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

Author

Daniel Constantin Mayer, Mar 06 2016

Keywords

• nonn
• easy

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.

Examples

			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).
		

Links

• 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.

Crossrefs

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

Programs

Magma

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;