MOV attack when $E(\mathbb{F}_q)$ is cyclic Suppose $P\in E(\mathbb{F}_q)$ and $R=dP$. 

In the MOV attack, we compute $\alpha=e(P,T)$ and $\beta=e(R,T)$ and try to solve the discrete logarithm problem for $\alpha$ and $\beta$ in the finite field $\mathbb{F}_{q^k}$ where $k$ is the embedding degree. 

But if $E(\mathbb{F}_q)$ is cyclic (which it can be even in the supersingular case), $\alpha,\beta$ are doomed to be $1$. 

So what am I missing here?
For the purposes of the MOV attack you choose $T$ to be a point in $E(\mathbb{F}_{q^k})$.