The current tension between the direct and the early-universe measurements of the Hubble constant, H_0_, requires detailed scrutiny of all the data and methods used in the studies on both sides of the debate. The Cepheids in the Type Ia supernova (SN Ia) host galaxy NGC5584 played a key role in the local measurement of H_0_. The SH0ES (Supernovae H_0_ for the Equation of State) project used the observations of this galaxy to derive a relation between the Cepheids' periods and ratios of their amplitudes in different optical bands of the Hubble Space Telescope and used these relations to analyze the light curves of the Cepheids in around half of the current sample of local SN Ia host galaxies. In this work, we present an independent detailed analysis of the Cepheids in NGC5584. We employ different tools for our photometric analysis and a completely different method for our light-curve analysis, and we do not find a systematic difference between our period and mean magnitude measurements compared to those reported by SH0ES. By adopting a period-luminosity relation calibrated by the Cepheids in the Milky Way, we measure a distance modulus {mu}=31.810+/-0.047 (mag), which is in agreement with {mu}=31.786+/-0.046 (mag) measured by SH0ES. In addition, the relations we find between the periods and amplitude ratios of the Cepheids in NGC5584 are significantly tighter than those of SH0ES, and their potential impact on the direct H0 measurement will be investigated in future studies.