def mynumerator(x):
  if parent(x) == R:
    return x
  return numerator(x)

class fastfrac:
  def __init__(self,top,bot=1):
    if parent(top) == ZZ or parent(top) == R:
      self.top = R(top)
      self.bot = R(bot)
    elif top.__class__ == fastfrac:
      self.top = top.top
      self.bot = top.bot * bot
    else:
      self.top = R(numerator(top))
      self.bot = R(denominator(top)) * bot
  def reduce(self):
    return fastfrac(self.top / self.bot)
  def sreduce(self):
    return fastfrac(I.reduce(self.top),I.reduce(self.bot))
  def iszero(self):
    return self.top in I and not (self.bot in I)
  def isdoublingzero(self):
    return self.top in J and not (self.bot in J)
  def __add__(self,other):
    if parent(other) == ZZ:
      return fastfrac(self.top + self.bot * other,self.bot)
    if other.__class__ == fastfrac:
      return fastfrac(self.top * other.bot + self.bot * other.top,self.bot * other.bot)
    return NotImplemented
  def __sub__(self,other):
    if parent(other) == ZZ:
      return fastfrac(self.top - self.bot * other,self.bot)
    if other.__class__ == fastfrac:
      return fastfrac(self.top * other.bot - self.bot * other.top,self.bot * other.bot)
    return NotImplemented
  def __neg__(self):
    return fastfrac(-self.top,self.bot)
  def __mul__(self,other):
    if parent(other) == ZZ:
      return fastfrac(self.top * other,self.bot)
    if other.__class__ == fastfrac:
      return fastfrac(self.top * other.top,self.bot * other.bot)
    return NotImplemented
  def __rmul__(self,other):
    return self.__mul__(other)
  def __div__(self,other):
    if parent(other) == ZZ:
      return fastfrac(self.top,self.bot * other)
    if other.__class__ == fastfrac:
      return fastfrac(self.top * other.bot,self.bot * other.top)
    return NotImplemented
  __truediv__ = __div__
  def __pow__(self,other):
    if parent(other) == ZZ:
      return fastfrac(self.top ^ other,self.bot ^ other)
    return NotImplemented

def isidentity(x):
  return x.iszero()

def isdoublingidentity(x):
  return x.isdoublingzero()

R.<ua3,ua,ux1,uy1,uZZ1,uX1,uY1,uZ1> = PolynomialRing(QQ,8,order='invlex')
I = R.ideal([
  mynumerator((uy1^2)-(ux1^3+3*ua*(ux1+1)^2))
, mynumerator((ux1)-(uX1/uZ1^2))
, mynumerator((uy1)-(uY1/uZ1^3))
, mynumerator((uZZ1)-(uZ1^2))
, mynumerator((ua3)-(3*ua))
])

ua3 = fastfrac(ua3)
ua = fastfrac(ua)
ux1 = fastfrac(ux1)
uy1 = fastfrac(uy1)
uZZ1 = fastfrac(uZZ1)
uX1 = fastfrac(uX1)
uY1 = fastfrac(uY1)
uZ1 = fastfrac(uZ1)


uA = ((uY1*uZ1))
uZ3 = ((fastfrac(2)*uA))
uB = ((fastfrac(4)*uY1^2*uX1))
uC = ((uB+fastfrac(6)*ua*uA^2))
uZZ3 = ((fastfrac(4)*uA^2))
uD = ((fastfrac(3)*uX1^2))
uE = ((uD+fastfrac(6)*ua*uZZ1*(uZZ1+uX1)))
uX3 = ((uE^2-fastfrac(2)*uB-ua3*uZZ3))
uY3 = ((uE*(uB-uX3)-fastfrac(8)*uY1^4))

ux3 = ((fastfrac(9)/(fastfrac(4)*uy1^2)*ux1^4+fastfrac(9)/uy1^2*ua*ux1^3+(fastfrac(9)/uy1^2*ua^2+fastfrac(9)/uy1^2*ua)*ux1^2+(fastfrac(18)/uy1^2*ua^2-fastfrac(2))*ux1+(fastfrac(9)/uy1^2*ua^2-fastfrac(3)*ua))).reduce()
uy3 = ((-fastfrac(27)/(fastfrac(8)*uy1^3)*ux1^6-fastfrac(81)/(fastfrac(4)*uy1^3)*ua*ux1^5+(-fastfrac(81)/(fastfrac(2)*uy1^3)*ua^2-fastfrac(81)/(fastfrac(4)*uy1^3)*ua)*ux1^4+(-fastfrac(27)/uy1^3*ua^3-fastfrac(81)/uy1^3*ua^2+fastfrac(9)/(fastfrac(2)*uy1))*ux1^3+(-fastfrac(81)/uy1^3*ua^3-fastfrac(81)/(fastfrac(2)*uy1^3)*ua^2+fastfrac(27)/(fastfrac(2)*uy1)*ua)*ux1^2+(-fastfrac(81)/uy1^3*ua^3+fastfrac(9)/uy1*ua^2+fastfrac(9)/uy1*ua)*ux1+(-fastfrac(27)/uy1^3*ua^3+fastfrac(9)/uy1*ua^2-uy1))).reduce()

print(isidentity((uy3^2)-(ux3^3+fastfrac(3)*ua*(ux3+fastfrac(1))^2)))
print(isidentity((ux3)-(uX3/uZ3^2)))
print(isidentity((uy3)-(uY3/uZ3^3)))
print(isidentity((uZZ3)-(uZ3^2)))