HAT-TR-318-007: A Double-lined M Dwarf Binary with Total Secondary Eclipses Discovered by HATNet and Observed by K2

We report the discovery by the HATNet survey of HAT-TR-318-007, a $P = 3.34395390\pm0.00000020$ d period detached double-lined M-dwarf binary with total secondary eclipses. We combine radial velocity (RV) measurements from TRES/FLWO 1.5 m, and time-series photometry from HATNet, FLWO 1.2 m, BOS 0.8 m and NASA K2 Campaign 5, to determine the masses and radii of the component stars: $M_{A} = 0.448\pm0.011$ $M_{\odot}$, $M_{B} = 0.2721^{+0.0041}_{-0.0042}$ $M_{\odot}$, $R_{A} = 0.4548^{+0.0035}_{-0.0036}$ $R_{\odot}$, and $R_{B} = 0.2913^{+0.0023}_{-0.0024}$ $R_{\odot}$. We obtained a FIRE/Magellan near-infrared spectrum of the primary star during a total secondary eclipse, and use this to obtain disentangled spectra of both components. We determine spectral types of ST$_{A} = {\rm M}3.71\pm0.69$ and ST$_{B} = {\rm M}5.01\pm0.73$, and effective temperatures of T$_{\rm eff,A} = 3190\pm110$ K and T$_{\rm eff,B} = 3100\pm110$ K, for the primary and secondary star, respectively. We also measure a metallicity of [Fe/H]$=+0.298\pm0.080$ for the system. We find that the system has a small, but significant, non-zero eccentricity of $0.0136\pm0.0026$. The K2 light curve shows a coherent variation at a period of $3.41315^{+0.00030}_{-0.00032}$ d, which is slightly longer than the orbital period, and which we demonstrate comes from the primary star. We interpret this as the rotation period of the primary. We perform a quantitative comparison between the Dartmouth stellar evolution models and the seven systems, including HAT-TR-318-007, that contain M dwarfs with $0.2 M_{\odot} < M < 0.5 M_{\odot}$, have metallicity measurements, and have masses and radii determined to better than 5% precision. Discrepancies between the predicted and observed masses and radii are found for three of the systems.