Remarks on the zeta function of some diagonal hyperelliptic curves

We compute the Jacobi sums and the zeta functions associated to the family of diagonal curves defined over image by y2=γx5+δ (with γδ≠0), then we prove that their Jacobians are simple. We discuss about the hardness of the associated discrete logarithm problem with respect to known attacks and we show that our family is suitable for cryptographic purpose.