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Let $K/k$ be a finite Galois extension, $G=\text{Gal}(K/k)$, $\Sigma$ be afan in a lattice $N$ and $X_{\Sigma}$ be an associated toric variety over $k$.It is well known that the set of $K/k$-forms of $X_{\Sigma}$ is in bijectionwith $H^1(G,\text{Aut}_{\Sigma}^T)$, where $\text{Aut}_{\Sigma}^T$ is analgebraic group of toric automorphisms of $X_{\Sigma}$. In this paper, wesuggest an algorithm to compute $H^1(G,\text{Aut}_{\Sigma}^T)$ and find thatfollowings can be classified via this algorithm : $K/k$-forms of all toricsurfaces, $K/k$-forms of all 3-dimensional affine toric varieties with no torusfactor, $K/k$-forms of all 3-dimensional quasi-projective toric varieties when$K/k$ is cyclic.

An algorithm for the classification of twisted forms of toric varieties