  We present a study of the neutron electric dipole moment ($\vec d_N$) within
the framework of lattice QCD with two flavors of dynamical lig ht quarks. The
dipole moment is sensitive to the topological structure of the gaug e fields,
and accuracy can only be achieved by using dynamical, or sea quark, calc
ulations. However, the topological charge evolves slowly in these calculations,
le ading to a relatively large uncertainty in $\vec d_N$. It is shown, using
quenched configurations, that a better sampling of the charge d istribution
reduces this problem, but because the CP even part of the fermion determinant
is absent, both the topological charge dis tribution and $\vec d_N$ are
pathological in the chiral limit. We discuss the statistical and systematic
uncertainties arising from the topological charge distr ibution and unphysical
size of the quark mass in our calculations and prospects fo r eliminating them.
  Our calculations employ the RBC collaboration two flavor domain wall fermion
and DBW2 gauge action lattices with inverse lattice spacing $a^{-1}\approx$ 1.7
GeV, physical volume $V\approx (2$ fm)$^3$, and light quark mass roughly equal
to the strange quark mass ($m_{sea}=0.03 $ and 0.04). We determine a value of
the electric dipole moment that is zero withi n (statistical) errors, $|\vec
d_N| = -0.04(20)$ e-$\theta$-fm at the smaller sea quark mass. Satisfactory
results for the magnetic and electric form factors of the proton and neutron
are also obtained and presented.
