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A269321
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Discriminants of real quadratic fields with 3-class tower group <81,7>
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1
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142097, 173944, 259653, 283673, 320785, 321053, 326945, 335229, 412277, 424236, 459964, 471713, 476152, 527068, 535441, 551384, 567473, 621749, 637820, 681276, 686977, 729293, 747496, 750376, 782737, 784997, 807937, 893029, 916181, 942961, 966053, 967928, 974157, 982049
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OFFSET
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1,1
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COMMENTS
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The Artin transfer homomorphisms of the assigned 3-class tower group G with SmallGroups identifier <81,7> [Besche, Eick, O'Brien], which is better known as the 3-Sylow subgroup Syl_3(A_9) of the alternating group of degree 9, determine the capitulation type (2,0,0,0) (TKT without fixed point) of the real quadratic field K in its four unramified cyclic cubic extensions N_i|K (i=1,...,4) and the abelian type invariants of the 3-class groups Cl(3,K)=(3,3) (whence A269321 is a subsequence of A269319) and [Cl(3,N_i)]=[(3,3,3),(3,3),(3,3),(3,3)] (TTT or IPAD). Conversely, the group G=<81,7> is determined uniquely not only by its Artin pattern AP(G)=(TTT,TKT) but even by the TTT component alone [Mayer, 2014, Fig.3.1, Tbl.4.1], where TKT, TTT, IPAD are abbreviations for transfer kernel type, transfer target type, index-p abelianization data, respectively [Mayer, 2016]. Consequently, it suffices that the MAGMA program only determines the TTT component of the Artin pattern. This is an instance of the "Principalization algorithm via class group structure" [Mayer, 2014] and saves a considerable amount of CPU time, since the determination of the TKT component is very delicate. In fact, G=<81,7> is the unique finite 3-group of coclass cc(G)=1 with a component (3,3,3) in its IPAD. Since the group G=<81,7> has derived length dl(G)=2, the Hilbert 3-class field tower of these real quadratic fields consists of exactly two stages.
The MAGMA program requires A269319 as its input list.
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LINKS
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H. U. Besche, B. Eick, and E. A. O'Brien, The SmallGroups Library - a library of groups of small order, 2005, an accepted and refereed GAP package, available also in MAGMA.
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EXAMPLE
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The two leading terms, 142097, 173944, were listed in [Mayer, 1991] (up to 2*10^5) without giving the Artin pattern. The first 34 terms of A269321 up to 10^6 have been published in [Mayer, 2009]. The first 698 terms up to 10^7 have been determined in [Mayer, 2012] and [Mayer, 2014] with erroneous counter 697 corrected by [Bush]. The 10244, resp. 122955, terms up to 10^8, resp. 10^9, have been computed by [Bush].
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PROG
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(Magma) SetClassGroupBounds("GRH"); p:=3; dList:=A269319; for d in dList do
ZX<X>:=PolynomialRing(Integers()); K:=NumberField(X^2-d); O:=MaximalOrder(K); C, mC:=ClassGroup(O); sS:=Subgroups(C: Quot:=[p]); sI:=[]; for j in [1..p+1] do Append(~sI, 0); end for; n:=Ngens(C); g:=(Order(C.(n-1)) div p)*C.(n-1); h:=(Order(C.n) div p)*C.n; ct:=0; for x in sS do ct:=ct+1; if g in x`subgroup then sI[1]:=ct; end if; if h in x`subgroup then sI[2]:=ct; end if; for e in [1..p-1] do if g+e*h in x`subgroup then sI[e+2]:=ct; end if; end for; end for;
sA:=[AbelianExtension(Inverse(mQ)*mC) where Q, mQ:=quo<C|x`subgroup>: x in sS];
sN:=[NumberField(x): x in sA]; sR:=[MaximalOrder(x): x in sA];
sF:=[AbsoluteField(x): x in sN]; sM:=[MaximalOrder(x): x in sF];
sM:=[OptimizedRepresentation(x): x in sF];
sA:=[NumberField(DefiningPolynomial(x)): x in sM]; sO:=[Simplify(LLL(MaximalOrder(x))): x in sA]; TTT:=[]; epsilon:=0; polarization1:=3; polarization2:=3; for j in [1..#sO] do CO:=ClassGroup(sO[j]); Append(~TTT, pPrimaryInvariants(CO, p));
if (3 eq #pPrimaryInvariants(CO, p)) then epsilon:=epsilon+1; end if;
val:=Valuation(Order(CO), p); if (2 eq val) then polarization2:=val; elif (4 le val) then if (3 eq polarization1) then polarization1:=val; else polarization2:=val; end if; end if; end for; if (2 eq polarization2) and (3 eq polarization1) and (1 eq epsilon) then printf "%o, ", d; end if; end for;
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CROSSREFS
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KEYWORD
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nonn,hard
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AUTHOR
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STATUS
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approved
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