A269319 -id:A269319 - cp's OEIS frontend

A379524 Minimal discriminants d of real quadratic number fields K = Q(sqrt(d)), d > 0, with elementary bicyclic 3-class group Cl_3(K)=(3,3) and second 3-class group M=Gal(F_3^2(K)/K) of assigned coclass cc(M)=1,2,3,4,...

32009, 214712, 710652, 8127208, 180527768

Offset: 1

Daniel Constantin Mayer, Dec 24 2024

The coclass cc(M) for the field K with discriminant d=a(n) is n, and for each field K with discriminant d < a(n), the coclass cc(M) is less than n.

The Magma program "RealCoClass.m" in the Links is independent of the data file "ipad_freq_real" by M. R. Bush. It computes the first five terms, 180527768 inclusively, in precisely 14 days of CPU time on an Intel Core i7 4790 quadcore processor with clock rate 4.0 GHz.

			We have cc(M)=1 for d=32009, cc(M)=2 for d=214712, cc(M)=3 for d=710652, cc(M)=4 for d=8127208, cc(M)=5 for d=180527768. The Magma script "SiftRealIPADs.m" produces a table "IpadFreqReal" of minimal discriminants for each IPAD from the file ipad_freq_real. This table admits the determination of the term a(n) of the sequence A379524. For instance: According to the FORMULA, the table contains three candidates for a(4) with cc(M)=4 and thus cc(M)+1=5=log_3(3^5)=log_3(#[9,27])=log_3(h_3(L_2)) with the second largest 3-class number h_3(L_2) in the IPAD. They are 8321505 and 8491713 and 8127208. Thus the minimal discriminant is a(4)=8127208.

M. R. Bush, ipad_freq_real, file with two lists, disclist and ipadlist, containing all IPADs of real quadratic fields K with 3-class group of rank 2 and discriminant d < 10^9, Washington and Lee Univ. Lexington, Virginia, 2015.

Daniel Constantin Mayer, The distribution of second p-class groups on coclass graphs, arXiv:1403.3833 [math.NT], 2014; J. Théor. Nombres Bordeaux 25 (2013), 401-456.
Daniel Constantin Mayer, Principalization algorithm via class group structure, arXiv:1403.3839 [math.NT], 2014; J. Théor. Nombres Bordeaux 26 (2014), 415-464.
Daniel Constantin Mayer, The second p-class group of a number field, arXiv:1403.3899 [math.NT], 2014; Int. J. Number Theory 8 (2012), no. 2, 471-505.
Daniel Constantin Mayer, M. R. Bush: data file ipad_freq_real
Daniel Constantin Mayer, Program "SiftRealIPADs.m" which extracts minimal discriminants for assigned IPADs from the file ipad_freq_real and arranges them in the table "IpadFreqReal"
Daniel Constantin Mayer, "IpadFreqReal": table of minimal discriminants for assigned IPADs
Daniel Constantin Mayer, Magma program "RealCoClass.m" with endless loop

Cf. A269318, A269319 (supersequences).

Magma
```
// See Links section.
```

According to Theorem 3.12 on page 435 of "The distribution of second p-class groups on coclass graphs", the coclass of the group M is given by cc(M)+1=log_3(h_3(L_2)), where h_3(L_2) is the second largest 3-class number among the four unramified cyclic cubic extensions L_1,..,L_4 of the quadratic field K. Thus, cc(M) is determined uniquely by the IPAD of K.

A269318 Discriminants of real quadratic number fields with 3-class rank 2.

Original entry on oeis.org

32009, 42817, 62501, 72329, 94636, 103809, 114889, 130397, 142097, 151141, 152949, 153949, 172252, 173944, 184137, 189237, 206776, 209765, 213913, 214028, 214712, 219461, 220217, 250748, 252977, 255973, 259653, 265245, 275881, 282461, 283673, 298849, 320785, 321053, 326945, 333656, 335229, 341724, 342664, 358285, 363397, 371965, 384369, 390876

