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A269323 Discriminants of real quadratic fields with second 3-class group <729,54>.

%I A269323 #12 Jun 02 2025 12:16:40
%S A269323 540365,945813,1202680,1695260,1958629,3018569,3236657,3687441,
%T A269323 4441560,5512252,5571377,5701693,6027557,6049356,6054060,6274609,
%U A269323 6366029,6501608,6773557,7573868,8243464,8251521,9054177,9162577,9967837
%N A269323 Discriminants of real quadratic fields with second 3-class group <729,54>.
%C A269323 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.
%C A269323 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.
%C A269323 The MAGMA program requires A269319 as its input list.
%H A269323 H. U. Besche, B. Eick, and E. A. O'Brien, <a href="http://www.icm.tu-bs.de/ag_algebra/software/small/small.html">The SmallGroups Library</a> - a library of groups of small order, 2005, an accepted and refereed GAP package, available also in MAGMA.
%H A269323 M. R. Bush, <a href="http://home.wlu.edu/~bushm/Research/research.html">private communication</a>, 11 July 2015.
%H A269323 D. C. Mayer, <a href="http://www.algebra.at/TargetsR66.htm">The real quadratic base field K with discriminant d=540365</a>, Targets 2007/2008.
%H A269323 D. C. Mayer, <a href="http://dx.doi.org/10.1142/S179304211250025X">The second p-class group of a number field</a>, Int. J. Number Theory 8 (2012), no. 2, 471-505.
%H A269323 D. C. Mayer, <a href="http://dx.doi.org/10.5802/jtnb.874">Principalization algorithm via class group structure</a>, J. Thèor. Nombres Bordeaux 26 (2014), no. 2, 415-464.
%H A269323 D. C. Mayer, <a href="http://www.degruyter.com/view/j/tmmp.2015.64.issue-1/tmmp-2015-0040/tmmp-2015-0040.xml">New number fields with known p-class tower</a>, Tatra Mt. Math. Publ. 64 (2015), 21-57.
%H A269323 D. C. Mayer, <a href="http://dx.doi.org/10.4236/apm.2016.62008">Artin transfer patterns on descendant trees of finite p-groups</a>, Adv. Pure Math. 6 (2016), no. 2, 66-104.
%e A269323 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].
%o A269323 (Magma) SetClassGroupBounds("GRH"); p:=3; dList:=A269319; for d in dList do
%o A269323 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;
%o A269323 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;
%o A269323 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];
%o A269323 sF:=[AbsoluteField(x): x in sN]; sM:=[MaximalOrder(x): x in sF];
%o A269323 sM:=[OptimizedRepresentation(x): x in sF];
%o A269323 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));
%o A269323 if (3 eq #pPrimaryInvariants(CO,p)) then epsilon:=epsilon+1; end if;
%o A269323 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;
%Y A269323 Subsequence of A269319
%K A269323 nonn,hard
%O A269323 1,1
%A A269323 _Daniel Constantin Mayer_, Mar 10 2016