  For $g>0$, we construct $g+1$ Legendrian embeddings of a surface of genus $g$
into $J^1(R^2)=R^5$ which lie in pairwise distinct Legendrian isotopy classes
and which all have $g+1$ transverse Reeb chords ($g+1$ is the conjecturally
minimal number of chords). Furthermore, for $g$ of the $g+1$ embeddings the
Legendrian contact homology DGA does not admit any augmentation over $Z/2Z$,
and hence cannot be linearized. We also investigate these surfaces from the
point of view of the theory of generating families. Finally, we consider
Legendrian spheres and planes in $J^1(S^2)$ from a similar perspective.
