*a0*a1^2*a2^5*a3*a5^3 + 620100*a0^2*a2^6*a3*a5^3 - 2865216*a1^5*a2^2*a3^2*a5^3 - 12410400*a0*a1^3*a2^3*a3^2*a5^3 - 1161000*a0^2*a1*a2^4*a3^2*a5^3 + 3166208*a1^6*a3^3*a5^3 + 11092800*a0*a1^4*a2*a3^3*a5^3 - 20550000*a0^2*a1^2*a2^2*a3^3*a5^3 - 6220000*a0^3*a2^3*a3^3*a5^3 + 46578000*a0^2*a1^3*a3^4*a5^3 + 30892500*a0^3*a1*a2*a3^4*a5^3 + 7252500*a0^4*a3^5*a5^3 + 873408*a1^5*a2^3*a4*a5^3 - 13180200*a0*a1^3*a2^4*a4*a5^3 - 24337500*a0^2*a1*a2^5*a4*a5^3 - 2787072*a1^6*a2*a3*a4*a5^3 + 60360000*a0*a1^4*a2^2*a3*a4*a5^3 + 194841000*a0^2*a1^2*a2^3*a3*a4*a5^3 + 30892500*a0^3*a2^4*a3*a4*a5^3 - 69678720*a0*a1^5*a3^2*a4*a5^3 - 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I18:
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8000000000*a0^5*a1^3*a3^4*a5^6 + 30000000000*a0^6*a1*a2*a3^4*a5^6 - 10000000000*a0^7*a3^5*a5^6 + 20480000*a1^11*a4*a5^6 - 102400000*a0*a1^9*a2*a4*a5^6 - 256000000*a0^2*a1^7*a2^2*a4*a5^6 + 2560000000*a0^3*a1^5*a2^3*a4*a5^6 - 5600000000*a0^4*a1^3*a2^4*a4*a5^6 + 4000000000*a0^5*a1*a2^5*a4*a5^6 - 768000000*a0^2*a1^8*a3*a4*a5^6 + 7680000000*a0^3*a1^6*a2*a3*a4*a5^6 - 28800000000*a0^4*a1^4*a2^2*a3*a4*a5^6 + 48000000000*a0^5*a1^2*a2^3*a3*a4*a5^6 - 30000000000*a0^6*a2^4*a3*a4*a5^6 - 102400000*a0*a1^10*a5^7 + 1280000000*a0^2*a1^8*a2*a5^7 - 6400000000*a0^3*a1^6*a2^2*a5^7 + 16000000000*a0^4*a1^4*a2^3*a5^7 - 20000000000*a0^5*a1^2*a2^4*a5^7 + 10000000000*a0^6*a2^5*a5^7



Delta:
a1^2*a2^2*a3^2*a4^2 - 4*a0*a2^3*a3^2*a4^2 - 4*a1^3*a3^3*a4^2 + 18*a0*a1*a2*a3^3*a4^2 - 27*a0^2*a3^4*a4^2 - 4*a1^2*a2^3*a4^3 + 16*a0*a2^4*a4^3 + 18*a1^3*a2*a3*a4^3 - 80*a0*a1*a2^2*a3*a4^3 - 6*a0*a1^2*a3^2*a4^3 + 144*a0^2*a2*a3^2*a4^3 - 27*a1^4*a4^4 + 144*a0*a1^2*a2*a4^4 - 128*a0^2*a2^2*a4^4 - 192*a0^2*a1*a3*a4^4 + 256*a0^3*a4^5 - 4*a1^2*a2^2*a3^3*a5 + 16*a0*a2^3*a3^3*a5 + 16*a1^3*a3^4*a5 - 72*a0*a1*a2*a3^4*a5 + 108*a0^2*a3^5*a5 + 18*a1^2*a2^3*a3*a4*a5 - 72*a0*a2^4*a3*a4*a5 - 80*a1^3*a2*a3^2*a4*a5 + 356*a0*a1*a2^2*a3^2*a4*a5 + 24*a0*a1^2*a3^3*a4*a5 - 630*a0^2*a2*a3^3*a4*a5 - 6*a1^3*a2^2*a4^2*a5 + 24*a0*a1*a2^3*a4^2*a5 + 144*a1^4*a3*a4^2*a5 - 746*a0*a1^2*a2*a3*a4^2*a5 + 560*a0^2*a2^2*a3*a4^2*a5 + 1020*a0^2*a1*a3^2*a4^2*a5 - 36*a0*a1^3*a4^3*a5 + 160*a0^2*a1*a2*a4^3*a5 - 1600*a0^3*a3*a4^3*a5 - 27*a1^2*a2^4*a5^2 + 108*a0*a2^5*a5^2 + 144*a1^3*a2^2*a3*a5^2 - 630*a0*a1*a2^3*a3*a5^2 - 128*a1^4*a3^2*a5^2 + 560*a0*a1^2*a2*a3^2*a5^2 + 825*a0^2*a2^2*a3^2*a5^2 - 900*a0^2*a1*a3^3*a5^2 - 192*a1^4*a2*a4*a5^2 + 1020*a0*a1^2*a2^2*a4*a5^2 - 900*a0^2*a2^3*a4*a5^2 + 160*a0*a1^3*a3*a4*a5^2 - 2050*a0^2*a1*a2*a3*a4*a5^2 + 2250*a0^3*a3^2*a4*a5^2 - 50*a0^2*a1^2*a4^2*a5^2 + 2000*a0^3*a2*a4^2*a5^2 + 256*a1^5*a5^3 - 1600*a0*a1^3*a2*a5^3 + 2250*a0^2*a1*a2^2*a5^3 + 2000*a0^2*a1^2*a3*a5^3 - 3750*a0^3*a2*a3*a5^3 - 2500*a0^3*a1*a4*a5^3 + 3125*a0^4*a5^4



