A379524 - OEIS

%I #24 Jan 13 2025 19:57:41

%S 32009,214712,710652,8127208,180527768

%N Minimal discriminants d of real quadratic number fields K = Q(sqrt(d)), d > 0, with elementary bicyclic 3-class group Cl_3(K)=(3,3) and second 3-class group M=Gal(F_3^2(K)/K) of assigned coclass cc(M)=1,2,3,4,...

%C The coclass cc(M) for the field K with discriminant d=a(n) is n, and for each field K with discriminant d < a(n), the coclass cc(M) is less than n.

%D M. R. Bush, ipad_freq_real, file with two lists, disclist and ipadlist, containing all IPADs of real quadratic fields K with 3-class group of rank 2 and discriminant d < 10^9, Washington and Lee Univ. Lexington, Virginia, 2015.

%H D. C. Mayer, <a href="http://arxiv.org/abs/1403.3833">The distribution of second p-class groups on coclass graphs</a>, arXiv:1403.3833 [math.NT], 2014; J. Théor. Nombres Bordeaux 25 (2013), 401-456.

%H D. C. Mayer, <a href="http://arxiv.org/abs/1403.3839">Principalization algorithm via class group structure</a>, arXiv:1403.3839 [math.NT], 2014; J. Théor. Nombres Bordeaux 26 (2014), 415-464.

%H D. C. Mayer, <a href="https://arxiv.org/abs/1403.3899">The second p-class group of a number field</a>, arXiv:1403.3899 [math.NT], 2014; Int. J. Number Theory 8 (2012), no. 2, 471-505.

%H Daniel Constantin Mayer, <a href="/A379524/a379524.m.txt">M. R. Bush: data file ipad_freq_real</a>

%H Daniel Constantin Mayer, <a href="/A379524/a379524_1.m.txt">Program "SiftRealIPADs.m" which extracts minimal discriminants for assigned IPADs from the file ipad_freq_real and arranges them in the table "IpadFreqReal"</a>

%H Daniel Constantin Mayer, <a href="/A379524/a379524.txt">"IpadFreqReal": table of minimal discriminants for assigned IPADs</a>

%F According to Theorem 3.12 on page 435 of "The distribution of second p-class groups on coclass graphs", the coclass of the group M is given by cc(M)+1=log_3(h_3(L_2)), where h_3(L_2) is the second largest 3-class number among the four unramified cyclic cubic extensions L_1,..,L_4 of the quadratic field K. Thus, cc(M) is determined uniquely by the IPAD of K.

%e We have cc(M)=1 for d=32009, cc(M)=2 for d=214712, cc(M)=3 for d=710652, cc(M)=4 for d=8127208, cc(M)=5 for d=180527768. The Magma script "SiftRealIPADs.m" produces a table "IpadFreqReal" of minimal discriminants for each IPAD from the file ipad_freq_real. This table admits the determination of the term a(n) of the sequence A379524. For instance: According to the FORMULA, the table contains three candidates for a(4) with cc(M)=4 and thus cc(M)+1=5=log_3(3^5)=log_3(#[9,27])=log_3(h_3(L_2)) with the second largest 3-class number h_3(L_2) in the IPAD. They are 8321505 and 8491713 and 8127208. Thus the minimal discriminant is a(4)=8127208.

%o (Magma) // See Links section.

%Y Cf. A269318, A269319 (supersequences).

%K nonn,hard,more,new

%O 1,1

%A _Daniel Constantin Mayer_, Dec 24 2024