A244574 Absolute discriminants of complex quadratic fields with 3-class rank 3 and thus with infinite class tower.
3321607, 3640387, 4019207, 4447704, 4472360, 4818916, 4897363, 5048347, 5067967, 5153431, 5288968, 5769988, 6562327, 7016747, 7060148, 7503391, 7546164, 8124503, 8180671, 8721735, 8819519, 8992363, 9379703, 9487991, 9778603
Offset: 1
Examples
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.
References
- F. Diaz y Diaz, Sur le 3-rang des corps quadratiques, Publ. math. d'Orsay, No. 78-11, Univ. Paris-Sud (1978).
Links
- D. A. Buell, Class groups of quadratic fields, Math. Comp. 30 (1976), no. 135, 610-623.
- Francisco Diaz y Diaz, Sur les corps quadratiques imaginaires dont le 3-rang du groupe des classes est supérieur à 1, Séminaire Delange-Pisot-Poitou, 1973/74, no. G15.
- H. Koch, B. B. Venkov, Über den p-Klassenkörperturm eines imaginär-quadratischen Zahlkörpers, Astérisque 24-25 (1975), 57-67.
- D. C. Mayer, Complex quadratic fields of type (3, 3, 3), 2014.
- Daniel C. Mayer, Index-p abelianization data of p-class tower groups, arXiv preprint arXiv:1502.03388, 2015
Programs
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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;
Comments