The host galaxy of a narrow-line Seyfert 1 galaxy, RE J1034+396, with X-ray quasi-periodic oscillations

Using simple stellar population synthesis, we model the bulge stellar contribution in the optical spectrum of a narrow-line Seyfert 1 galaxy, RE J1034+396. We find that its bulge stellar velocity dispersion is 67.7 +/- 8 km s-1. The supermassive black hole (SMBH) mass is about (1-4) x 106 M(circle dot) if it follows the well-known M(BH)-Sigma(*) relation found in quiescent galaxies. We also derive the SMBH mass from the H beta second moment, which is consistent with that from its bulge stellar velocity dispersion. The SMBH mass of (1-4) x 106 M(circle dot) implies that the X-ray quasi-periodic oscillation (QPO) of RE J1034+396 can be scaled to a high-frequency QPO at 27-108 Hz found in Galactic black hole binaries with a 10-M(circle dot) black hole. With the mass distribution in different age stellar populations, we find that the mean specific star formation rate (SSFR) over the past 0.1 Gyr is 0.0163 +/- 0.0011 Gyr-1, the stellar mass in the logarithm is 10.155 +/- 0.06 in units of solar mass and the current star formation rate is 0.23 +/- 0.016 M(circle dot) yr-1. For RE J1034+396, there is no relation between the Eddington ratio and the SSFR as suggested by Chen et al., despite a larger scatter in their relation. We also suggest that about 7.0 per cent of the total H alpha luminosity and 50 per cent of the total [O ii] luminosity come from the star formation process.