An algorithm to compute relative cubic fields
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Abstract:
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].References
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Additional Information
- Anna Morra
- Affiliation: Université Rennes 1, IRMAR, 263 avenue du Général Leclerc, CS74205, 35042 Rennes Cedex, France
- Received by editor(s): March 21, 2011
- Received by editor(s) in revised form: August 26, 2011, and February 5, 2012
- Published electronically: March 14, 2013
- © Copyright 2013 American Mathematical Society
- Journal: Math. Comp. 82 (2013), 2343-2361
- MSC (2010): Primary 11R16, 11Y40
- DOI: https://doi.org/10.1090/S0025-5718-2013-02686-5
- MathSciNet review: 3073205