  We report 59Co nuclear quadrupole resonance (NQR) studies of bilayer water
intercalated sodium cobalt oxides NaxCoO2.yH2O (BLH) with the superconducting
transition temperatures, 2 K < T_c <= 4.6 K, as well as a magnetic BLH sample
without superconductivity. We obtained a magnetic phase diagram of T_c and the
magnetic ordering temperature T_M against the peak frequency nu_3 59Co NQR
transition I_z = +- 5/2 <=> +-7/2 and found a dome shape superconducting phase.
The 59Co NQR spectrum of the non-superconducting BLH shows a broadening below
T_M without the critical divergence of 1/T_1 and 1/T_2, suggesting an
unconventional magnetic ordering. The degree of the enhancement of 1/T_1T at
low temperatures increases with the increase of nu_3 though the optimal
nu_3~12.30 MHz. In the NaxCoO2.yH2O system, the optimal-T_c superconductivity
emerges close to the magnetic instability. T_c is suppressed near the phase
boundary at nu_3~12.50 MHz, which is not a conventional magnetic quantum
critical point.
