The Jupiter Trojans are a large group of asteroids that are coorbiting with Jupiter near its L4 and L5 Lagrange points. The study of Jupiter Trojans is crucial for testing different models of planet formation that are directly related to our understanding of solar system evolution. In this work, we select known Jupiter Trojans listed by the Minor Planet Center from the full six years data set (Y6) of the Dark Energy Survey (DES) to analyze their photometric properties. The DES data allow us to study Jupiter Trojans with a fainter magnitude limit than previous studies in a homogeneous survey with griz band measurements. We extract a final catalog of 573 unique Jupiter Trojans. Our sample include 547 asteroids belonging to L5. This is one of the largest analyzed samples for this group. By comparing with the data reported by other surveys we found that the color distribution of L5 Trojans is similar to that of L4 Trojans. We find that L5 Trojans' g - i and g - r colors become less red with fainter absolute magnitudes, a trend also seen in L4 Trojans. Both the L4 and L5 clouds consistently show such a color-size correlation over an absolute magnitude range 11 < H < 18. We also use DES colors to perform taxonomic classifications. C- and P-type asteroids outnumber D-type asteroids in the L5 Trojans DES sample, which have diameters in the 5-20 km range. This is consistent with the color-size correlation.