H:
-22*a1^2*a2^2*a3^8 + 88*a0*a2^3*a3^8 + 88*a1^3*a3^9 - 396*a0*a1*a2*a3^9 + 594*a0^2*a3^10 + 264*a1^2*a2^3*a3^6*a4 - 1056*a0*a2^4*a3^6*a4 - 1100*a1^3*a2*a3^7*a4 + 4928*a0*a1*a2^2*a3^7*a4 + 132*a0*a1^2*a3^8*a4 - 7920*a0^2*a2*a3^8*a4 - 22*a2^8*a3^2*a4^2 + 264*a1*a2^6*a3^3*a4^2 - 2626*a1^2*a2^4*a3^4*a4^2 + 5752*a0*a2^5*a3^4*a4^2 + 9274*a1^3*a2^2*a3^5*a4^2 - 30468*a0*a1*a2^3*a3^5*a4^2 - 4364*a1^4*a3^6*a4^2 + 10068*a0*a1^2*a2*a3^6*a4^2 + 47978*a0^2*a2^2*a3^6*a4^2 - 17214*a0^2*a1*a3^7*a4^2 + 88*a2^9*a4^3 - 1100*a1*a2^7*a3*a4^3 + 9274*a1^2*a2^5*a3^2*a4^3 - 17164*a0*a2^6*a3^2*a4^3 - 34259*a1^3*a2^3*a3^3*a4^3 + 108460*a0*a1*a2^4*a3^3*a4^3 + 27532*a1^4*a2*a3^4*a4^3 - 97405*a0*a1^2*a2^2*a3^4*a4^3 - 151270*a0^2*a2^3*a3^4*a4^3 + 22140*a0*a1^3*a3^5*a4^3 + 104979*a0^2*a1*a2*a3^5*a4^3 + 55685*a0^3*a3^6*a4^3 - 4364*a1^2*a2^6*a4^4 + 21856*a0*a2^7*a4^4 + 27532*a1^3*a2^4*a3*a4^4 - 160832*a0*a1*a2^5*a3*a4^4 - 12127*a1^4*a2^2*a3^2*a4^4 + 177740*a0*a1^2*a2^3*a3^2*a4^4 + 200000*a0^2*a2^4*a3^2*a4^4 - 35750*a1^5*a3^3*a4^4 - 10250*a0*a1^3*a2*a3^3*a4^4 + 10160*a0^2*a1*a2^2*a3^3*a4^4 - 206025*a0^2*a1^2*a3^4*a4^4 - 445480*a0^3*a2*a3^4*a4^4 - 35750*a1^4*a2^3*a4^5 + 175872*a0*a1^2*a2^4*a4^5 - 30080*a0^2*a2^5*a4^5 + 81603*a1^5*a2*a3*a4^5 - 397264*a0*a1^3*a2^2*a3*a4^5 - 574400*a0^2*a1*a2^3*a3*a4^5 + 184191*a0*a1^4*a3^2*a4^5 + 516528*a0^2*a1^2*a2*a3^2*a4^5 + 993856*a0^3*a2^2*a3^2*a4^5 + 582272*a0^3*a1*a3^3*a4^5 - 24435*a1^6*a4^6 - 33192*a0*a1^4*a2*a4^6 + 927296*a0^2*a1^2*a2^2*a4^6 - 470528*a0^3*a2^3*a4^6 - 686976*a0^2*a1^3*a3*a4^6 - 1858048*a0^3*a1*a2*a3*a4^6 - 353280*a0^4*a3^2*a4^6 + 915968*a0^3*a1^2*a4^7 + 942080*a0^4*a2*a4^7 + 88*a2^8*a3^3*a5 - 1056*a1*a2^6*a3^4*a5 + 5752*a1^2*a2^4*a3^5*a5 - 4000*a0*a2^5*a3^5*a5 - 17164*a1^3*a2^2*a3^6*a5 + 32640*a0*a1*a2^3*a3^6*a5 + 21856*a1^4*a3^7*a5 - 66364*a0*a1^2*a2*a3^7*a5 - 26472*a0^2*a2^2*a3^7*a5 + 99216*a0^2*a1*a3^8*a5 - 396*a2^9*a3*a4*a5 + 4928*a1*a2^7*a3^2*a4*a5 - 30468*a1^2*a2^5*a3^3*a4*a5 + 32640*a0*a2^6*a3^3*a4*a5 + 108460*a1^3*a2^3*a3^4*a4*a5 - 282520*a0*a1*a2^4*a3^4*a4*a5 - 160832*a1^4*a2*a3^5*a4*a5 + 629712*a0*a1^2*a2^2*a3^5*a4*a5 + 196356*a0^2*a2^3*a3^5*a4*a5 - 85832*a0*a1^3*a3^6*a4*a5 - 752884*a0^2*a1*a2*a3^6*a4*a5 - 248040*a0^3*a3^7*a4*a5 + 132*a1*a2^8*a4^2*a5 + 10068*a1^2*a2^6*a3*a4^2*a5 - 66364*a0*a2^7*a3*a4^2*a5 - 97405*a1^3*a2^4*a3^2*a4^2*a5 + 629712*a0*a1*a2^5*a3^2*a4^2*a5 + 177740*a1^4*a2^2*a3^3*a4^2*a5 - 