Euclidean minimum of the cubic field of discriminant -87

Determine the Euclidean minimum of the cubic number field of discriminant $-87$, and establish whether it equals $1/3$ and is attained at infinitely many irrational points.

Background

The paper presents numerical and exact Euclidean-minimum data for cubic fields of small discriminant. For discriminant −87-87, it explicitly states that the minimum had not been determined and gives a tentative value together with a conjectural description of its attainment set.

References

The Euclidean minimum of the field of discriminant $-87$ I have not yet been able to determine; probably (?), however, $M(K) = 1/3$, where this minimum, besides at the two rational points given above, is also attained at infinitely many irrational points (as in $(\sqrt{13})$).

— Euclidean Rings  (2608.23216 - Lemmermeyer, 24 Aug 2026) in Section 4, paragraph following the table of cubic Euclidean minima