# Sage code for "A nonabelian conjecture of Birch and Swinnerton-Dyer type
# for hyperbolic curves", by J. Balakrishnan, I. Dan-Cohen, M. Kim,
# S. Wewers"

# We check that Li_2(zeta_6)<>0 for all primes p=1 mod 3, p<=N,
# see loc.cit., p. 18f 


#N=100000

print "Checking that Li_2(zeta_6)<>0 for all p=1 mod 3, p<=",N
print

check=True
badprimes=set()

for p in prime_range(N):

    if p.mod(3)==1:

      k=FiniteField(p)
      R.<v>=k[]
      #print "p=",p

      g0=v-v^p
      g1=-R(g0/(v-v^2)).integral()
      g2=-R(g1/(v-v^2)).integral()
      #print "g_2=",g2

      a=k.primitive_element()
      b=a^((p-1)/6)
      #print "zeta_6=",b

      if g2(b)==0:
          print "for p=",p,", zeta=",b,"is a root of g_2"
          check=False
          badprimes.add(p)

 
if check:
    print "Check was successful for all p<=",N 
else:
    print "Check unsuccessful; bad primes are:",badprimes