A269318 Discriminants of real quadratic number fields with 3-class rank 2

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, 255973, 259653, 265245, 275881, 282461, 283673, 298849, 320785, 321053, 326945, 333656, 335229, 341724, 342664, 358285, 363397, 371965, 384369, 390876

Offset

1,1

Comments

The number of unramified cyclic extensions N|K of relative degree p of a quadratic field K with p-class rank r (p an odd prime) is given by the multiplicity formula m=(p^r-1)/(p-1) [Mayer, Theorem 3.1]. Here, we have p=3, r=2, and thus m=4. Consequently, the terms of A269318 characterize all 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), 61 in [Ennola, Turunen] (up to 5*10^5), and 2870 in [Llorente, Quer] (up to 10^7). The number 2879 in the first and third line below Table 4 [Llorente, Quer] is erroneous, since the 9 quartets in Table 6 [Llorente, Quer] are ramified and satisfy d(L_i)=f^2*d(K) with various conductors f>1. (We point out misprints in the caption and in the header of Table 6 [Llorente, Quer], where our Fuehrer f is denoted by T and should correctly be given by 3^m*T_0.) The most recent and most extensive computation is due to [Bush]. He found 481756 unramified quartets up to 10^9, which are obviously very sparse with absolute density~0.05%. The density~0.16% with respect to the asymptotic number (3/pi^2)*10^9~303963551 of all positive fundamental discriminants is slightly bigger. Compare the Cohen-Lenstra heuristics [Cohen, Martinet].

Links

Table of n, a(n) for n=1..44.

I. O. Angell, A table of totally real cubic fields, Math. Comp. 30 (1976), no. 133, 184-187.

M. R. Bush, private communication, 11 July 2015.

H. Cohen and J. Martinet, Class groups of number fields: numerical heuristics, Math. Comp. 48 (1987), no. 177, 123-137.

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, Quadratic p-ring spaces for counting dihedral fields, Int. J. Number Theory 10 (2014), no. 8, 2205-2242.

Example

The execution of the MAGMA program yields the 161 leading terms of A269318 up to 10^6 and requires 9200 seconds on a single thread of an Intel i7 4-core processor with clock frequency 4GHz. The computation is slow because 303957 discriminants have to be checked for the structure of their associated 3-class groups. Among the 161 3-class groups of 3-rank 2, there are 149 of type (3,3) and 12 of type (9,3). Parallelization (for instance, 4 threads processing ranges of length 250000) would reduce the CPU-time.

Prog

(Magma) SetClassGroupBounds("GRH"); p:=3;
for d:=0 to 10^6 do if ((d gt 1) and IsFundamental(d)) then
Q:=QuadraticField(d); O:=MaximalOrder(Q); C:=ClassGroup(O);
if (2 eq #pPrimaryInvariants(C, p)) then printf "%o, ", d;
end if; end if; end for;

Crossrefs

Subsequence A269319

Sequence in context: A255079 A354079 A235309 * A269319 A197114 A224621

Adjacent sequences: A269315 A269316 A269317 * A269319 A269320 A269321

Keyword

nonn,easy

Author

Daniel Constantin Mayer, Mar 06 2016

Status

approved