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Negative fundamental discriminants with non-isomorphic class groups (negated).
2

%I #16 Dec 26 2015 02:21:26

%S 3,15,23,39,47,71,84,87,95,119,167,191,199,215,231,239,260,311,327,

%T 335,383,399,407,420,431,455,479,551,591,647,671,695,719,759,776,791,

%U 831,839,887,935,959,983,1031,1079,1140,1151,1199,1239,1271,1295,1319,1391,1439,1487,1511,1559,1679,1751,1799,1847,1959,1991,2015

%N Negative fundamental discriminants with non-isomorphic class groups (negated).

%C More precisely, the least absolute values of negative fundamental discriminants with class groups non-isomorphic to all class groups for negative fundamental discriminants with smaller absolute values.

%C The n-th line of the linked file gives the invariant factor decomposition of the class group corresponding to the fundamental discriminant -a(n).

%H Rick L. Shepherd and Charles R Greathouse IV, <a href="/A225365/b225365.txt">Table of n, a(n) for n = 1..10000</a> (first 7194 terms from Shepherd)

%H 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 Rick L. Shepherd, <a href="/A225365/a225365.txt">Invariant factor decompositions for corresponding class groups</a>

%e The fundamental discriminant -3 corresponds to the trivial class group. The fundamental discriminant -15 is the first negative fundamental discriminant encountered (least absolute value) whose class group has a different structure, isomorphic to Z_2. The fundamental discriminant -84 is the one with least absolute value whose class group is isomorphic to the Klein-4 group.

%o (PARI)

%o {allocatemem(32000000);

%o \\ Increase precision to more than 100 digits to go beyond 7194 terms.

%o default(realprecision, 100);

%o terms_wanted = 7194;

%o G = Set(); k = 0;

%o while(length(G)<terms_wanted,

%o k++;

%o if(isfundamental(-k),

%o F = bnfinit(quadpoly(-k,x), , [6,6,4]); \\ Without optional 3rd argument, Generalized Riemann Hypothesis assumed

%o if(bnfcertify(F)<>1, print("Certify failed for ", -k, " -- exiting (", length(G), " terms found)"); break);

%o if(setsearch(G, F.clgp.cyc)==0,

%o G = setunion(G, [F.clgp.cyc]);

%o write("b225365.txt", length(G), " ", k);

%o write("a225365.txt", length(G), " ", F.clgp.cyc);

%o if(length(G)%100==0, print1("...", length(G), "... ")))))}

%Y Cf. A003657.

%K nonn

%O 1,1

%A _Rick L. Shepherd_, May 05 2013