  We use a sample of 809 photometrically classified type Ia supernovae (SNe Ia)
discovered by the Dark Energy Survey (DES) along with 40415 field galaxies to
calculate the rate of SNe Ia per galaxy in the redshift range $0.2 < z <0.6$.
We recover the known correlation between SN Ia rate and galaxy stellar mass
across a broad range of scales $8.5 \leq \log(M_*/\mathrm{M}_{\odot}) \leq
11.25$. We find that the SN Ia rate increases with stellar mass as a power-law
with index $0.63 \pm 0.02$, which is consistent with previous work. We use an
empirical model of stellar mass assembly to estimate the average star-formation
histories (SFHs) of galaxies across the stellar mass range of our measurement.
Combining the modelled SFHs with the SN Ia rates to estimate constraints on the
SN Ia delay time distribution (DTD), we find the data are fit well by a
power-law DTD with slope index $\beta = -1.13 \pm 0.05$ and normalisation $A =
2.11 \pm0.05 \times
10^{-13}~\mathrm{SNe}~{\mathrm{M}_{\odot}}^{-1}~\mathrm{yr}^{-1}$, which
corresponds to an overall SN Ia production efficiency $N_{\mathrm{Ia}}/M_* =
0.9~_{-0.7}^{+4.0} \times 10^{-3}~\mathrm{SNe}~\mathrm{M}_{\odot}^{-1}$. Upon
splitting the SN sample by properties of the light curves, we find a strong
dependence on DTD slope with the SN decline rate, with slower-declining SNe
exhibiting a steeper DTD slope. We interpret this as a result of a relationship
between intrinsic luminosity and progenitor age, and explore the implications
of the result in the context of SN Ia progenitors.
