$H_0$ tension or $M$ overestimation?
  There is a strong discrepancy between the value of the Hubble parameter
$H_0^P$ obtained from large scale observations such as the Planck mission, and
the small scale value $H_0^R$, obtained from low redshift supernovae (SNe). The
value of the absolute magnitude $M^{hom}$ used as prior in analyzing
observational data is obtained from low-redshift SNe, assuming a homogeneous
Universe, but the distance of the anchors used to calibrate the SNe to obtain
$M$ would be affected by a local inhomogeneity, making it inconsistent to test
the Copernican principle using $M^{hom}$, since $M$ estimation itself is
affected by local inhomogeneities.
  We perform an analysis of the luminosity distance of low redshift SNe, using
different values of $M$, $\{M^P,M^R\}$, corresponding to different values of
$H_0$, $\{H_0^P,H_0^R\}$, obtained from the model independent consistency
relation between $H_0$ and $M$ which can be derived from the definition of the
distance modulus. We find that the value of $M$ can strongly affect the
evidence of a local inhomogeneity. We analyze data from the Pantheon catalog,
finding no significant statistical evidence of a local inhomogeneity using the
parameters $\{M^R,H_0^R\}$, confirming previous studies, while with
$\{M^P,H_0^P\}$ we find evidence of a small local void, which cause an
overestimation of $M^R$ with respect to $M^P$.
  An inhomogeneous model with the parameters $\{M^P,H_0^P\}$ fits the data
better then a homogeneous model with $\{M^R,H_0^R\}$, resolving the apparent
$H_0$ tension. Using $\{M^P,H_0^P\}$, we obtain evidence of a local
inhomogeneity with a density contrast $-0.140 \pm 0.042 $, extending up to a
redshift of $z_v =0.056 \pm 0.0002$, in good agreement with recent results of
galaxy catalogs analysis.