1474535*a0*a1^2*a2^3*a3^3*a4^2*a5 - 327060*a0^2*a2^4*a3^3*a4^2*a5 + 175872*a1^5*a3^4*a4^2*a5 - 21500*a0*a1^3*a2*a3^4*a4^2*a5 + 1133325*a0^2*a1*a2^2*a3^4*a4^2*a5 + 896184*a0^2*a1^2*a3^5*a4^2*a5 + 2036615*a0^3*a2*a3^5*a4^2*a5 + 22140*a1^3*a2^5*a4^3*a5 - 85832*a0*a1*a2^6*a4^3*a5 - 10250*a1^4*a2^3*a3*a4^3*a5 - 21500*a0*a1^2*a2^4*a3*a4^3*a5 - 95040*a0^2*a2^5*a3*a4^3*a5 - 397264*a1^5*a2*a3^2*a4^3*a5 + 2174550*a0*a1^3*a2^2*a3^2*a4^3*a5 + 675160*a0^2*a1*a2^3*a3^2*a4^3*a5 - 907160*a0*a1^4*a3^3*a4^3*a5 - 2620510*a0^2*a1^2*a2*a3^3*a4^3*a5 - 4643600*a0^3*a2^2*a3^3*a4^3*a5 - 1980550*a0^3*a1*a3^4*a4^3*a5 + 184191*a1^5*a2^2*a4^4*a5 - 907160*a0*a1^3*a2^3*a4^4*a5 + 566080*a0^2*a1*a2^4*a4^4*a5 - 33192*a1^6*a3*a4^4*a5 + 431195*a0*a1^4*a2*a3*a4^4*a5 - 1374960*a0^2*a1^2*a2^2*a3*a4^4*a5 + 2555200*a0^3*a2^3*a3*a4^4*a5 + 2493780*a0^2*a1^3*a3^2*a4^4*a5 + 3940000*a0^3*a1*a2*a3^2*a4^4*a5 + 1328000*a0^4*a3^3*a4^4*a5 + 865818*a0*a1^5*a4^5*a5 - 4693600*a0^2*a1^3*a2*a4^5*a5 + 1947520*a0^3*a1*a2^2*a4^5*a5 + 4913280*a0^3*a1^2*a3*a4^5*a5 - 1779200*a0^4*a2*a3*a4^5*a5 - 12928000*a0^4*a1*a4^6*a5 + 594*a2^10*a5^2 - 7920*a1*a2^8*a3*a5^2 + 47978*a1^2*a2^6*a3^2*a5^2 - 26472*a0*a2^7*a3^2*a5^2 - 151270*a1^3*a2^4*a3^3*a5^2 + 196356*a0*a1*a2^5*a3^3*a5^2 + 200000*a1^4*a2^2*a3^4*a5^2 - 327060*a0*a1^2*a2^3*a3^4*a5^2 - 112250*a0^2*a2^4*a3^4*a5^2 - 30080*a1^5*a3^5*a5^2 - 95040*a0*a1^3*a2*a3^5*a5^2 + 182610*a0^2*a1*a2^2*a3^5*a5^2 + 357140*a0^2*a1^2*a3^6*a5^2 + 413400*a0^3*a2*a3^6*a5^2 - 17214*a1^2*a2^7*a4*a5^2 + 99216*a0*a2^8*a4*a5^2 + 104979*a1^3*a2^5*a3*a4*a5^2 - 752884*a0*a1*a2^6*a3*a4*a5^2 + 10160*a1^4*a2^3*a3^2*a4*a5^2 + 1133325*a0*a1^2*a2^4*a3^2*a4*a5^2 + 182610*a0^2*a2^5*a3^2*a4*a5^2 - 574400*a1^5*a2*a3^3*a4*a5^2 + 675160*a0*a1^3*a2^2*a3^3*a4*a5^2 + 1163725*a0^2*a1*a2^3*a3^3*a4*a5^2 + 566080*a0*a1^4*a3^4*a4*a5^2 - 3231900*a0^2*a1^2*a2*a3^4*a4*a5^2 - 4177125*a0^3*a2^2*a3^4*a4*a5^2 - 2612500*a0^3*a1*a3^5*a4*a5^2 - 206025*a1^4*a2^4*a4^2*a5^2 + 896184*a0*a1^2*a2^5*a4^2*a5^2 + 357140*a0^2*a2^6*a4^2*a5^2 + 516528*a1^5*a2^2*a3*a4^2*a5^2 - 2620510*a0*a1^3*a2^3*a3*a4^2*a5^2 - 3231900*a0^2*a1*a2^4*a3*a4^2*a5^2 + 927296*a1^6*a3^2*a4^2*a5^2 - 1374960*a0*a1^4*a2*a3^2*a4^2*a5^2 - 3067950*a0^2*a1^2*a2^2*a3^2*a4^2*a5^2 + 8987500*a0^3*a2^3*a3^2*a4^2*a5^2 + 4337500*a0^2*a1^3*a3^3*a4^2*a5^2 + 27800250*a0^3*a1*a2*a3^3*a4^2*a5^2 - 360625*a0^4*a3^4*a4^2*a5^2 - 686976*a1^6*a2*a4^3*a5^2 + 2493780*a0*a1^4*a2^2*a4^3*a5^2 + 