  The infrared dynamics of $2+1$ dimensional quantum electrodynamics (QED$_3$)
with a large number $N$ of fermion flavors is governed by an interacting CFT
that can be studied in the $1/N$ expansion. We use the $1/N$ expansion to
calculate the scaling dimensions of all the lowest three scalar operators that
transform under the $SU(N)$ flavor symmetry as a Young diagram with two columns
of not necessarily equal heights and that have vanishing topological charge. In
the case of $SU(N)$ singlets, we study the mixing of $(\bar \psi_i \psi^i)(\bar
\psi_j \psi^j)$ and $F_{\mu\nu} F^{\mu\nu}$, which are the lowest dimension
parity-even singlets. Our results suggest that these operators are irrelevant
for all $N>1$.