Offset: 1

Daniel Constantin Mayer, Mar 06 2016

The number of unramified cyclic extensions N|K of relative degree p of a quadratic field K with p-class rank r (p an odd prime) is given by the multiplicity formula m = (p^r-1)/(p-1) [Mayer, Theorem 3.1]. Here, we have p=3, r=2, and thus m=4. Consequently, the terms of A269318 characterize all quartets (L_1, ..., L_4) of pairwise non-isomorphic non-Galois cubic fields sharing a common fundamental discriminant d(L_i) = d(K). There occur 5 of these quartets in [Angell] (up to 10^5), 61 in [Ennola, Turunen] (up to 5*10^5), and 2870 in [Llorente, Quer] (up to 10^7). The number 2879 in the first and third line below Table 4 [Llorente, Quer] is erroneous, since the 9 quartets in Table 6 [Llorente, Quer] are ramified and satisfy d(L_i) = f^2*d(K) with various conductors f > 1. (We point out misprints in the caption and in the header of Table 6 [Llorente, Quer], where our Fuehrer f is denoted by T and should correctly be given by 3^m*T_0.) The most recent and most extensive computation is due to [Bush]. He found 481756 unramified quartets up to 10^9, which are obviously very sparse with absolute density ~0.05%. The density ~0.16% with respect to the asymptotic number (3/Pi^2)*10^9 ~ 303963551 of all positive fundamental discriminants is slightly bigger. Compare the Cohen-Lenstra heuristics [Cohen, Martinet].

			The execution of the Magma program yields the 161 leading terms of A269318 up to 10^6 and requires 9200 seconds on a single thread of an Intel i7 4-core processor with clock frequency 4GHz. The computation is slow because 303957 discriminants have to be checked for the structure of their associated 3-class groups. Among the 161 3-class groups of 3-rank 2, there are 149 of type (3,3) and 12 of type (9,3). Parallelization (for instance, 4 threads processing ranges of length 250000) would reduce the CPU time.

I. O. Angell, A table of totally real cubic fields, Math. Comp. 30 (1976), no. 133, 184-187.
M. R. Bush, private communication, 11 July 2015.
H. Cohen and J. Martinet, Class groups of number fields: numerical heuristics, Math. Comp. 48 (1987), no. 177, 123-137.
V. Ennola and R. Turunen, On totally real cubic fields, Math. Comp. 44 (1985), no. 170, 495-518.
P. Llorente and J. Quer, On totally real cubic fields with discriminant D < 10^7, Math. Comp. 50 (1988), no. 182, 581-594.
D. C. Mayer, Quadratic p-ring spaces for counting dihedral fields, Int. J. Number Theory 10 (2014), no. 8, 2205-2242.

Subsequence A269319.

SetClassGroupBounds("GRH"); p:=3;
for d:=0 to 10^6 do if ((d gt 1) and IsFundamental(d)) then
Q:=QuadraticField(d); O:=MaximalOrder(Q); C:=ClassGroup(O);
if (2 eq #pPrimaryInvariants(C,p)) then printf "%o,",d;
end if; end if; end for;

A329786 Discriminants of totally real cubic fields with 2 associated nonconjugate fields.

Original entry on oeis.org

3969, 8281, 13689, 17689, 29241, 37300, 38612, 45684, 46548, 47089, 55700, 61009, 66825, 67081, 69012, 77841, 83700, 90601, 92340, 110889, 113940, 115668, 138996, 148372, 149769, 155412, 157300, 162324, 162409, 164052, 168372, 173556, 181300, 182329, 182868, 185652, 186516, 189972, 191700

Offset: 1

N. J. A. Sloane, Nov 30 2019

nonn

			a(1) = 3969 as there are exactly two distinct totally real cubic fields both with discriminant 3969, namely Q[x]/(x^3 - 21*x - 28) and Q[x]/(x^3 - 21*x - 35). - _Robin Visser_, Apr 18 2025

Robin Visser, Table of n, a(n) for n = 1..700
V. Ennola and R. Turunen, On totally real cubic fields, Math. Comp. 44 (1985), no. 170, 495-518.

Cf. A329769, A329787, A269319.

More terms from Robin Visser, Apr 18 2025

A329787 Discriminants of totally real cubic fields with 3 associated nonconjugate fields.

Original entry on oeis.org

22356, 28212, 31425, 41332, 47860, 54324, 57588, 58077, 62004, 62644, 63028, 65908, 77844, 82484, 86485, 86828, 89073, 95992, 97844, 98132, 99860, 101876, 105192, 108729, 109396, 119604, 122300, 123860, 129164, 136628, 138388, 144212, 144532, 146452, 150164, 152212, 153981, 156244, 161844

Offset: 1

N. J. A. Sloane, Nov 30 2019

nonn

			a(1) = 22356 as there are exactly three distinct totally real cubic fields all with discriminant 22356, namely Q[x]/(x^3 - 36*x - 60), Q[x]/(x^3 - 36*x - 78) and Q[x]/(x^3 - 18*x - 6). - _Robin Visser_, Apr 18 2025

Robin Visser, Table of n, a(n) for n = 1..1500
V. Ennola and R. Turunen, On totally real cubic fields, Math. Comp. 44 (1985), no. 170, 495-518.