4337500*a0^2*a1^2*a2^3*a4^3*a5^2 - 5248000*a0^3*a2^4*a4^3*a5^2 - 4693600*a0*a1^5*a3*a4^3*a5^2 + 17174550*a0^2*a1^3*a2*a3*a4^3*a5^2 - 30014000*a0^3*a1*a2^2*a3*a4^3*a5^2 - 51149750*a0^3*a1^2*a3^2*a4^3*a5^2 - 6930000*a0^4*a2*a3^2*a4^3*a5^2 - 1435525*a0^2*a1^4*a4^4*a5^2 + 20042000*a0^3*a1^2*a2*a4^4*a5^2 - 2200000*a0^4*a2^2*a4^4*a5^2 + 56560000*a0^4*a1*a3*a4^4*a5^2 + 32320000*a0^5*a4^5*a5^2 + 55685*a1^3*a2^6*a5^3 - 248040*a0*a1*a2^7*a5^3 - 445480*a1^4*a2^4*a3*a5^3 + 2036615*a0*a1^2*a2^5*a3*a5^3 + 413400*a0^2*a2^6*a3*a5^3 + 993856*a1^5*a2^2*a3^2*a5^3 - 4643600*a0*a1^3*a2^3*a3^2*a5^3 - 4177125*a0^2*a1*a2^4*a3^2*a5^3 - 470528*a1^6*a3^3*a5^3 + 2555200*a0*a1^4*a2*a3^3*a5^3 + 8987500*a0^2*a1^2*a2^2*a3^3*a5^3 + 2785625*a0^3*a2^3*a3^3*a5^3 - 5248000*a0^2*a1^3*a3^4*a5^3 - 2092500*a0^3*a1*a2*a3^4*a5^3 + 4835000*a0^4*a3^5*a5^3 + 582272*a1^5*a2^3*a4*a5^3 - 1980550*a0*a1^3*a2^4*a4*a5^3 - 2612500*a0^2*a1*a2^5*a4*a5^3 - 1858048*a1^6*a2*a3*a4*a5^3 + 3940000*a0*a1^4*a2^2*a3*a4*a5^3 + 27800250*a0^2*a1^2*a2^3*a3*a4*a5^3 - 2092500*a0^3*a2^4*a3*a4*a5^3 + 1947520*a0*a1^5*a3^2*a4*a5^3 - 30014000*a0^2*a1^3*a2*a3^2*a4*a5^3 - 49096250*a0^3*a1*a2^2*a3^2*a4*a5^3 + 43545000*a0^3*a1^2*a3^3*a4*a5^3 - 31031250*a0^4*a2*a3^3*a4*a5^3 + 915968*a1^7*a4^2*a5^3 + 4913280*a0*a1^5*a2*a4^2*a5^3 - 51149750*a0^2*a1^3*a2^2*a4^2*a5^3 + 43545000*a0^3*a1*a2^3*a4^2*a5^3 + 20042000*a0^2*a1^4*a3*a4^2*a5^3 + 43018750*a0^3*a1^2*a2*a3*a4^2*a5^3 + 69750000*a0^4*a2^2*a3*a4^2*a5^3 + 28912500*a0^4*a1*a3^2*a4^2*a5^3 - 17152500*a0^3*a1^3*a4^3*a5^3 - 117500000*a0^4*a1*a2*a4^3*a5^3 - 202000000*a0^5*a3*a4^3*a5^3 - 353280*a1^6*a2^2*a5^4 + 1328000*a0*a1^4*a2^3*a5^4 - 360625*a0^2*a1^2*a2^4*a5^4 + 4835000*a0^3*a2^5*a5^4 + 942080*a1^7*a3*a5^4 - 1779200*a0*a1^5*a2*a3*a5^4 - 6930000*a0^2*a1^3*a2^2*a3*a5^4 - 31031250*a0^3*a1*a2^3*a3*a5^4 - 2200000*a0^2*a1^4*a3^2*a5^4 + 69750000*a0^3*a1^2*a2*a3^2*a5^4 + 61440625*a0^4*a2^2*a3^2*a5^4 - 106743750*a0^4*a1*a3^3*a5^4 - 12928000*a0*a1^6*a4*a5^4 + 56560000*a0^2*a1^4*a2*a4*a5^4 + 28912500*a0^3*a1^2*a2^2*a4*a5^4 - 106743750*a0^4*a2^3*a4*a5^4 - 117500000*a0^3*a1^3*a3*a4*a5^4 - 114165625*a0^4*a1*a2*a3*a4*a5^4 + 266859375*a0^5*a3^2*a4*a5^4 + 181821875*a0^4*a1^2*a4^2*a5^4 + 298375000*a0^5*a2*a4^2*a5^4 + 32320000*a0^2*a1^5*a5^5 - 202000000*a0^3*a1^3*a2*a5^5 + 266859375*a0^4*a1*a2^2*a5^5 + 298375000*a0^4*a1^2*a3*a5^5 - 444765625*a0^5*a2*a3*a5^5 - 602343750*a0^5*a1*a4*a5^5 + 501953125*a0^6*a5^6

