cp's OEIS Frontend

A227735 Negative fundamental discriminants with cyclic class groups of composite order (negated).

%I A227735 #15 Dec 26 2015 02:29:53
%S A227735 39,55,56,68,87,95,104,111,116,119,136,143,152,155,159,164,183,184,
%T A227735 199,203,212,215,219,239,244,247,248,259,287,291,292,295,296,299,303,
%U A227735 319,323,327,328,335,339,344,355,356,367,371,376,388,391,395,404,407,411
%N A227735 Negative fundamental discriminants with cyclic class groups of composite order (negated).
%C A227735 Absolute values of fundamental discriminants of imaginary quadratic fields whose class groups are cyclic of composite order. Of course every class group of squarefree order is necessarily cyclic. (This means the negatives of negative fundamental discriminants with class groups of composite squarefree orders are a proper subsequence.)
%C A227735 The n-th line of the linked file gives the order of the class group (the class number) corresponding to the fundamental discriminant -a(n).
%C A227735 The negative of each term is either a negative fundamental discriminant or the product of exactly one positive prime discriminant and one negative prime discriminant where the product contains at most one factor in {-8, -4, 8} and is unique disregarding order. In the first case, the class number is odd (because the 2-rank of the class group is 0); in the second, even (because the 2-rank is 1).
%H A227735 Rick L. Shepherd, <a href="/A227735/b227735.txt">Table of n, a(n) for n = 1..10000</a>
%H A227735 Rick L. Shepherd, <a href="http://libres.uncg.edu/ir/uncg/f/Shepherd_uncg_0154M_11099.pdf">Binary quadratic forms and genus theory</a>, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.
%H A227735 Rick L. Shepherd, <a href="/A227735/a227735.txt">Orders of corresponding class groups</a>
%H A227735 <a href="/index/Qua#quadfield">Index entries for sequences related to quadratic fields</a>
%e A227735 The fundamental discriminant -39 = (-3)(13) has a cyclic class group of order 4, which is composite (but not squarefree). The fundamental discriminant -104 = (-8)(13) has a cyclic class group of order 6, which is composite. The fundamental discriminant -239 is itself a prime discriminant with cyclic class group of order 15, also composite (but not divisible by 2).
%o A227735 (PARI)
%o A227735 {default(realprecision, 100);
%o A227735 terms_wanted = 10000;
%o A227735 t = 0; k = 0;
%o A227735 while(t < terms_wanted,
%o A227735   k++;
%o A227735   if(isfundamental(-k),
%o A227735     F = bnfinit(quadpoly(-k, x), , [6, 6, 4]);
%o A227735     if(bnfcertify(F) <> 1,
%o A227735       print("Certify failed for ", -k, " -- exiting (",
%o A227735         t, " terms found)"); break);
%o A227735     if(length(F.clgp.cyc) == 1 &&
%o A227735       isprime(F.clgp.cyc[1]) == 0,
%o A227735       t++;
%o A227735       write("b227735.txt", t, " ", k);
%o A227735       write("a227735.txt", t, " ", F.clgp.cyc[1]))))}
%Y A227735 Cf. A227734.
%K A227735 nonn
%O A227735 1,1
%A A227735 _Rick L. Shepherd_, Jul 29 2013