This paper focuses on carbon that is one of the most abundant elements in the Universe and is of high importance in the field of nucleosynthesis and galactic and stellar evolution. Even nowadays, the origin of carbon and the relative importance of massive and low- to intermediate-mass stars in producing it is still a matter of debate. In this paper we aim at better understanding the origin of carbon by studying the trends of [C/H], [C/Fe],and [C/Mg] versus [Fe/H], and [Mg/H] for 2133 FGK dwarf stars from the fifth Gaia-ESO Survey internal data release (GES iDR5). The availability of accurate parallaxes and proper motions from Gaia DR2 and radial velocities from GES iDR5 allows us to compute Galactic velocities, orbits and absolute magnitudes and, for 1751 stars, ages via a Bayesian approach. Three different selection methodologies have been adopted to discriminate between thin and thick disk stars. In all the cases, the two stellar groups show different abundance ratios, [C/H], [C/Fe], and [C/Mg], and span different age intervals, with the thick disk stars being, on average, older than those in the thin disk. The behaviours of [C/H], [C/Fe], and [C/Mg] versus [Fe/H], [Mg/H], and age all suggest that C is primarily produced in massive stars like Mg. The increase of [C/Mg] for young thin disk stars indicates a contribution from low-mass stars or the increased C production from massive stars at high metallicities due to the enhanced mass loss. The analysis of the orbital parameters Rmed and |Zmax| support an inside-out and upside-down formation scenario for the disks of Milky Way.
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