cp's OEIS Frontend

A359872 Absolute discriminants of imaginary quadratic number fields with elementary bicyclic 7-class group (7,7).

%I A359872 #26 Feb 25 2023 18:07:38
%S A359872 63499,118843,124043,149519,159592,170679,183619,185723,220503,226691,
%T A359872 227387,227860,236931,240347,240655,247252,260111,268739,272179,
%U A359872 275636,294935,299627,301211,308531,318547,346883,361595,366295,373655,465719,489576,491767,501576,506551,511988,518879,528243,546792,553791
%N A359872 Absolute discriminants of imaginary quadratic number fields with elementary bicyclic 7-class group (7,7).
%C A359872 The maximal unramified pro-7-extension, that is, the Hilbert 7-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 A359872 Daniel 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 A359872 Tobias 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 A359872 Daniel C. Mayer, <a href="http://www.algebra.at/Heptadic2012.htm">Heptadic quantum class groups</a>
%H A359872 Daniel 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. 3.5.4, pp. 450-451)
%e A359872 On 06 January 2012, Daniel C. Mayer determined the abelian type invariants (ATI), and thus indirectly the coarse capitulation type, of the eight unramified cyclic septic extensions for all 70 discriminants in the range between -63499 and -751288. On page 133 of his 2012 Ph.D. thesis, Tobias Bembom independently recomputed the capitulation for the two discriminants -63499 and -159592. In the time between 09 and 16 August 2014, Daniel C. Mayer computed the fine capitulation type of all 94 discriminants in the range -63499 and -991720 without any hit of the identity capitulation. Since the fine capitulation requires much more CPU time than the ATI, Mayer conducted an extensive search for the identity capitulation, identified by eight ATI of the shape (7,7,7), in the range from 10^6 to 6578723, with an eventual successful hit of the identity capitulation for -5073691 (the 555th term of A359872) on 26 October 2019 (see A359296).
%o A359872 (Magma)
%o A359872 for d := 2 to 10^6 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 ([7,7] eq pPrimaryInvariants(C,7)) then d, ", "; end if; end if; end for;
%Y A359872 Cf. A359296 (subsequence), A242863 (3,3), A359871 (5,5).
%K A359872 easy,nonn
%O A359872 1,1
%A A359872 _Daniel Constantin Mayer_, Jan 16 2023