Let $\mathbb F_2$ denote the binary field. Write
$\mathscr A$ for the Steenrod ring over
$\mathbb F_2.$ In this paper, we study Singer’s
conjecture [Singer] for the algebraic transfers of ranks 5 and
6 in the generic families of internal degrees. The Singer algebraic
transfer stands as a valuable instrument for unraveling the intricate
structure of the cohomology ${\rm
Ext}_{\mathscr A}^{s,k} := {\rm
Ext}_{\mathscr A}^{s}(\mathbb
F_2, \Sigma^{k}\mathbb F_2)$ of
the (mod-2) Steenrod ring. Remarkably, we have shown that the
indecomposable element $y\in {\rm
Ext}_{\mathscr A}^{6,44}$ is not in the image
of the sixth algebraic transfer.