Aspects of this question have been thoroughly treated in The Classification of Finite Simple Groups, Number 3, by Gorenstein-Lyons-Solomon, AMS, 1994: see especially their Table 3.3.1 for the Chevalley groups. In your notation (which differs somewhat from theirs), the $p$-rank is $16a$. This gives only the rank of a maximal elementary abelian $p$-subgroup, however.

P.S. Concerning maximal abelian $p$-subgroups of $E_6$, the relevant table in Vdovin's thesis (linked by Nick Gill) seems to give the same answer $p^{16a}$.

Aspects of this question have been thoroughly treated in The Classification of Finite Simple Groups, Number 3, by Gorenstein-Lyons-Solomon, AMS, 1994: see especially their Table 3.3.1 for the Chevalley groups. In your notation (which differs somewhat from theirs), the $p$-rank is $16a$. This gives only the rank of a maximal elementary abelian $p$-subgroup, however.

P.S. Concerning maximal abelian $p$-subgroups of $E_6$, the relevant table in Vdovin's thesis (linked by Nick Gill) seems to give the same answer $p^{16a}$. Probably the point here is that the 16 "commuting" positive roots yield the only possible maximal abelian $p$-subgroups in a Chevalley group, automatically elementary abelian because of the structure of root groups. The emphasis on $p$-rank comes mainly from the connection with cohomological support varieties and such. Of course, Sylow $p$-subgroups are all conjugate, so their subgroup structure is what one needs to know.

Aspects of this question have been thoroughly treated in The Classification of Finite Simple Groups, Number 3, by Gorenstein-Lyons-Solomon, AMS, 1994: see especially their Table 3.3.1 for the Chevalley groups. In your notation (which differs somewhat from theirs), the $p$-rank is $16a$. This gives only the rank of a maximal elementary abelian $p$-subgroup, however.

Aspects of this question have been thoroughly treated in The Classification of Finite Simple Groups, Number 3, by Gorenstein-Lyons-Solomon, AMS, 1994: see especially their Table 3.3.1 for the Chevalley groups. In your notation (which differs somewhat from theirs), the $p$-rank is $16a$. This gives only the rank of a maximal elementary abelian $p$-subgroup, however.

P.S. Concerning maximal abelian $p$-subgroups of $E_6$, the relevant table in Vdovin's thesis (linked by Nick Gill) seems to give the same answer $p^{16a}$.