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

Author

Daniel Constantin Mayer, Jun 30 2014

Keywords

• hard
• more
• nonn

Comments

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.

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

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

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

Crossrefs

Cf. A242862, A244575 (a subsequence).

Programs

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;