This is similar to a question I recently asked about. It is from Ireland's Number theory book, ch.8, ex.27 b,c. I think I can do the first part of this question, but I think there might be a trick here that is stumping me.
Let $p=1\mod{3}$, $\chi$ a character of order 3, $\rho$ the Legendre symbol. Show
b.) $N(y^{2}+x^{3}=1)=p+2ReJ(\chi,\rho)$
c.)$2a-b\equiv -(^{(p-1)/2}_{(p-1)/3})(p)$ where $J(\chi,\rho)=a+b\omega$
If this is too similar to my last question I would still appreciate a hint. Thanks.