Using the \(\mathrm{GL}_2\) action we can assume that three of the five distinct roots of the quintic are \(0\) , \(1\) and \(\infty\). So the we may assume that our quintic is of the form

\[\Large{f(x,y) = xy(x-y) { \color{red} {(x-\lambda_1 y) (x - \lambda_2 y)}}}\]

Functions to compute the invariants in terms of the two roots \(\lambda_1\) and \(\lambda_2\)

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

# importing essential packages
from sage.rings.invariants.invariant_theory import AlgebraicForm, transvectant


# Making the universal coordinate ring

_.<a0,a1,a2,a3,a4,a5, mu1,mu2, t1,t2, x,y> = QQ[]


# a universal quintic in terms of coefficients to check that H reduces to the discriminant.

#p = a0*x^5 + a1*x^4*y + a2*x^3*y^2 + a3*x^2*y^3 + a4*x*y^4 + a5*y^5


def InvariantsFromRoots(lambda1,lambda2):

    p = x * (x - y) * y * (x - lambda1*y) * (x - lambda2*y)

    qDelta = p(a0,a1,a2,a3,a4,a5, mu1,mu2, t1,t2, x,1)
    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

    q18_1, q18_2, q18_3 = p^5, p^6, p^7
    f18_1 = AlgebraicForm(2, 25, q18_1, x, y)
    f18_2 = AlgebraicForm(2, 30, q18_2, x, y)
    T18 = transvectant(f18_1, f18_2, 10, scale='none')
    f18_3 = AlgebraicForm(2, 35, q18_3, 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,
            22*H]

