Eridanus II (EriII) is an ultra-faint dwarf (UFD) galaxy (M_V=-7.1) located at a distance close to the Milky Way virial radius. Early shallow color-magnitude diagrams (CMD) indicated that it possibly hosted an intermediate-age or even young stellar population, which is unusual for a galaxy of this mass. In this paper, we present new ACS/HST CMDs reaching the oldest main sequence turnoff with excellent photometric precision, and derive a precise star formation history (SFH) for this galaxy through CMD-fitting. This SFH shows that the bulk of the stellar mass in Eri II formed in an extremely short star formation burst at the earliest possible time. The derived star formation rate profile has a width at half maximum of 500 Myr and reaches a value compatible with null star formation 13 Gyr ago. However, tests with mock stellar populations and with the CMD of the globular cluster M92 indicate that the star formation period could be shorter than 100 Myr. From the quantitative determination of the amount of mass turned into stars in this early star formation burst (~2x10^5 Msun) we infer the number of SNe events and the corresponding energy injected into the interstellar medium. For reasonable estimates of the EriII virial mass and values of the coupling efficiency of the SNe energy, we conclude that EriII could be quenched by SNe feedback alone, thus casting doubts on the need to invoke cosmic reionization as the preferred explanation for the early quenching of old UFD galaxies.
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