Revealing the nature of the ULX and X-ray population of the spiral galaxy NGC 4088

We present the first $\mathit{\text{Chandra}}$ and $\mathit{\text{Swift}}$ X-ray study of the spiral galaxy NGC,4088 and its ultraluminous X-ray source (ULX N4088–X1). We also report very long baseline interferometry (VLBI) observations at 1.6 and 5 GHz performed quasi-simultaneously with the $\mathit{\text{Swift}}$ and $\mathit{\text{Chandra}}$ observations, respectively. Fifteen X-ray sources are detected by $\mathit{\text{Chandra}}$ within the D25 ellipse of NGC,4088, from which we derive the X-ray luminosity function (XLF) of this galaxy. We find the XLF is very similar to those of star-forming galaxies and estimate a star-formation rate of 4.5 $M_{\odot }$ yr${}^{-1}$. The $\mathit{\text{Chandra}}$ detection of the ULX yields its most accurate X-ray position, which is spatially coincident with compact radio emission at 1.6 GHz. The ULX $\mathit{\text{Chandra}}$ X-ray luminosity, $L_{0.2-10.0\mathrm{k}\mathrm{e}\mathrm{V}}=3.4\times$ 10${}^{39}$ erg s${}^{-1}$, indicates that N4088–X1 could be located at the high-luminosity end of the high-mass X-ray binary (HMXB) population of NGC,4088. The estimates of the black hole mass and ratio of radio to X-ray luminosity of N4088–X1 rule out a supermassive black hole nature. The $\mathit{\text{Swift}}$ X-ray spectrum of N4088–X1 is best described by a thermal Comptonization model and presents a statistically significant high-energy cut-off. We conclude that N4088–X1 is most likely a stellar remnant black hole in a HMXB, probably fed by Roche lobe overflow, residing in a super-Eddington ultraluminous state. The 1.6 GHz VLBI source is consistent with radio emission from possible ballistic jet ejections in this state.