A269318 -id:A269318 - cp's OEIS frontend

A269319 Discriminants of real quadratic fields with 3-class group of type (3,3).

32009, 42817, 62501, 72329, 94636, 103809, 114889, 130397, 142097, 151141, 152949, 153949, 172252, 173944, 184137, 189237, 206776, 209765, 213913, 214028, 214712, 219461, 220217, 250748, 252977, 259653, 265245, 275881, 283673, 298849, 320785, 321053, 326945, 333656, 335229, 341724, 342664, 358285, 363397, 371965, 390876

Offset: 1

Daniel Constantin Mayer, Mar 06 2016

As explained in the comments in A269318, the terms of A269319 are discriminants of quadratic fields K which correspond to certain quartets (L_1,...L_4) of pairwise non-isomorphic non-Galois cubic fields sharing a common fundamental discriminant d(L_i)=d(K). There occur 5 of these quartets in [Angell] (up to 10^5), 58 in [Ennola, Turunen] (up to 5*10^5), and 2576 in [Llorente, Quer] (up to 10^7). It should be pointed out that, whereas [Angell] does not contain other quartets than the 5 corresponding to type (3,3), there occur 3 further quartets associated with type (9,3) in [Ennola, Turunen], namely 255973, 282461, 384369. In [Llorente, Quer], we have 271 additional quartets of type (9,3), 20 of type (27,3), 1 of type (81,3), and 2 of type (9,9). The splitting 2879-9=2870=2576+271+20+1+2 was computed in [Mayer, 2010] and is not contained in [Llorente, Quer]. The number 2576 was published in [Mayer, 2012] and is not mentioned in [Llorente, Quer]. The most recent and most extensive information is due to [Bush], who showed that there are 415698 quartets associated with type (3,3) up to the bound 10^9.

			The execution of the MAGMA program requires the supersequence A269318 as its input list, and yields the 149 leading terms of A269319 up to 10^6, sifting out 12 terms with associated 3-class group of type (9,3).

I. O. Angell, A table of totally real cubic fields, Math. Comp. 30 (1976), no. 133, 184-187.
M. R. Bush, private communication, on 11 July 2015.
V. Ennola and R. Turunen, On totally real cubic fields, Math. Comp. 44 (1985), no. 170, 495-518.
P. Llorente and J. Quer, On totally real cubic fields with discriminant D < 10^7, Math. Comp. 50 (1988), no. 182, 581-594.
D. C. Mayer, Top down capitulation algorithm, Scientific Research 2010.
D. C. Mayer, The second p-class group of a number field, Int. J. Number Theory 8 (2012), no. 2, 471-505.

Subsequence of A269318, contains disjoint subsequences A269320,...,A269323

SetClassGroupBounds("GRH"); p:=3; dList:=A269318; for d in dList do
ZX:=PolynomialRing(Integers()); K:=NumberField(X^2-d); O:=MaximalOrder(K); C:=ClassGroup(O); if ([p,p] eq pPrimaryInvariants(C,p)) then printf "%o,",d; end if; end for;

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

Daniel Constantin Mayer, Dec 24 2024

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.

			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.

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.

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

Cf. A269318, A269319 (supersequences).

Magma
```
// See Links section.
```

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.

cp's OEIS Frontend

A269319 Discriminants of real quadratic fields with 3-class group of type (3,3).

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