  Type Ia supernovae (SNe Ia) play a key role in measuring cosmological
parameters, in which the Phillips relation is adopted. However, the origin of
the relation is still unclear. Several parameters are suggested, e.g. the
relative content of carbon to oxygen (C/O) and the central density of the white
dwarf (WD) at ignition. These parameters are mainly determined by the WD's
initial mass and its cooling time, respectively. Using the progenitor model
developed by Meng and Yang, we present the distributions of the initial WD mass
and the cooling time. We do not find any correlation between these parameters.
However, we notice that the range of the WD's mass decreases, while its average
value increases with the cooling time. These results could provide a constraint
when simulating the SN Ia explosion, i.e. the WDs with a high C/O ratio usually
have a lower central density at ignition, while those having the highest
central density at ignition generally have a lower C/O ratio. The cooling time
is mainly determined by the evolutionary age of secondaries, and the scatter of
the cooling time decreases with the evolutionary age. Our results may indicate
that WDs with a long cooling time have more uniform properties than those with
a short cooling time, which may be helpful to explain why SNe Ia in elliptical
galaxies have a more uniform maximum luminosity than those in spiral galaxies.
