cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A379524 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,...

Original entry on oeis.org

32009, 214712, 710652, 8127208, 180527768
Offset: 1

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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.

Examples

			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.
		

References

  • 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.

Crossrefs

Cf. A269318, A269319 (supersequences).

Programs

  • Magma
    // See Links section.

Formula

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.