  The PHENIX experiment has measured the spin alignment for inclusive
$J/\psi\rightarrow e^{+}e^{-}$ decays in $p$+$p$ collisions at $\sqrt{s}=510$
GeV at midrapidity. The angular distributions have been measured in three
different polarization frames, and the three decay angular coefficients have
been extracted in a full two-dimensional analysis. Previously, PHENIX saw large
longitudinal net polarization at forward rapidity at the same collision energy.
This analysis at midrapidity, complementary to the previous PHENIX results,
sees no sizable polarization in the measured transverse momentum range of
$0.0<p_T<10.0$ GeV/$c$. The results are consistent with a previous
one-dimensional analysis at midrapidity at $\sqrt{s}=200$ GeV. The
transverse-momentum-dependent cross section for midrapidity $J/\psi$ production
has additionally been measured, and after comparison to world data we find a
simple logarithmic dependence of the cross section on $\sqrt{s}$.
