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A359871 Absolute discriminants of imaginary quadratic number fields with elementary bicyclic 5-class group (5,5).

%I A359871 #15 Jun 02 2025 15:26:40
%S A359871 11199,12451,17944,30263,33531,37363,38047,39947,42871,53079,54211,
%T A359871 58424,61556,62632,63411,64103,65784,66328,67031,67063,67128,69811,
%U A359871 72084,74051,75688,83767,84271,85099,85279,87971,89751,90795,90868,92263,98591,99031,99743
%N A359871 Absolute discriminants of imaginary quadratic number fields with elementary bicyclic 5-class group (5,5).
%C A359871 The maximal unramified pro-5-extension, that is, the Hilbert 5-class field tower, of these imaginary quadratic fields must have a Schur sigma-group as its Galois group. The tower has an unbounded number of stages at least equal to two, and may even be infinite.
%D A359871 F.-P. Heider, B. Schmithals, Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen, J. reine angew. Math. 336 (1982), 1 - 25.
%D A359871 D. C. Mayer, The distribution of second p-class groups on coclass graphs, J. Théor. Nombres Bordeaux 25 (2013), no. 2, 401-456. (Sec. 3.5.2, p. 448)
%H A359871 T. Bembom, <a href="https://ediss.uni-goettingen.de/handle/11858/00-1735-0000-000D-F05F-8">The capitulation problem in class field theory</a>, Dissertation, Univ. Göttingen, 2012. (Sec. 6.3, p. 128)
%H A359871 D. C. Mayer, <a href="http://www.algebra.at/Pentadic2012.htm">Pentadic quantum class groups</a>
%H A359871 D. C. Mayer, <a href="http://arxiv.org/abs/1403.3833">The distribution of second p-class groups on coclass graphs</a>, arXiv:1403.3833 [math.NT], 2014. (Sec. 3.5.2-3.5.3, pp. 448-450)
%e A359871 On page 22 of their 1982 paper, Franz-Peter Heider and Bodo Schmithals gave the smallest prime discriminant -12451 and determined two of the six capitulation kernels in unramified cyclic quintic extensions. On 03 November 2011, Daniel C. Mayer determined the abelian type invariants, and thus indirectly the coarse capitulation type, of these six extensions for all 37 discriminants in the range between -11199 and -99743, with computational aid by Claus Fieker. In particular, -89751 was the minimal occurrence of the identity capitulation (see A359291). In his 2012 Ph.D. thesis, Tobias Bembom independently recomputed the capitulation in this range, without being able to detect the identity capitulation for -89751. It must be pointed out that in his table on pages 129 and 130, the minimal discriminant -11199=-3*3733 is missing, whereas the discriminant -81287 is superfluous and must be cancelled, since its 5-class group is non-elementary bicyclic of type (25,5).
%o A359871 (Magma)
%o A359871 for d := 2 to 10^5 do a := false; if (3 eq d mod 4) and IsSquarefree(d) then a := true; end if; if (0 eq d mod 4) then r := d div 4; if IsSquarefree(r) and ((2 eq r mod 4) or (1 eq r mod 4)) then a := true; end if; end if; if (true eq a) then K := QuadraticField(-d); C := ClassGroup(K); if ([5,5] eq pPrimaryInvariants(C,5)) then d, ", "; end if; end if; end for;
%Y A359871 Cf. A359291 (subsequence), A242863 (3,3), A359872 (7,7).
%K A359871 easy,nonn
%O A359871 1,1
%A A359871 _Daniel Constantin Mayer_, Jan 16 2023