Unmarked tree type I

In this case, we have

1. \(v(\lambda_i) = 0\) for \(i = 1,2\),

2. \(v(\lambda_i - 1) = 0\) for \(i = 1,2\), and

3. \(v(\lambda_1 - \lambda_2) = 0\).

lambda1 = mu1
lambda2 = mu2

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



################################# Computing the reduction of the

I4_red    = Reduction(I4)
I8_red    = Reduction(I8)
I12_red   = Reduction(I12)
I18_red   = Reduction(I18)
Delta_red = Reduction(Delta)
H_red     = Reduction(H)



print("I4_red: \n")
print(Factor(I4_red))
print("\n\n\n")

print("I8_red: \n")
print(Factor(I8_red))
print("\n\n\n")

print("I12_red: \n")
print(Factor(I12_red))
print("\n\n\n")


print("I18_red: \n")
print(Factor(I18_red))
print("\n\n\n")

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


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

I4_red:

(-1) * (2*mu1^4*mu2^2 - 2*mu1^3*mu2^3 + 2*mu1^2*mu2^4 - 2*mu1^4*mu2 - mu1^3*mu2^2 - mu1^2*mu2^3 - 2*mu1*mu2^4 + 2*mu1^4 - mu1^3*mu2 + 6*mu1^2*mu2^2 - mu1*mu2^3 + 2*mu2^4 - 2*mu1^3 - mu1^2*mu2 - mu1*mu2^2 - 2*mu2^3 + 2*mu1^2 - 2*mu1*mu2 + 2*mu2^2)




I8_red:

14*mu1^8*mu2^4 - 28*mu1^7*mu2^5 + 42*mu1^6*mu2^6 - 28*mu1^5*mu2^7 + 14*mu1^4*mu2^8 - 28*mu1^8*mu2^3 + 14*mu1^7*mu2^4 - 28*mu1^6*mu2^5 - 28*mu1^5*mu2^6 + 14*mu1^4*mu2^7 - 28*mu1^3*mu2^8 + 42*mu1^8*mu2^2 - 28*mu1^7*mu2^3 + 149*mu1^6*mu2^4 - 88*mu1^5*mu2^5 + 149*mu1^4*mu2^6 - 28*mu1^3*mu2^7 + 42*mu1^2*mu2^8 - 28*mu1^8*mu2 - 28*mu1^7*mu2^2 - 88*mu1^6*mu2^3 - 66*mu1^5*mu2^4 - 66*mu1^4*mu2^5 - 88*mu1^3*mu2^6 - 28*mu1^2*mu2^7 - 28*mu1*mu2^8 + 14*mu1^8 + 14*mu1^7*mu2 + 149*mu1^6*mu2^2 - 66*mu1^5*mu2^3 + 282*mu1^4*mu2^4 - 66*mu1^3*mu2^5 + 149*mu1^2*mu2^6 + 14*mu1*mu2^7 + 14*mu2^8 - 28*mu1^7 - 28*mu1^6*mu2 - 88*mu1^5*mu2^2 - 66*mu1^4*mu2^3 - 66*mu1^3*mu2^4 - 88*mu1^2*mu2^5 - 28*mu1*mu2^6 - 28*mu2^7 + 42*mu1^6 - 28*mu1^5*mu2 + 149*mu1^4*mu2^2 - 88*mu1^3*mu2^3 + 149*mu1^2*mu2^4 - 28*mu1*mu2^5 + 42*mu2^6 - 28*mu1^5 + 14*mu1^4*mu2 - 28*mu1^3*mu2^2 - 28*mu1^2*mu2^3 + 14*mu1*mu2^4 - 28*mu2^5 + 14*mu1^4 - 28*mu1^3*mu2 + 42*mu1^2*mu2^2 - 28*mu1*mu2^3 + 14*mu2^4




I12_red:

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




I18_red:

(-1) * (-mu1 + mu2 - 1) * (-mu1 + mu2 + 1) * (mu1 + mu2 - 1) * (mu2^2 - mu1) * (mu2^2 + mu1 - 2*mu2) * (-2*mu1*mu2 + mu2^2 + mu1) * (mu1*mu2 - 1) * (mu1*mu2 - 2*mu2 + 1) * (mu1*mu2 - 2*mu1 + 1) * (mu1*mu2 - mu1 - mu2) * (mu1*mu2 - mu1 + mu2) * (mu1*mu2 + mu1 - mu2) * (-mu1^2 + mu2) * (-mu1^2 + 2*mu1*mu2 - mu2) * (mu1^2 - 2*mu1 + mu2)




