A269321 Discriminants of real quadratic fields with 3-class tower group <81,7>.

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

Offset: 1

Author

Daniel Constantin Mayer, Mar 10 2016

Keywords

• nonn
• hard

Comments

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.

Examples

			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].
		

Links

• 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.
• M. R. Bush, private communication, 11 July 2015.
• D. C. Mayer, List of discriminants less than 200000 of totally real cubic fields, 1991, ResearchGate.
• D. C. Mayer, All known examples for principalization types, Memorial 2009.
• D. C. Mayer, The second p-class group of a number field, Int. J. Number Theory 8 (2012), no. 2, 471-505.
• D. C. Mayer, Principalization algorithm via class group structure, J. Thèor. Nombres Bordeaux 26 (2014), no. 2, 415-464.
• D. C. Mayer, Artin transfer patterns on descendant trees of finite p-groups, Adv. Pure Math. 6 (2016), no. 2, 66-104.

Crossrefs

Subsequence of A269319

Programs

Magma

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