The BL Lacertae object 1ES 1440+122 was observed in the energy range from 85 GeV to 30 TeV by the VERITAS array of imaging atmospheric Cherenkov telescopes. The observations, taken between 2008 May and 2010 June and totalling 53 hours, resulted in the discovery of $gamma$-ray emission from the blazar, which has a redshift $z$=0.163. 1ES 1440+122 is detected at a statistical significance of 5.5 standard deviations above the background with an integral flux of (2.8$pm0.7_{mathrm{stat}}pm0.8_{mathrm{sys}}$) $times$ 10$^{-12}$ cm$^{-2}$ s$^{-1}$ (1.2% of the Crab Nebulas flux) above 200 GeV. The measured spectrum is described well by a power law from 0.2 TeV to 1.3 TeV with a photon index of 3.1 $pm$ 0.4$_{mathrm{stat}}$ $pm$ 0.2$_{mathrm{sys}}$. Quasi-simultaneous multi-wavelength data from the Fermi Large Area Telescope (0.3--300 GeV) and the Swift X-ray Telescope (0.2--10 keV) are additionally used to model the properties of the emission region. A synchrotron self-Compton model produces a good representation of the multi-wavelength data. Adding an external-Compton or a hadronic component also adequately describes the data.
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