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One of the main scientific goals of the TESS mission is the discovery oftransiting small planets around the closest and brightest stars in the sky.Here, using data from the CARMENES, MAROON-X, and HIRES spectrographs, togetherwith TESS, we report the discovery and mass determination of a planetary systemaround the M1.5 V star GJ 806 (TOI-4481). GJ 806 is a bright (V=10.8 mag, J=7.3mag) and nearby (d=12 pc) M dwarf that hosts at least two planets. Theinnermost planet, GJ 806 b, is transiting and has an ultra-short orbital periodof 0.93 d, a radius of 1.331+-0.023 Re, a mass of 1.90+-0.17 Me, a mean densityof 4.40+-0.45 g/cm3, and an equilibrium temperature of 940+-10 K. We detect asecond, non-transiting, super-Earth planet in the system, GJ 806c, with anorbital period of 6.6 d, a minimum mass of 5.80+-0.30 Me, and an equilibriumtemperature of 490+-5 K. The radial velocity data also shows evidence for athird periodicity at 13.6 d, although the current dataset does not providesufficient evidence to unambiguously distinguish between a third super-Earthmass (Msin(i)=8.50+-0.45 Me) planet or stellar activity. Additionally, wereport one transit observation of GJ 806 b taken with CARMENES in search for apossible extended atmosphere of H or He, but we can only place upper limits toits existence. This is not surprising as our evolutionary models support theidea that any possible primordial H/He atmosphere that GJ 806 b might have had,would long have been lost. However, GJ 806b's bulk density makes it likely thatthe planet hosts some type of volatile atmosphere. In fact, with a transmissionspectroscopy metrics (TSM) of 44 and an emission spectroscopy metrics (ESM) of24, GJ 806 b the third-ranked terrestrial planet around an M dwarf suitable fortransmission spectroscopy studies, and the most promising terrestrial planetfor emission spectroscopy studies.

GJ 806 (TOI-4481): A bright nearby multi-planetary system with a transiting hot, low-density super-Earth