

A269318


Discriminants of real quadratic number fields with 3class rank 2


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
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OFFSET

1,1


COMMENTS

The number of unramified cyclic extensions NK of relative degree p of a quadratic field K with pclass rank r (p an odd prime) is given by the multiplicity formula m=(p^r1)/(p1) [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 nonisomorphic nonGalois 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 CohenLenstra heuristics [Cohen, Martinet].


LINKS



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 4core processor with clock frequency 4GHz. The computation is slow because 303957 discriminants have to be checked for the structure of their associated 3class groups. Among the 161 3class groups of 3rank 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 CPUtime.


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



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



