  We carry out numerical studies of static packings of frictionless
superellipsoidal particles in three spatial dimensions. We consider more than
$200$ different particle shapes by varying the three shape parameters that
define superellipsoids. We characterize the structural and mechanical
properties of both disordered and ordered packings using two packing-generation
protocols. We perform athermal quasi-static compression simulations starting
from either random, dilute configurations (Protocol 1) or thermalized, dense
configurations (protocol $2$), which allows us to tune the orientational order
of the packings. In general, we find that the contact numbers at jamming onset
for superellipsoid packings are hypostatic, with $z_J < z_{\rm iso}$, where
$z_{\rm iso} = 2d_f$ and $d_f = 5$ or $6$ depending on whether the particles
are axi-symmetric or not. Over the full range of orientational order, we find
that the number of quartic modes of the dynamical matrix for the packings
always matches the number of missing contacts relative to the isostatic value.
This result suggests that there are no mechanically redundant contacts for
ordered, yet hypostatic packings of superellipsoidal particles. Additionally,
we find that the packing fraction at jamming onset for disordered packings of
superellipsoidal particles can be collapsed using two particle shape
parameters, e.g. the asphericity $\mathcal{A}$ and reduced aspect ratio $\beta$
of the particles.
