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.<ud,ux1,uy1,ux2,uy2> = PolynomialRing(QQ,5,order='invlex')
I = R.ideal([
  mynumerator((ux1^3+uy1^3+1)-(3*ud*ux1*uy1))
, mynumerator((ux2^3+uy2^3+1)-(3*ud*ux2*uy2))
])

J = I + R.ideal([0
, ux1-ux2
, uy1-uy2
])

ud = fastfrac(ud)
ux1 = fastfrac(ux1)
uy1 = fastfrac(uy1)
ux2 = fastfrac(ux2)
uy2 = fastfrac(uy2)

ux3 = (((uy1^2*ux2-uy2^2*ux1)/(ux2*uy2-ux1*uy1))).reduce()
uy3 = (((ux1^2*uy2-ux2^2*uy1)/(ux2*uy2-ux1*uy1))).reduce()
ux4 = ((uy1*(fastfrac(1)-ux1^3)/(ux1^3-uy1^3))).reduce()
uy4 = ((ux1*(uy1^3-fastfrac(1))/(ux1^3-uy1^3))).reduce()
a0 = fastfrac((fastfrac(1)))
a1 = fastfrac((fastfrac(0)))
a2 = fastfrac((fastfrac(0)))
a3 = fastfrac((fastfrac(0)))
a4 = fastfrac((-fastfrac(27)*ud*(ud^3+fastfrac(8))))
a6 = fastfrac((fastfrac(54)*(ud^6-fastfrac(20)*ud^3-fastfrac(8))))
wu1 = ((fastfrac(12)*(ud^3-fastfrac(1))/(ud+ux1+uy1)-fastfrac(9)*ud^2)).reduce().sreduce()
wv1 = ((fastfrac(36)*(uy1-ux1)*(ud^3-fastfrac(1))/(ud+ux1+uy1))).reduce().sreduce()
print(isidentity(a0*(wv1^2)+a1*(wu1*wv1)+a3*wv1-(((wu1+a2)*wu1+a4)*wu1+a6)))
print(isidentity(ux1-((fastfrac(36)*(ud^3-fastfrac(1))-wv1)/(fastfrac(6)*(wu1+fastfrac(9)*ud^2))-ud/fastfrac(2))))
print(isidentity(uy1-((wv1+fastfrac(36)*(ud^3-fastfrac(1)))/(fastfrac(6)*(wu1+fastfrac(9)*ud^2))-ud/fastfrac(2))))
wu2 = ((fastfrac(12)*(ud^3-fastfrac(1))/(ud+ux2+uy2)-fastfrac(9)*ud^2)).reduce().sreduce()
wv2 = ((fastfrac(36)*(uy2-ux2)*(ud^3-fastfrac(1))/(ud+ux2+uy2))).reduce().sreduce()
wu3 = ((fastfrac(12)*(ud^3-fastfrac(1))/(ud+ux3+uy3)-fastfrac(9)*ud^2)).reduce().sreduce()
wv3 = ((fastfrac(36)*(uy3-ux3)*(ud^3-fastfrac(1))/(ud+ux3+uy3))).reduce().sreduce()
wu4 = ((fastfrac(12)*(ud^3-fastfrac(1))/(ud+ux4+uy4)-fastfrac(9)*ud^2)).reduce().sreduce()
wv4 = ((fastfrac(36)*(uy4-ux4)*(ud^3-fastfrac(1))/(ud+ux4+uy4))).reduce().sreduce()
slope = ((wv2-wv1)/(wu2-wu1)).reduce().sreduce()
print(isidentity(a0*slope^2+a1*slope-a2-wu1-wu2-wu3))
print(isidentity(slope*(wu1-wu3)-wv1-a1*wu3-a3-wv3))
slope = ((fastfrac(3)*wu1^2+fastfrac(2)*a2*wu1+a4-a1*wv1)/(fastfrac(2)*a0*wv1+a1*wu1+a3)).reduce().sreduce()
print(isidentity(a0*slope^2+a1*slope-a2-wu1-wu1-wu4))
print(isidentity(slope*(wu1-wu4)-wv1-a1*wu4-a3-wv4))