We present Keck/MOSFIRE (Multi-Object Spectrometer for InfraRed Exploration) and Keck/LRIS (Low Resolution Imaging Spectrometer) spectroscopy of A1689-217, a lensed (magnification ∼7.9), star-forming (SFR ∼ 16 M yr-1), dwarf (log(M ∗/M ) = 8.07-8.59) Lyα emitter (EW0 ∼ 138 Å) at z = 2.5918. Dwarf galaxies similar to A1689-217 are common at high redshift and likely responsible for reionization, yet few have been studied with detailed spectroscopy. We report a 4.2σ detection of the electron-temperature-sensitive [O iii] λ4363 emission line and use this line to directly measure an oxygen abundance of 12 + log(O/H) = 8.06 ± 0.12 (∼1/4 Z ). A1689-217 is the lowest-mass galaxy at z > 2 with an [O iii] λ4363 detection. Using the rest-optical emission lines, we measure A1689-217's other nebular conditions, including electron temperature (T e([O iii]) ∼ 14,000 K), electron density (n e ∼ 220 cm-3), and reddening (E(B-V) ∼ 0.39). We study relations between strong-line ratios and direct metallicities with A1689-217 and other galaxies with [O iii] λ4363 detections at z ∼ 0-3.1, showing that the locally calibrated, oxygen-based, strong-line relations are consistent from z ∼ 0 to 3.1. We also show additional evidence that the O32 versus R23 excitation diagram can be utilized as a redshift-invariant, direct-metallicity-based, oxygen abundance diagnostic out to z ∼ 3.1. From this excitation diagram and the strong-line ratio-metallicity plots, we observe that the ionization parameter at fixed O/H is consistent with no redshift evolution. Although A1689-217 is metal-rich for its M ∗ and star formation rate, we find it to be consistent within the large scatter of the low-mass end of the fundamental metallicity relation.
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