Co nuclear-quadrupole-resonance (NQR) measurements were performed on various bilayered hydrate cobaltate Na_x(H_3O)_zCoO_2cdot yH_2O with different values of the superconducting and magnetic-ordering temperatures, T_c and T_M, respectively. From measurements of the temperature and sample dependence of the NQR frequency, it was revealed that the NQR frequency is changed by the change of the electric field gradient (EFG) along the c axis u_{zz} rather than the asymmetry of EFG within the ab-plane. In addition, it is considered that the change of u_{zz} is gaverned mainly by the trigonal distortion of the CoO_2 block layers along the c axis, from the relationships between u_{zz} and the various physical parameters. We found the tendency that samples with u_{zz} larger than 4.2 MHz show magnetic ordering, whereas samples with lower u_{zz} show superconductivity. We measured the nuclear spin-lattice relaxation rate 1/T_1 in these samples, and found that magnetic fluctuations depend on samples. The higher- u_{zz} sample has stronger magnetic fluctuations at T_c. From the relationship between u_{zz} and T_c or T_M, we suggest that the NQR frequency can be regarded as a tuning parameter to determine the ground state of the system, and develop the phase diagram using u_{zz}. This phase diagram shows that the highest-T_c sample is located at the point where T_M is considered to be zero, which suggests that the superconductivity is induced by quantum critical fluctuations. We strongly advocate that the hydrate cobaltate superconductor presents an example of the magnetic-fluctuation-mediated superconductivity argued in the heavy-fermion compounds. The coexistence of superconductivity and magnetism observed in the sample with the highest u_{zz} is also discussed on the basis of the results of our experiments.
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