supersymmetry - A certain $cal{N}=2$ superconformal theory (or is it?)

I want to look at the following theory in $1+1$ dimensions with $\Phi$ being the chiral superfield,

$L = \int d^2x d^4\theta \bar{\Phi}\Phi - \int d^2x d^2\theta \frac{\Phi^{k+2}}{k+2} - \int d^2x d^2\bar{\theta} \frac{\bar{\Phi}^{k+2}}{k+2} $

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  How does one show that the above theory has the $\cal{N}=2$ superconformal symmetry? (..I guess that is a claim that I see in various literature..)

  
  


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  How does one calculate the charge of the chiral primary states in this theory and which is claimed to be $\frac{n}{k+2}$ for $n=0,1,2,..,k$? And can one explicitly enumerate those states?

  
  


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  How does one show that the index $Tr(-1)^F$ for the potential $\frac{\Phi^{k+2}}{k+2}$ is $k+1$?