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Mathematics of Computation

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An algorithm to compute relative cubic fields

Author: Anna Morra
Journal: Math. Comp. 82 (2013), 2343-2361
MSC (2010): Primary 11R16, 11Y40
Published electronically: March 14, 2013
MathSciNet review: 3073205
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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].

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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
Article copyright: © Copyright 2013 American Mathematical Society

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