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
  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.<ud,ux1,uy1,uX1,uY1,uZ1> = PolynomialRing(QQ,6,order='invlex')
I = R.ideal([
  mynumerator((ux1^3+uy1^3+1)-(3*ud*ux1*uy1))
, mynumerator((ux1)-(uX1/uZ1))
, mynumerator((uy1)-(uY1/uZ1))
])

ud = fastfrac(ud)
ux1 = fastfrac(ux1)
uy1 = fastfrac(uy1)
uX1 = fastfrac(uX1)
uY1 = fastfrac(uY1)
uZ1 = fastfrac(uZ1)


uR0 = ((uX1^2))
uR1 = ((uY1^2))
uR2 = ((uZ1^2))
uR3 = ((uR0+uR1))
uR4 = ((uR0+uR2))
uR5 = ((uR1+uR2))
uR0 = ((uX1+uY1))
uR0 = ((uR0^2))
uR0 = ((uR0-uR3))
uR1 = ((uX1+uZ1))
uR1 = ((uR1^2))
uR1 = ((uR1-uR4))
uR2 = ((uY1+uZ1))
uR3 = ((fastfrac(2)*uR3))
uR2 = ((uR2^2))
uR4 = ((fastfrac(2)*uR4))
uR2 = ((uR2-uR5))
uR5 = ((fastfrac(2)*uR5))
uX3 = ((uR2-uR0))
uR4 = ((uR1+uR4))
uX3 = ((uX3*uR4))
uY3 = ((uR0-uR1))
uR5 = ((uR2+uR5))
uY3 = ((uY3*uR5))
uZ3 = ((uR1-uR2))
uR0 = ((uR0+uR3))
uZ3 = ((uZ3*uR0))

ux3 = ((uy1*(fastfrac(1)-ux1^3)/(ux1^3-uy1^3))).reduce()
uy3 = ((ux1*(uy1^3-fastfrac(1))/(ux1^3-uy1^3))).reduce()

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