Delta_red:

mu2^2 * (mu2 - 1)^2 * (-mu1 + mu2)^2 * mu1^2 * (mu1 - 1)^2




H_red:

(-1) * (22*mu1^10*mu2^8 - 44*mu1^9*mu2^9 + 22*mu1^8*mu2^10 - 88*mu1^10*mu2^7 + 88*mu1^9*mu2^8 + 88*mu1^8*mu2^9 - 88*mu1^7*mu2^10 + 22*mu1^12*mu2^4 - 88*mu1^11*mu2^5 + 690*mu1^10*mu2^6 - 1278*mu1^9*mu2^7 + 1330*mu1^8*mu2^8 - 1278*mu1^7*mu2^9 + 690*mu1^6*mu2^10 - 88*mu1^5*mu2^11 + 22*mu1^4*mu2^12 - 44*mu1^12*mu2^3 + 88*mu1^11*mu2^4 - 1278*mu1^10*mu2^5 + 1639*mu1^9*mu2^6 - 493*mu1^8*mu2^7 - 493*mu1^7*mu2^8 + 1639*mu1^6*mu2^9 - 1278*mu1^5*mu2^10 + 88*mu1^4*mu2^11 - 44*mu1^3*mu2^12 + 22*mu1^12*mu2^2 + 88*mu1^11*mu2^3 + 1330*mu1^10*mu2^4 - 493*mu1^9*mu2^5 - 868*mu1^8*mu2^6 + 458*mu1^7*mu2^7 - 868*mu1^6*mu2^8 - 493*mu1^5*mu2^9 + 1330*mu1^4*mu2^10 + 88*mu1^3*mu2^11 + 22*mu1^2*mu2^12 - 88*mu1^11*mu2^2 - 1278*mu1^10*mu2^3 - 493*mu1^9*mu2^4 + 458*mu1^8*mu2^5 + 785*mu1^7*mu2^6 + 785*mu1^6*mu2^7 + 458*mu1^5*mu2^8 - 493*mu1^4*mu2^9 - 1278*mu1^3*mu2^10 - 88*mu1^2*mu2^11 + 690*mu1^10*mu2^2 + 1639*mu1^9*mu2^3 - 868*mu1^8*mu2^4 + 785*mu1^7*mu2^5 - 2952*mu1^6*mu2^6 + 785*mu1^5*mu2^7 - 868*mu1^4*mu2^8 + 1639*mu1^3*mu2^9 + 690*mu1^2*mu2^10 - 88*mu1^10*mu2 - 1278*mu1^9*mu2^2 - 493*mu1^8*mu2^3 + 458*mu1^7*mu2^4 + 785*mu1^6*mu2^5 + 785*mu1^5*mu2^6 + 458*mu1^4*mu2^7 - 493*mu1^3*mu2^8 - 1278*mu1^2*mu2^9 - 88*mu1*mu2^10 + 22*mu1^10 + 88*mu1^9*mu2 + 1330*mu1^8*mu2^2 - 493*mu1^7*mu2^3 - 868*mu1^6*mu2^4 + 458*mu1^5*mu2^5 - 868*mu1^4*mu2^6 - 493*mu1^3*mu2^7 + 1330*mu1^2*mu2^8 + 88*mu1*mu2^9 + 22*mu2^10 - 44*mu1^9 + 88*mu1^8*mu2 - 1278*mu1^7*mu2^2 + 1639*mu1^6*mu2^3 - 493*mu1^5*mu2^4 - 493*mu1^4*mu2^5 + 1639*mu1^3*mu2^6 - 1278*mu1^2*mu2^7 + 88*mu1*mu2^8 - 44*mu2^9 + 22*mu1^8 - 88*mu1^7*mu2 + 690*mu1^6*mu2^2 - 1278*mu1^5*mu2^3 + 1330*mu1^4*mu2^4 - 1278*mu1^3*mu2^5 + 690*mu1^2*mu2^6 - 88*mu1*mu2^7 + 22*mu2^8 - 88*mu1^5*mu2^2 + 88*mu1^4*mu2^3 + 88*mu1^3*mu2^4 - 88*mu1^2*mu2^5 + 22*mu1^4*mu2^2 - 44*mu1^3*mu2^3 + 22*mu1^2*mu2^4)

Notice that \(\mathrm{val}(\Delta) = 0\) and every invariant \(I\) is a polynomial in \(t_1,t_2,\mu_1,\mu_2\) so

\[8\mathrm{val}(I) \geq \deg(I)\mathrm{val}(\Delta), \quad \text{for any invariant } I\]

