On a conjecture for the algebraic transfer in generic families of internal degrees

The binary field will be symbolized as $\mathbb F_2$ in the customary manner, whilst the Steenrod algebra over this field shall be written as $\mathscr A.$ In this paper, we study Singer's conjecture [Math. Z. \textbf{202} (1989), 493-523] 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 $\mathscr A.$ 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.