Cf. A329769, A329786, A269319.

More terms from Robin Visser, Apr 18 2025

A269320 Discriminants of real quadratic fields with 3-class tower group <81,10>.

Original entry on oeis.org

72329, 94636, 153949, 189237, 206776, 209765, 214028, 219461, 275881, 390876, 400369, 431761, 460817, 486581, 548549, 551692, 552392, 602521, 698556, 775480, 775661, 781177, 782876, 804648, 831484, 836493, 893689, 907629, 907709, 957484, 959629, 980108, 993349, 994008

Offset: 1

Daniel Constantin Mayer, Mar 09 2016

The Artin transfer homomorphisms of the assigned 3-class tower group G with SmallGroups identifier <81,10> [Besche, Eick, O'Brien] determine the capitulation type (1,0,0,0) (TKT with fixed point 1) 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 A269320 is a subsequence of A269319) and [Cl(3,N_i)]=[(9,3),(3,3),(3,3),(3,3)] (TTT or IPAD). Conversely, the group G=<81,10> is determined uniquely by its Artin pattern AP(G)=(TTT,TKT) [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]. The MAGMA program has to determine both components of the Artin pattern, since there are infinitely many 3-groups with TKT a.2, (1,0,0,0), and there are three groups with IPAD [(3,3);(9,3),(3,3),(3,3),(3,3)]. (This is one of the few cases where the "Principalization algorithm via class group structure" [Mayer, 2014] is unable to distinguish between TKT a.2, (1,0,0,0), and a.3, (2,0,0,0). A zero always denotes a total capitulation.) Since the group G=<81,10> has derived length dl(G)=2, the Hilbert 3-class field tower of these real quadratic fields consists of exactly two stages.
It must be pointed out that the MAGMA program must be executed on a machine with Linux operating system, since the MAGMA versions starting with V2.21-8 are not available for Mac OS and MS Windows. MAGMA version V2.21-7 will fail at discriminant 751657. (Bug corrected 13 November 2015 by the MAGMA group, Univ. of Sydney, on our request. See the Change Log of V2.21-8.)
The MAGMA program requires A269319 as its input data.

			The leading two terms, 72329, 94636, have been identified by [Heider, Schmithals] (up to 10^5). The first 34 terms up to 10^6 have been determined in the time between 2006 and 2009 [Mayer, 2009]. The 535 terms up to 10^7, computed 13 January 2016, are not published officially yet. They constitute a refinement of the numerical results in [Mayer, 2014].

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.
F.-P. Heider and B. Schmithals, Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen, J. Reine Angew. Math. 336 (1982), 1-25.
D. C. Mayer, All known examples for principalization types, Memorial 2009.
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.

Subsequence of A269319.

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; TKT:=[]; for j in [1..#sR] do Collector:=[]; I:=sR[j]!!mC(g); if IsPrincipal(I) then Append(~Collector,sI[1]); end if;
I:=sR[j]!!mC(h); if IsPrincipal(I) then Append(~Collector,sI[2]); end if;
for e in [1..p-1] do I := sR[j]!!mC(g+e*h); if IsPrincipal(I) then Append(~Collector,sI[e+2]); end if; end for;
if (2 le #Collector) then Append(~TKT,0); else Append(~TKT,Collector[1]); end if; end for; TAB:=[]; image:=[]; fixedpoints:=0; capitulations:=0;
for j in [1..#TKT] do if (j eq TKT[j]) then Append(~TAB,"A"); fixedpoints:=fixedpoints+1;
elif (0 eq TKT[j]) then Append(~TAB,"A"); capitulations:=capitulations+1;
else Append(~TAB,"B"); end if;
if not (TKT[j] in image) then Append(~image,TKT[j]); end if; end for;
if (2 eq polarization2) and (3 eq polarization1) and (0 eq epsilon) and (1 eq fixedpoints) then printf "%o, ",d; end if; end for;
// On 04 April 2016, MAGMA version V2.21-11 was released for Mac OS, and is able to execute the PROG. - Daniel Constantin Mayer, Apr 16 2016

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

Original entry on oeis.org

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

Offset: 1

Daniel Constantin Mayer, Mar 10 2016

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.

			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].

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, The real quadratic base field K with discriminant d=540365, Targets 2007/2008.
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, New number fields with known p-class tower, Tatra Mt. Math. Publ. 64 (2015), 21-57.
D. C. Mayer, Artin transfer patterns on descendant trees of finite p-groups, Adv. Pure Math. 6 (2016), no. 2, 66-104.