Unmarked tree type II

In this case, we can write

\[\lambda_1 = 1 + t_1 \mu_1 \quad \text{and} \quad \lambda_2 = 1 + t_1 \mu_2\]

where

1. \(v(\mu_i) = 0\) for \(i = 1,2\),

2. \(v(\mu_1 - \mu_2) = 0\), and

3. \(\lambda_i = 1 + t_1\mu_i\) for \(i = 1,2\). Here \(t_1\) is …, and \(\mu_1\) and \(\mu_2\) …

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



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


k = QQ

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

print("I4 : ")
print(Factor(I4.change_ring(k)))
print("\n \n")

print("I8 : ")
print(Factor(I8.change_ring(k)))
print("\n \n")

print("I12 : ")
print(Factor(I12.change_ring(k)))
print("\n \n")

print("I18 : ")
print(Factor(I18.change_ring(k)))
print("\n \n")

print("Delta : ")
print(Factor(Delta.change_ring(k)))
print("\n \n")

print("H : ")
print(Factor(H.change_ring(k)))
print("\n \n")


################################# Computing the reduction of the


I4_red    = Reduction(I4/t1^2)
I8_red    = Reduction(I8/t1^4)
I12_red   = Reduction(I12/t1^6)
I18_red   = Reduction(I18/t1^9)
Delta_red = Reduction(Delta/t1^6)
H_red     = Reduction(H/t1^6)


## we are translating with the vector - (4,8,12,18,8,12)*val(t1)/2    ##


print("-----------------")

print("I4_red: ")
print(Factor(I4_red))
print("\n \n")

print("I8_red: ")
print(Factor(I8_red))
print("\n \n")

print("I12_red: ")
print(Factor(I12_red))
print("\n \n")


print("I18_red: ")
print(Factor(I18_red))
print("\n \n")

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


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

----------------------



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



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



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



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



Delta :
mu2^2 * (-mu1 + mu2)^2 * mu1^2 * t1^6 * (mu2*t1 + 1)^2 * (mu1*t1 + 1)^2



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



-----------------
I4_red:
(-2) * (mu1^2 - mu1*mu2 + mu2^2)



I8_red:
(14) * (mu1^2 - mu1*mu2 + mu2^2)^2



I12_red:
(-11) * (44*mu1^6 - 132*mu1^5*mu2 + 267*mu1^4*mu2^2 - 314*mu1^3*mu2^3 + 267*mu1^2*mu2^4 - 132*mu1*mu2^5 + 44*mu2^6)



I18_red:
(-1) * (mu1 - 2*mu2) * (mu1 + mu2) * (2*mu1 - mu2) * mu2^2 * mu1^2 * (mu1 - mu2)^2



Delta_red:
mu2^2 * mu1^2 * (mu1 - mu2)^2



H_red:
(-22) * mu2^2 * mu1^2 * (mu1 - mu2)^2

Unmarked tree type III

In this case, we can write

\[\lambda_1 = t_1 \mu_1 \quad \text{and} \quad \lambda_2 = 1 + t_2 \mu_2\]

1. \(v(t_1) > 0\) and \(v(t_2) > 0\).

2. \(v(\mu_1) = v(\mu_2) = 0\)

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

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

################################# Computing the reduction of the

k = QQ

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

print("I4 : ")
print(Factor(I4.change_ring(k)))
print("\n \n")

print("I8 : ")
print(Factor(I8.change_ring(k)))
print("\n \n")

print("I12 : ")
print(Factor(I12.change_ring(k)))
print("\n \n")

print("I18 : ")
print(Factor(I18.change_ring(k)))
print("\n \n")

print("Delta : ")
print(Factor(Delta.change_ring(k)))
print("\n \n")

print("H : ")
print(Factor(H.change_ring(k)))
print("\n \n")


################################# Computing the reduction of the

I4_red    = Reduction(I4/t1^0)
I8_red    = Reduction(I8/t1^0)
I12_red   = Reduction(I12/t1^0)
I18_red   = Reduction(I18/t1^0)
Delta_red = Reduction(Delta/t1^0)
H_red     = Reduction(H/t1^0)

print("-----------------")

print("I4_red: ")
print(Factor(I4_red))
print("\n \n")

print("I8_red: ")
print(Factor(I8_red))
print("\n \n")

print("I12_red: ")
print(Factor(I12_red))
print("\n \n")


print("I18_red: ")
print(Factor(I18_red))
print("\n \n")

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


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

----------------------



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 * (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)



-----------------
I4_red:
-1 * 2



I8_red:
2 * 7



I12_red:
-1 * 2^2 * 11^2



I18_red:
0



Delta_red:
0



H_red:
0