Cubic field and the corresponding cubic binary form I am currently reading about binary cubic forms and cubic number fields (mainly about using binary cubic forms with integer coefficients to parametrize orders in the cubic field) and I thought it might be good to do some computations to understand how this theory works by taking a concrete example.
Given a cubic field I want to find the corresponding binary cubic form associated with it or rather with the ring of integer of $K$ should I say? 
Suppose we take  $$K=\mathbb{Q}(\sqrt[3]{2})\supset\mathbb{Q}$$ 
Since  $$x^{3}-2\in\mathbb{Q}[x]$$ is irreducible over $\mathbb{Q}$ by Eisenstein's criterion with $p=2$ we get 
$$[K:\mathbb{Q}]=3$$
The ring of integers of $K$ is $$\mathcal{O}_{k}=\mathbb{Z}[\alpha]=\mathbb{Z}[\sqrt[3]{2}]$$ with integral basis $\{1,\alpha,\alpha^2\}$ hence 
$$\mathbb{Z}[\sqrt[3]{2}]=\{a+b\sqrt[3]{2}+c\sqrt[3]{2^2}|a,b,c\in\mathbb{Z}\}$$
In the next step I would like to find (if possible) the corresponding binary cubic form. Will this form correspond to the field of to the ring?
Following Belabas and Cohen ''Binary cubic forms and cubic number fields'':
Let $K$ be number field defined by a root $\theta$ of the polynomial $x^3+px^2+qx+r$ with $p,q,r \in\mathbb{Z}$ such that there exists an integral basis of the form $(1,\theta,(\theta^2+t\theta+u)/f)$ with $t,u,f\in \mathbb{Z}$ and $f=[\mathbb{Z}_{k}:\mathbb{Z}[\theta]]$ Then we choose $\alpha=\theta$ and $\beta=(\theta^2+t\theta+u)/f)$ we have explicitly:
$$\begin{split}
F_{B}(x,y)&=((t^3-2t^2p+t(q+p^2)+r-pq)/f^2)x^3\\&+((-3t^2+4tp-(p^2+q))/f))x^2y\\
&+(3t-2p)xy^2-fy^3\end{split}$$
In case of $x^{3}-2$ we have $\begin{cases}q=0\\r=-2\\p=0\end{cases}$
And $f=[\mathbb{Z}_{k}:\mathbb{Z}[\theta]]=1$
Plugging everything in I get something like this
$F_{B}(x,y)=-2x^3-y^3$
Does this work? Or is it completely incorrect? Is there any other algorithm for finding such form? 
Thank you.
 A: I get the form $$\mathcal{C}(x,y) = (a, b, c, d) = (1, 3, 3, -1) = x^3 + 3 x^2 y + 3 x y^2 - y^3$$ corresponding to the image of the Davenport-Heilbronn map but there are infinitely many acceptable binary cubic forms, in a class corresponding to the image, related to each other by a substitution $\begin{pmatrix}{x}\\{y}\end{pmatrix} = M \begin{pmatrix}{x'}\\{y'}\end{pmatrix}$, where $M = \begin{pmatrix}{\alpha}&{\beta}\\{\gamma}&{\delta}\end{pmatrix} \in GL_2(\mathbb{Z})$. The form $\mathcal{C}(x, y)$ is transformed into $\mathcal{C}'(x',y') = -2 x'^3 - y'^3$ using $M = \begin{pmatrix}{-1}&{-1}\\{1}&{0}\end{pmatrix}$ so your calculations are correct. 
I do not remember whether the $GL_2(\mathbb{Z})$ class is said to correspond to the field or to the ring but it certainly contains useful information about both the field $K$ and the ring of integers. The ring of integers can be read from the the coefficients of the form. The integral basis can be expressed as $\{1, a \zeta , a \zeta^2 + b \zeta \}$, where $\zeta$ satisfies $\mathcal{C}(\zeta , 1) = 0$. Also the field $ \mathbb{Q} (\zeta)$ is isomorphic to $\mathbb{Q}(2^{1/3})$.
Yes there is an algorithm for computing such a form: 


*

*Calculate an integral basis: $\{ 1, \omega_2, \omega_3 \}$ for the ring of integers $\mathcal{O}_K$ of $K$.

*Calculate the discriminant $\Delta$ of the cubic field $K$.

*Calculate a form
$ \frac{1}{\sqrt{\Delta }} \prod_{(i, j) \in \Theta } (\omega_2^{\tau^i} - \omega_2^{\tau^j}) x + (\omega_3^{\tau^i} - \omega_3^{\tau^j}) y $, where $\Theta = \{ (0, 1), (0, 2), (1, 2) \}$ and $\tau^i  , \tau^j $ are the embeddings of $K$.

