I would like to know if one can compute all the cohomology sheaves of the cotangent complex of a subvariety of the affine space once a resolution of its ideal sheaf is given?

In my precise situation, I work with $X \subset \mathbb{A}^n$ Gorenstein of codimension 3 and I have a resolution:

$$0 \rightarrow \mathcal{O}_{\mathbb{A}^n} \rightarrow E^* \rightarrow E \rightarrow I_X \rightarrow 0$$

Is it possible to compute the cotangent complex of the closed immersion $ X \hookrightarrow A^n$ from this resolution?

Many thanks!

I would like to know if one can compute all the cohomology sheaves of the cotangent complex of a subvariety of the affine space once a resolution of its ideal sheaf is given?

In my precise situation, I work with $X \subset \mathbb{A}^n$ Gorenstein of codimension 3 and I have a resolution:

$$0 \rightarrow \mathcal{O}_{\mathbb{A}^n} \rightarrow E^* \rightarrow E \rightarrow I_X \rightarrow 0$$

Is it possible to compute the cotangent complex of the closed immersion $ X \hookrightarrow \mathbb{A}^n$ from this resolution?

Many thanks!

EDIT : I really want the cotangent complex of the closed immersion $X \hookrightarrow \mathbb{A}^n$ (and not of the map $X \rightarrow spec(k)$).