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A269319
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Discriminants of real quadratic fields with 3-class group of type (3,3)
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9
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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).
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PROG
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(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;
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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