for q in range(40):
  q = Integer(q)
  if q < 2: continue
  if not q.is_prime_power(): continue
  if q.is_power_of(2): continue

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

  for A in Fq:
    for B in Fq:
      if A*B*(A^2-4*B) == 0: continue
      for u in Fq:
        if u.is_square(): continue
        R = [r for r in Fq if 1+u*r^2!=0 and A^2*u*r^2!=B*(1+u*r^2)^2]
        for r in R:
          if r != 0:
            print 1+u*r^2 != 0
            v = -A/(1+u*r^2)
            eps = chi(v^3+A*v^2+B*v)
            x = eps*v-(1-eps)*A/2
            print (x^3+A*x^2+B*x).is_square()
            y = -eps*sqrt(x^3+A*x^2+B*x)
            print v != 0
            print eps != 0
            print x != 0
            print y != 0
            print y^2 == x^3+A*x^2+B*x
        if q % 4 == 1:
          if not (A^2-4*B).is_square():
            print len(R) == len(Fq)