Integral basis of the cubic field of discriminant minus 307
= Integral basis of the cubic field of discriminant minus 307
The elements
$$
1,\qquad\alpha,\qquad\beta=\frac{\alpha+\alpha^2}{2}
$$
form an integral basis. Relative to this basis, multiplication by $\beta$ has integer matrix
$$
\begin{pmatrix}
0&-4&-4\\
0&2&-1\\
1&1&3
\end{pmatrix}.
$$
The basis discriminant is one quarter of the discriminant $-4\cdot307$ of $(1,\alpha,\alpha^2)$, hence is the square-free integer $-307$.