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Empirical calibrations of the stellar age-rotation-activity relation (ARAR)rely on observations of the co-eval populations of stars in open clusters. Weused the Chandra X-ray Observatory to study M37, a 500-Myr-old open clusterthat has been extensively surveyed for rotation periods ($P_{\rm rot}$). M37was observed almost continuously for five days, for a total of 440.5 ksec, tomeasure stellar X-ray luminosities ($L_{\mathrm{X}}$), a proxy for coronalactivity, across a wide range of masses. The cluster's membership catalog wasrevisited to calculate updated membership probabilities from photometric dataand each star's distance to the cluster center. The result is a comprehensivesample of 1699 M37 members: 426 with $P_{\rm rot}$, 278 with X-ray detections,and 76 with both. We calculate Rossby numbers, $R_o = P_{\rm rot}/\tau$, where$\tau$ is the convective turnover time, and ratios of the X-ray-to-bolometricluminosity, $L_{\rm X}/L_{\rm bol}$, to minimize mass dependencies in ourcharacterization of the rotation-coronal activity relation at 500 Myr. We findthat fast rotators, for which $R_o<0.09\pm0.01$, show saturated levels ofactivity, with log($L_{\rm X}/L_{\rm bol}$)$=-3.06\pm0.04$. For$R_o\geq0.09\pm0.01$, activity is unsaturated and follows a power law of theform $R_o^\beta$, where $\beta$=$-2.03_{-0.14}^{+0.17}$. This is the largestsample available for analyzing the dependence of coronal emission on rotationfor a single-aged population, covering stellar masses in the range 0.4$-$1.3$M_{\odot}$, $P_{\rm rot}$ in the range 0.4$-$12.8 d, and $L_{\rm X}$ in therange 10$^{28.4-30.5}$ erg s$^{-1}$. Our results make M37 a new benchmark opencluster for calibrating the ARAR at ages of $\approx$500 Myr.

Linking Stellar Coronal Activity and Rotation at 500 Myr: A Deep Chandra Observation of M37