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A244575 Absolute discriminants of complex quadratic fields with 3-class group of type (3,3,3), thus having an infinite class tower. 1
4447704, 4472360, 4818916, 4897363, 5067967, 5769988, 7060148, 8180671, 8721735, 8819519, 8992363, 9379703, 9487991, 9778603 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.

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.

REFERENCES

F. Diaz y Diaz, Sur le 3-rang des corps quadratiques, Publ. math. d'Orsay, No. 78-11, Univ. Paris-Sud (1978).

LINKS

Table of n, a(n) for n=1..14.

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, 2015

D. Shanks, Class groups of the quadratic fields found by Diaz y Diaz, Math. Comp. 30 (1976), 173-178.

EXAMPLE

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).

PROG

(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;

CROSSREFS

Cf. A242863, A244574 (a supersequence).

Sequence in context: A210297 A019288 A237910 * A105649 A015337 A183679

Adjacent sequences:  A244572 A244573 A244574 * A244576 A244577 A244578

KEYWORD

hard,more,nonn

AUTHOR

Daniel Constantin Mayer, Jun 30 2014

STATUS

approved

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Last modified November 20 13:59 EST 2017. Contains 294972 sequences.