Cepheid metallicity in the Leavitt law (C-MetaLL) survey – III. Simultaneous derivation of the<i>Gaia</i>parallax offset and period–luminosity–metallicity coefficients

Classical Cepheids (DCEPs) are the most important standard candles in extra-galactic distance scale thanks to Period-Luminosity ($\rm PL$), Period-Luminosity-Color PLC$) and Period-Wesenheit PW$) relations that hold for these objects. The advent of {\it Gaia} mission, particular Early Data Release 3 (EDR3) provided accurate parallaxes calibrate relations. In order fully exploit measurements, zero point (ZP) should be determined with an accuracy a few $\rm \mu as$. individual ZP corrections by team depend on magnitude position sky target. this paper, we use implicit method relies Cepheid PL$ PW$ evaluate ensemble parallax point. best inferred estimation offset value needed additionally correct (after correction) present DCEP sample, amounts -22\pm 4\, This is agreement recent literature values confirms correction proposed over-corrected parallaxes.\\ As further application our results, derive estimate Large Magellanic Cloud \mu_0=18.49\pm 0.06\, mag$), very good currently accepted obtained through geometric methods.