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A244574 Absolute discriminants of complex quadratic fields with 3-class rank 3 and thus with infinite class tower. 1


%S 3321607,3640387,4019207,4447704,4472360,4818916,4897363,5048347,

%T 5067967,5153431,5288968,5769988,6562327,7016747,7060148,7503391,

%U 7546164,8124503,8180671,8721735,8819519,8992363,9379703,9487991,9778603

%N Absolute discriminants of complex quadratic fields with 3-class rank 3 and thus with infinite class tower.

%C Diaz y Diaz discovered a(1), a(2) and three other terms in 1973. However, Buell was the first who proved minimality of a(1). According to Koch and Venkov, 3-class rank 3 ensures an infinite Hilbert (3-)class field tower.

%C The first 25 terms were computed with MAGMA over 18 hours of CPU time.

%C With exception of a(16)=7503391, all terms below 10^7 and lots of further terms below 10^8 are given in Appendice 1, pp. 66-77, of the Thesis of F. Diaz y Diaz (1978). - _Daniel Constantin Mayer_, Sep 27 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 Francisco 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 H. Koch, B. B. Venkov, <a href="http://www.numdam.org/item/AST_1975__24-25__57_0/">Über den p-Klassenkörperturm eines imaginär-quadratischen Zahlkörpers</a>, Astérisque 24-25 (1975), 57-67.

%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

%e 3-class group of type (9,3,3) for a(1)=3321607, and of type (3,3,3) for a(4)=4447704. Unique 3-class group of type (27,3,3) for a(10)=5153431.

%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 eq #pPrimaryInvariants(C,3)) then d,","; end if; end if; end for;

%Y Cf. A242862, A244575 (a subsequence).

%K hard,more,nonn

%O 1,1

%A _Daniel Constantin Mayer_, Jun 30 2014

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