  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 (SNIa) host galaxy NGC 5584 played a key role in the local
measurement of $H_0$. The SH0ES project used the observations of this galaxy to
derive a relation between Cepheids' periods and ratios of their amplitudes in
different optical bands of the Hubble Space Telescope (HST), and used these
relations to analyse the light curves of the Cepheids in around half of the
current sample of local SNIa host galaxies. In this work, we present an
independent detailed analysis of the Cepheids in NGC 5584. 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 $\mu=31.810\pm0.047$ (mag) which is in
agreement with $\mu=31.786\pm0.046$ (mag) measured by SH0ES. In addition, the
relations we find between periods and amplitude ratios of the Cepheids in NGC
5584 are significantly tighter than those of SH0ES and their potential impact
on the direct $H_0$ measurement will be investigated in future studies.
