def test_lemma_3_4(m):
    a = (4*m-27).sqrt()
    if not a in ZZ:
        print "m must be of the form 1/4(a^2+27)"
        return None
    Q.<x>=QQ[]
    if a % 3 != 0:
        fa = x^3 + x^2 + (1-m)/3*x - (m*(3+a)-1)/27
    else:
        fa = x^3 - m/3*x-(a*m)/27
    K.<alpha>=NumberField(fa)
    if a % 3 != 0:
        w = a^2/36 + a*alpha/3 + alpha^2 +a/9 + 2/3*alpha+31/36
    else:
        w = a^2/36 + alpha*a/3 + alpha^2 + 3/4
    gm = x^3-m*x^2+2*m*x-m 
    if gm(w) !=0:
        print "error, w is not a root of gm"
        print w,gm.roots(K)
        return None
    if a % 3 != 0:
        eta_ = alpha^2+(a-1)/3*alpha+a^2/36-a/18+1/36
    else:
        eta_ = alpha^2 + (a-3)*alpha/3+a^2/36-a/6+1/4  
    mu_a = x^3 - (2*m-3-a)/2*x^2 + (2*m-3+a)/2*x - 1
    if nu_a(eta_) != 0:
        print "error, eta is not a root of nu_a"
        return None 
    nu_a = f=x^3+(m-3)*x^2+3*x-1
    A = (x^2*(4*a^2+12)+x*(a^4+18*a^2-8*a+21)+(-2*a^3+6*a^2-30*a-6))/(16*a)  
    assert uu_a(x=A) % nu_a == 0
    print "everythng ok"
  


test_lemma_3_4(19)
test_lemma_3_4(9)
"""
test_lemma_3_4(19)
"""