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A269319 Discriminants of real quadratic fields with 3-class group of type (3,3) 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..41.

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.

EXAMPLE

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).

PROG

(MAGMA) SetClassGroupBounds("GRH"); p:=3; dList:=A269318; for d in dList do

ZX<X>:=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;

CROSSREFS

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

Sequence in context: A255079 A235309 A269318 * A197114 A224621 A231613

Adjacent sequences:  A269316 A269317 A269318 * A269320 A269321 A269322

KEYWORD

nonn,easy

AUTHOR

Daniel Constantin Mayer, Mar 06 2016

STATUS

approved

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Last modified November 19 09:04 EST 2018. Contains 317348 sequences. (Running on oeis4.)