  The electric dipole moment (EDM) of electron is studied in the supersymmetric
$\rm A_4$ modular invariant theory of flavors with CP invariance. The CP
symmetry of the lepton sector is broken by fixing the modulus $\tau$. Lepton
mass matrices are completely consistent with observed lepton masses and mixing
angles in our model. In this framework, a fixed $\tau$ also causes the CP
violation in the soft SUSY breaking terms. The elecrton EDM arises from the CP
non-conserved soft SUSY breaking terms. The experimental upper bound of the
electron EDM excludes the SUSY mass scale below $2-6$ TeV for five cases of the
lepton mass matrices. In order to see the effect of CP phase of the modulus
$\tau$, we examine the correlation between the electron EDM and the decay rate
of the $\mu \rightarrow e \gamma$ decay, which is also predicted by the soft
SUSY breaking terms. The correlations are clearly predicted in contrast to
models of the conventional flavor symmetry. The SUSY mass scale will be
constrained by the future sensitivity of the electron EDM, $|d_e/e| \simeq
10^{-30}$. Indeed, it could probe the SUSY mass range of $10-20$ TeV in our
model. Thus, the electron EDM provides a severe test of the CP violation via
the modulus $\tau$ in the supersymmetric modular invariant theory of flavors.
