An algorithm to compute relative cubic fields

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 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=Q(i), and we compare our results with ray class field algorithm ones, and with asymptotic heuristics, based on a generalization of Roberts' conjecture.
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Contributor : Anna Morra <>
Submitted on : Tuesday, March 22, 2011 - 9:56:28 AM
Last modification on : Thursday, November 15, 2018 - 11:56:21 AM

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Anna Morra. An algorithm to compute relative cubic fields. Mathematics of Computation, American Mathematical Society, 2013, 82 (284), pp.2343-2361. 〈10.1090/S0025-5718-2013-02686-5〉. 〈hal-00578750〉

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