# copy paste these into sage

def count(K, n):
  # K is a number field
	# counts the number of ideals of norm <= n
	return sum([len(x) for x in K.ideals_of_bdd_norm(n).values()])

def estimate(K):
  r = len(K.real_embeddings())
  s = (len(K.complex_embeddings())-r)/2
  R = K.regulator()
  w = K.number_of_roots_of_unity()
  d = K.discriminant()
  h = K.class_number()
  return n(2^r * (2*pi)^s * R * h /(w * sqrt(abs(d))))
	
K = QQ[sqrt(23)]

L = QQ[7^(1/9)]

M = CyclotomicField(17)


count(K,1000)-round(estimate(K)*1000)

count(L,1000)-round(estimate(L)*1000)

count(M,1000)-round(estimate(M)*1000)