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Determination of all imaginary abelian sextic number fields with class number \(\leq 11\). (English) Zbl 0889.11036

Louboutin and Yamamura proved that there are exactly 17 imaginary abelian sextic number fields with class number one, and that their conductors are \(\leq 129\) [cf. S. Louboutin, Acta Arith. 62, No. 2, 109-124 (1992; Zbl 0761.11041) and K. Yamamura, Math. Comput. 62, No. 206, 899-921 (1994; Zbl 0798.11046)]. The present authors intend to determine all imaginary abelian sextic number fields with class number \(\leq 11\).
For this purpose, they first improve the lower bounds for the relative class number established by Louboutin, and then using these give reasonable upper bounds for the conductors of abelian sextic CM-fields with small class number. Thus, they make a list of all possible conductors for a given class number \(\leq 11\) by using the divisibility properties of the relative class number. Finally, they determine all fields with relative class number \(\leq 4\) and all fields with class number \(\leq 11\).
Reviewer: H.Yokoi (Iwasaki)

MSC:

11R29 Class numbers, class groups, discriminants
11R20 Other abelian and metabelian extensions
11-04 Software, source code, etc. for problems pertaining to number theory
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