for q in range(200):
  q = Integer(q)
  if q < 2: continue
  if not q.is_prime_power(): continue
  if q % 4 != 3: continue

  Fq.<alpha> = GF(q)
  def chi(a): return a^((q-1)//2)

  for s in Fq:
    if s == 0: continue
    if (s^2-2)*(s^2+2) == 0: continue
    c = 2/s^2
    r = c+1/c
    d = -(c+1)^2/(c-1)^2

    def phi(t):
      if t == 1: return (Fq(0),Fq(1))
      if t == -1: return (Fq(0),Fq(1))
      u = (1-t)/(1+t)
      v = u^5 + (r^2-2)*u^3 + u
      X = chi(v)*u
      Y = (chi(v)*v)^((q+1)//4)*chi(v)*chi(u^2+1/c^2)
      x = (c-1)*s*X*(1+X)/Y
      y = (r*X-(1+X)^2)/(r*X+(1+X)^2)
      return (x,y)

    phigraph = [(t,phi(t)) for t in Fq]

    for (t,p) in phigraph:
      preimages = [r2 for (r2,p2) in phigraph if p2 == p]
      if t == 0:
        print preimages == [0]
      else:
        print preimages == [t,-t] or preimages == [-t,t]

    images = set()
    for x in Fq:
      for y in Fq:
        if x^2+y^2==1+d*x^2*y^2:
          if y + 1 == 0: continue
          eta = (y-1)/(2*(y+1))
          if not ((1+eta*r)^2-1).is_square(): continue
          if eta*r == -2 and x != 2*s*(c-1)*chi(c)/r: continue
          images.add((x,y))

    phiFq = set()
    for t in Fq:
      phiFq.add(phi(t))

    print phiFq == images

    for x,y in phiFq:
      print y+1 != 0
      eta = (y-1)/(2*(y+1))
      X = -(1+eta*r)+((1+eta*r)^2-1)^((q+1)//4)
      z = chi((c-1)*s*X*(1+X)*x*(X^2+1/c^2))
      u = z*X
      print 1+u != 0
      t = (1-u)/(1+u)
      print phi(t) == (x,y)