Context : When type Ia supernovae ( SNe Ia ) were chosen as distance indicator to measure cosmological parameters , Phillips relation was applied .
However , the origin of the scatter of the maximum luminosity of SNe Ia ( or the variation of the production of ^ { 56 } Ni ) is still unclear .
The metallicity and the carbon abundance of white dwarf ( WD ) before supernova explosion are possible key , but neither of them has an ability to interpret the scatter of the maximum luminosity of SNe Ia .

Aims : In this paper , we want to check whether or not the carbon abundance can be affected by initial metallicity .

Methods : We calculated a series of stellar evolution .

Results : We found that when Z \leq 0.02 , the carbon abundance is almost independent of metallicity if it is plotted against the initial WD mass .
However , when Z > 0.02 , the carbon abundance is not only a function of the initial WD mass , but also metallicity , i.e .
for a given initial WD mass , the higher the metallicity , the lower the carbon abundance .
Based on some previous studies , i.e .
both a high metallicity and a low carbon abundance lead to a lower production of ^ { 56 } Ni formed during SN Ia explosion , the effects of the carbon abundance and the metallicity on the amount of ^ { 56 } Ni are enhanced by each other , which may account for the variation of maximum luminosity of SNe Ia , at least qualitatively .

Conclusions : Considering that the central density of WD before supernova explosion may also play a role on the production of ^ { 56 } Ni and the carbon abundance , the metallicity and the central density are all determined by the initial parameters of progenitor system , i.e .
the initial WD mass , metallicity , orbital period and secondary mass , the amount of ^ { 56 } Ni might be a function of the initial parameters .
Then , our results might construct a bridge linking the progenitor model and the explosion model of SNe Ia .