Subsequence of A269319

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 (3 eq polarization2) and (4 eq polarization1) and (0 eq epsilon) then printf "%o, ",d; end if; end for;

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

Original entry on oeis.org

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

Daniel Constantin Mayer, Mar 10 2016

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.

			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].

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.

Subsequence of A269319

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;

A269322 Discriminants of real quadratic fields with second 3-class group <729,49>.

Original entry on oeis.org

534824, 1030117, 2661365, 2733965, 3194013, 3259597, 3268781, 3928632, 4006033, 4593673, 5180081, 5250941, 5327080, 5489661, 5909813, 6115852, 6290549, 7102277, 7712184, 7738629, 7758589, 7857048, 7943761, 8243113, 8747997, 8899661, 9583736, 9907837

Offset: 1

Daniel Constantin Mayer, Mar 09 2016

The Artin transfer homomorphisms of the assigned second 3-class group M with SmallGroups identifier <729,49> [Besche, Eick, O'Brien] determine the capitulation type (0,1,2,2) (TKT without fixed points) 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 A269322 is a subsequence of A269319) and [Cl(3,N_i)]=[(9,9),(3,3,3),(9,3),(9,3)] (TTT or IPAD). Conversely, the group M=<729,49> 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, the MAGMA program has to determine only 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=<729,49> 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 <2187,284> and <2187,291> [Besche, Eick, O'Brien] are permitted for G, and the decision is possible with the aid of iterated IPADs of second order (which require computing 3-class groups of number fields with absolute degree 18). Since the derived length of both groups is equal to 3, the Hilbert 3-class field tower of all these real quadratic fields has exact length 3.
The MAGMA program requires A269319 as its input data.

			The leading term, 534824, and thus the first real quadratic field K with capitulation type c.18, (0,1,2,2), has been identified on 20 August 2009. However, it required six further years to determine the pro-3 Galois group G=<2187,291>, with metabelianization M=G/G''=<729,49>, of the Hilbert 3-class field tower of K in August 2015. The first 28 terms of A269322 up to 10^7 have been determined in [Mayer, 2012] and [Mayer, 2014]. The 347, resp. 4318, terms up to 10^8, resp. 10^9, have been computed by [Bush].
Concerning the two possibilities for the 3-class tower group, 534824 is the smallest term with associated group G=<2187,291> and 1030117 is the smallest term with associated group G=<2187,284>. (See [Mayer, 2015] for more details.)

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, 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, New number fields with known p-class tower, Tatra Mt. Math. Publ. 64 (2015), 21-57.
D. C. Mayer, Artin transfer patterns on descendant trees of finite p-groups, Adv. Pure Math. 6 (2016), no. 2, 66-104.

Subsequence of A269319

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 (3 eq polarization2) and (4 eq polarization1) and (1 eq epsilon) then printf "%o, ",d; end if; end for;

A329785 Discriminants of totally real cubic fields with noncyclic class group.

Original entry on oeis.org

26569, 35537, 76729, 121801, 128357, 146853, 151717, 157609, 210649, 229577, 240149, 277429, 299209, 312709, 314369, 347485, 368449, 376712, 394609, 395177, 409533, 412277, 424148, 428657, 431649, 442489, 444412, 455700, 461041, 468892, 474949

Offset: 1

N. J. A. Sloane, Nov 29 2019

nonn

In Ennola et al., 431649 is listed twice, because there are two such fields with that discriminant.

V. Ennola and R. Turunen, On totally real cubic fields, Math. Comp. 44 (1985), no. 170, 495-518.

Cf. A269319, A329786, A329787.

cp's OEIS Frontend

A379524 Minimal discriminants d of real quadratic number fields K = Q(sqrt(d)), d > 0, with elementary bicyclic 3-class group Cl_3(K)=(3,3) and second 3-class group M=Gal(F_3^2(K)/K) of assigned coclass cc(M)=1,2,3,4,...

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Formula

A269318 Discriminants of real quadratic number fields with 3-class rank 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

A329786 Discriminants of totally real cubic fields with 2 associated nonconjugate fields.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

A329787 Discriminants of totally real cubic fields with 3 associated nonconjugate fields.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

A269320 Discriminants of real quadratic fields with 3-class tower group <81,10>.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

A269322 Discriminants of real quadratic fields with second 3-class group <729,49>.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

A329785 Discriminants of totally real cubic fields with noncyclic class group.