  For the extremely bright lensed galaxy SDSS J1723+3411 at z=1.3293 , we
analyze spatially integrated MMT, Keck, and Hubble Space Telescope spectra that
fully cover the rest-frame wavelength range of 1400 to 7200 Angstroms. We also
analyze near-IR spectra from Gemini that cover H alpha for a portion of the
lensed arc. We report fluxes for 42 detected emission lines, and upper limits
for an additional 22. This galaxy has extreme emission line ratios and high
equivalent widths that are characteristic of extreme emission-line galaxies. We
compute strong emission line diagnostics from both the rest-frame optical and
rest-frame ultraviolet (UV), to constrain physical conditions and test the
spectral diagnostics themselves. We tightly determine the nebular physical
conditions using the most reliable diagnostics, and then compare to results
from other diagnostics. We find disappointing performance from the UV--only
diagnostics: they either are unable to measure the metallicity or dramatically
under-estimate it; they over-estimate the pressure; and the UV diagnostic of
ionization parameter has a strong metallicity dependence in this regime. Based
on these results, we suggest that upcoming James Webb Space Telescope
spectroscopic surveys of galaxies in the reionization epoch should invest the
additional integration time to capture the optical [O II] and [O III] emission
lines, and not rely solely on the rest-frame UV emission lines. We make
available the spectra; they represent one of the highest-quality emission line
spectral atlases of star-forming galaxy available beyond the local universe,
and will aid planning observations with JWST.
