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A269323
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Discriminants of real quadratic fields with second 3-class group <729,54>
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2
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540365, 945813, 1202680, 1695260, 1958629, 3018569, 3236657, 3687441, 4441560, 5512252, 5571377, 5701693, 6027557, 6049356, 6054060, 6274609, 6366029, 6501608, 6773557, 7573868, 8243464, 8251521, 9054177, 9162577, 9967837
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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 second 3-class group M with SmallGroups identifier <729,54> [Besche, Eick, O'Brien] determine the capitulation type (2,0,3,4) (TKT with two fixed points 3 and 4) 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 A269323 is a subsequence of A269319) and [Cl(3,N_i)]=[(9,9),(9,3),(9,3),(9,3)] (TTT or IPAD). Conversely, the group M=<729,54> is determined uniquely not only by its Artin pattern AP(G)=(TTT,TKT) but even by the TTT component alone [Mayer, 2015], 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.
An important Theorem by I.R. Shafarevich [Mayer, 2015, Thm.5.1] disables the metabelian group M=<729,54> as a candidate for the 3-class tower group G, since the relation rank of M is too big. In [Mayer, 2015] it is proved that exactly the two non-metabelian groups <2187,307> and <2187,308> [Besche, Eick, O'Brien] are permitted for G, and the decision is possible with the aid of multi-layered iterated IPADs of second order (which require computing 3-class groups of number fields with absolute degree 54). Since the derived length of both groups is equal to 3, the Hilbert 3-class field tower of all these real quadratic fields has certainly exact length 3.
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 leading term, 540365, and thus the first real quadratic field K with capitulation type c.21, (2,0,3,4), has been identified on 01 January 2008 [Mayer, 2007/2008]. However, it required seven further years to determine the pro-3 Galois group G=<2187,307|308>, with metabelianization M=G/G''=<729,54>, of the Hilbert 3-class field tower of K in August 2015. (See [Mayer, 2015] for more details.) The first 25 terms of A269323 up to 10^7 have been determined in [Mayer, 2012] and [Mayer, 2014]. The 358, resp. 4377, 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 (3 eq polarization2) and (4 eq polarization1) and (0 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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