On computation of Jacobi sum [closed]
I’m computing Jacobi sum of cubic residue symbol as follows:
N=3
x=polygen(ZZ,'x')
K.<z>=CyclotomicField(N)
p=-4-3*z
P = K.ideal(p)
sum([K(a).residue_symbol(P,N)*K(1-a).residue_symbol(P,N) for a in [2..P.norm()-1] if a not in P and 1-a not in P])
The output is -3*z - 4. This is correct.
But I wrote another equivalent code:
def Jacobi(P,l,j):
x=sum([K(a).residue_symbol(P,l)^j*K(1-a).residue_symbol(P,l)^j for a in range(2, P.norm()-1) if a not in P and 1-a not in P])
return x
N=3
x=polygen(ZZ,'x')
K.<z>=CyclotomicField(N)
p=-4-3*z
P = K.ideal(p)
Jacobi(P,N,1)
Then the output is -4*z - 4, this is odd. Where is wrong?
You have different summation ranges.
rangestops just before its upper boundOh my goodness.Thank you.
On a different note - there is no need to create a list to sum up its elements, that is, instead of
sum([...])you can do simplysum(...), which may save some memory when the number of terms is large.And same for
[2..P.norm()-1]- there is no need to create a list here, generator(2..P.norm()-1)(orrange()) is well sufficient for summation purposes.