A244575 Absolute discriminants of complex quadratic fields with 3-class group of type (3,3,3), thus having an infinite class tower.

4447704, 4472360, 4818916, 4897363, 5067967, 5769988, 7060148, 8180671, 8721735, 8819519, 8992363, 9379703, 9487991, 9778603

Offset: 1

Author

Daniel Constantin Mayer, Jun 30 2014

Keywords

• hard
• more
• nonn

Comments

I do not know who actually discovered a(1)=4447704. It is mentioned neither 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.

Meanwhile, it came to my attention 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.

Examples

			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).
		

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.
• F. 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.
• 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 [math.NT], 2015.
• D. Shanks, Class groups of the quadratic fields found by Diaz y Diaz, Math. Comp. 30 (1976), 173-178.

Crossrefs

Cf. A242863, A244574 (a supersequence).

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 ([3,3,3] eq pPrimaryInvariants(C,3)) then d,","; end if; end if; end for;