  The generation of the right amount of baryon asymmetry $\eta$ of the Universe
from supersymmetric leptogenesis is studied within the type-I seesaw framework
with three heavy singlet Majorana neutrinos $N_i\,\,(i = 1,2,3)$ and their
superpartners. We assume the occurrence of four zeroes in the neutrino Yukawa
coupling matrix $Y_\nu$, taken to be $\mu\tau$ symmetric, in the weak basis
where $N_i$ (with real masses $M_i>0$) and the charged leptons $l_\alpha\,\,
(\alpha = e,\mu,\tau)$ are mass diagonal. The quadrant of the single nontrivial
phase, allowed in the corresponding light neutrino mass matrix $m_\nu$, gets
fixed and additional constraints ensue from the requirement of matching $\eta$
with its observed value. Special attention is paid to flavor effects in the
washout of the lepton asymmetry. We also comment on the role of small
departures from high scale $\mu\tau$ symmetry due to RG evolution.
