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

Author

Daniel Constantin Mayer, Mar 06 2016

Keywords

• nonn
• easy

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].

Examples

			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.
		

Links

• 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.

Crossrefs

Subsequence A269319.

Programs

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;