$p$ is a prime and $M$ is an $F_p[X]-$ module. Given $(X-1)^3M=0, \vert (X-1)^2M\vert=p, \vert (X-1)M\vert=p^3$ and $\vert M\vert =p^7$, determine $M$ as an $F_p[X]-$module, up to isomorphism.
Using the structure theorem for modules over a PID, we get
$M\cong F_p[X]/(x-1)^{a_1}\oplus F_p[X]/((x-1)^2)^{a_2}\oplus F_p[X] /((x-1)^3)^{a_3}$
From the given info we can deduce $a_3=0,a_2=2,a_1=3$ so that the invariant factors are $x-1,x-1,x-1,(x-1)^2, (x-1)^2$. Is my work correct? What exactly is the relationship between the orders of $\vert (X-1)^kM\vert$ and $a_k$?