An algorithm to compute relative cubic fields

Let $K$ be an imaginary quadratic number field with class number $1$. We describe a new, essentially linear-time algorithm, to list all isomorphism classes of cubic extensions $L/K$ up to a bound $X$ on the norm of the relative discriminant ideal. The main tools are Taniguchi's [18] generalization of Davenport-Heilbronn parametrisation of cubic extensions, and reduction theory for binary cubic forms over imaginary quadratic fields. Finally, we give numerical data for $K=\mathbb{Q}(i)$, and we compare our results with ray class field algorithm results, and with asymptotic heuristics, based on a generalization of Roberts' conjecture [19].