This is a somewhat embarrassing question, but still I will ask it. Let $V$ be a vector space over $\mathbb C$ of dimension $d$. Let $X$ be the dg-preimage of $0$ under the natural map $V\to Sym^2(V)$ (i.e. impose equations of the form $f=0$ for all homogeneous polynomials of degree 2 on $V$; we have $d\choose{2}$ equations, so $X$ always has a non-trivial dg-part for $d>1$).

$\mathbf{Question:}$ How does the cotangent complex of $X$ look like? More precisely, I want to know its restriction to the unique $\mathbb{C}$-point of $X$. I would be happy to understand the case $d=2$.