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Discriminants of real quadratic fields with 3-class tower group <81,7>
1

%I #11 Sep 08 2022 08:46:15

%S 142097,173944,259653,283673,320785,321053,326945,335229,412277,

%T 424236,459964,471713,476152,527068,535441,551384,567473,621749,

%U 637820,681276,686977,729293,747496,750376,782737,784997,807937,893029,916181,942961,966053,967928,974157,982049

%N Discriminants of real quadratic fields with 3-class tower group <81,7>

%C 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.

%C The MAGMA program requires A269319 as its input list.

%H 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 M. R. Bush, <a href="http://home.wlu.edu/~bushm/Research/research.html">private communication</a>, 11 July 2015.

%H D. C. Mayer, <a href="https://www.researchgate.net/publication/234063636">List of discriminants less than 200000 of totally real cubic fields</a>, 1991, ResearchGate.

%H D. C. Mayer, <a href="http://www.algebra.at/Memorial2009Principalization.htm">All known examples for principalization types</a>, Memorial 2009.

%H 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 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 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 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].

%o (Magma) SetClassGroupBounds("GRH"); p:=3; dList:=A269319; for d in dList do

%o 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;

%o sA:=[AbelianExtension(Inverse(mQ)*mC) where Q,mQ:=quo<C|x`subgroup>: x in sS];

%o sN:=[NumberField(x): x in sA]; sR:=[MaximalOrder(x): x in sA];

%o sF:=[AbsoluteField(x): x in sN]; sM:=[MaximalOrder(x): x in sF];

%o sM:=[OptimizedRepresentation(x): x in sF];

%o 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 if (3 eq #pPrimaryInvariants(CO,p)) then epsilon:=epsilon+1; end if;

%o 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;

%Y Subsequence of A269319

%K nonn,hard

%O 1,1

%A _Daniel Constantin Mayer_, Mar 10 2016