1,1

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.

The Magma program "RealCoClass.m" in the Links is independent of the data file "ipad_freq_real" by M. R. Bush. It computes the first five terms, 180527768 inclusively, in precisely 14 days of CPU time on an Intel Core i7 4790 quadcore processor with clock rate 4.0 GHz.

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.

Siham Aouissi and Daniel C. Mayer, Coclass of the second 3-class group, arXiv:2508.17510 [math.NT], 2025. See p. 10.

Daniel Constantin Mayer, The distribution of second p-class groups on coclass graphs, arXiv:1403.3833 [math.NT], 2014; J. Théor. Nombres Bordeaux 25 (2013), 401-456.

Daniel Constantin Mayer, Principalization algorithm via class group structure, arXiv:1403.3839 [math.NT], 2014; J. Théor. Nombres Bordeaux 26 (2014), 415-464.

Daniel Constantin Mayer, The second p-class group of a number field, arXiv:1403.3899 [math.NT], 2014; Int. J. Number Theory 8 (2012), no. 2, 471-505.

Daniel Constantin Mayer, M. R. Bush: data file ipad_freq_real

Daniel Constantin Mayer, Program "SiftRealIPADs.m" which extracts minimal discriminants for assigned IPADs from the file ipad_freq_real and arranges them in the table "IpadFreqReal"

Daniel Constantin Mayer, "IpadFreqReal": table of minimal discriminants for assigned IPADs

Daniel Constantin Mayer, Magma program "RealCoClass.m" with endless loop

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.

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.

(Magma) // See Links section.

Cf. A269318, A269319 (supersequences).

Sequence in context: A269318 A269319 A197114 * A224621 A231613 A054038

Adjacent sequences: A379521 A379522 A379523 * A379525 A379526 A379527

nonn,hard,more

Daniel Constantin Mayer, Dec 24 2024

approved