This is the second paper of a series in which we present measurements of the Supernova (SN) rates from the SUDARE survey. In this paper, we study the trend of the SN rates with the intrinsic colours, the star formation activity and the mass of the parent galaxies. We have considered a sample of about 130000 galaxies and a SN sample of about 50 events. We found that the SN Ia rate per unit mass is higher by a factor of six in the star-forming galaxies with respect to the passive galaxies. The SN Ia rate per unit mass is also higher in the less massive galaxies that are also younger. These results suggest a distribution of the delay times (DTD) less populated at long delay times than at short delays. The CC SN rate per unit mass is proportional to both the sSFR and the galaxy mass. The trends of the Type Ia and CC SN rates as a function of the sSFR and the galaxy mass that we observed from SUDARE data are in agreement with literature results at different redshifts. The expected number of SNe Ia is in agreement with the observed one for all four DTD models considered both in passive and star-forming galaxies so we can not discriminate between different progenitor scenarios. The expected number of CC SNe is higher than the observed one, suggesting a higher limit for the minimum progenitor mass. We also compare the expected and observed trends of the SN Ia rate with the intrinsic U - J colour of the parent galaxy, assumed as a tracer of the age distribution. While the slope of the relation between the SN Ia rate and the U - J color in star-forming galaxies can be reproduced well by all four DTD models considered, only the steepest of them is able to account for the rates and colour in star-forming and passive galaxies with the same value of the SN Ia production efficiency.
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