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%I #26 Sep 08 2022 08:46:08
%S 4447704,4472360,4818916,4897363,5067967,5769988,7060148,8180671,
%T 8721735,8819519,8992363,9379703,9487991,9778603
%N Absolute discriminants of complex quadratic fields with 3-class group of type (3,3,3), thus having an infinite class tower.
%C I do not know who actually discovered a(1)=4447704. It is neither mentioned in Diaz y Diaz (1973) nor in Buell (1976). Maybe it can be found in Shanks (1976). MAGMA required 18 hours CPU time for the first 14 terms.
%C Meanwhile, it came to my knowledge that a(1)=4447704 and all the other terms below 10^7 are given in Appendice 1, pp. 66-77, of the Thesis of Diaz y Diaz (1978). a(1) is not contained in Shanks (1976). - _Daniel Constantin Mayer_, Sep 28 2014.
%D F. Diaz y Diaz, Sur le 3-rang des corps quadratiques, Publ. math. d'Orsay, No. 78-11, Univ. Paris-Sud (1978).
%H D. A. Buell, <a href="http://dx.doi.org/10.1090/S0025-5718-1976-0404205-X">Class groups of quadratic fields</a>, Math. Comp. 30 (1976), no. 135, 610-623.
%H F. Diaz y Diaz, <a href="http://www.numdam.org/item?id=SDPP_1973-1974__15_2_A10_0">Sur les corps quadratiques imaginaires dont le 3-rang du groupe des classes est supérieur à 1</a>, Séminaire Delange-Pisot-Poitou, 1973/74, no. G15
%H D. C. Mayer, <a href="http://www.algebra.at/FieldsOf3Rank3.pdf">Complex quadratic fields of type (3, 3, 3)</a>, 2014.
%H Daniel C. Mayer, <a href="http://arxiv.org/abs/1502.03388">Index-p abelianization data of p-class tower groups</a>, arXiv preprint arXiv:1502.03388, 2015
%H D. Shanks, <a href="http://dx.doi.org/10.1090/S0025-5718-1976-0399039-9">Class groups of the quadratic fields found by Diaz y Diaz</a>, Math. Comp. 30 (1976), 173-178.
%e a(1)=4447704 is the minimal absolute discriminant with elementary abelian 3-class group of type (3,3,3), whereas the smaller A244574(1)=3321607 has non-elementary (9,3,3).
%o (Magma) for d := 1 to 10^7 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 ([3,3,3] eq pPrimaryInvariants(C,3)) then d,","; end if; end if; end for;
%Y Cf. A242863, A244574 (a supersequence).
%K hard,more,nonn
%O 1,1
%A _Daniel Constantin Mayer_, Jun 30 2014
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