Extreme high-energy peaked BL Lac objects (EHBLs) are an emerging class of blazars with exceptional spectral properties. The non-thermal emission of the relativistic jet peaks in the spectral energy distribution (SED) plot with the synchrotron emission in X-rays and with the gamma-ray emission in the TeV range or above. These high photon energies may represent a challenge for the standard modeling of these sources. They are important for the implications on the indirect measurements of the extragalactic background light, the intergalactic magnetic field estimate, and the possible origin of extragalactic high-energy neutrinos. In this paper, we perform a comparative study of the multi-wavelength spectra of 32 EHBL objects detected by the Swift-BAT telescope in the hard X-ray band and by the Fermi-LAT telescope in the high-energy gamma-ray band. The source sample presents uniform spectral properties in the broad-band SEDs, except for the TeV gamma-ray band where an interesting bimodality seems to emerge. This suggests that the EHBL class is not homogeneous, and a possible sub-classification of the EHBLs may be unveiled. Furthermore, in order to increase the number of EHBLs and settle their statistics, we discuss the potential detectability of the 14 currently TeV gamma-ray undetected sources in our sample by the Cherenkov telescopes.
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