Properties

Label 27.1.199...291.1
Degree $27$
Signature $[1, 13]$
Discriminant $-1.998\times 10^{49}$
Root discriminant $66.98$
Ramified primes $7, 419$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $D_{27}$ (as 27T8)

Related objects

Downloads

Learn more about

Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 2*x^26 + 16*x^25 + 62*x^24 + 342*x^23 + 1648*x^22 + 6094*x^21 + 15834*x^20 + 34184*x^19 + 61822*x^18 + 112334*x^17 + 237408*x^16 + 637440*x^15 + 1828206*x^14 + 5099608*x^13 + 12463122*x^12 + 26430338*x^11 + 48978608*x^10 + 80603954*x^9 + 116977534*x^8 + 149538616*x^7 + 167922122*x^6 + 164525402*x^5 + 137815936*x^4 + 94764543*x^3 + 50090044*x^2 + 17954520*x + 3418472)
 
gp: K = bnfinit(x^27 - 2*x^26 + 16*x^25 + 62*x^24 + 342*x^23 + 1648*x^22 + 6094*x^21 + 15834*x^20 + 34184*x^19 + 61822*x^18 + 112334*x^17 + 237408*x^16 + 637440*x^15 + 1828206*x^14 + 5099608*x^13 + 12463122*x^12 + 26430338*x^11 + 48978608*x^10 + 80603954*x^9 + 116977534*x^8 + 149538616*x^7 + 167922122*x^6 + 164525402*x^5 + 137815936*x^4 + 94764543*x^3 + 50090044*x^2 + 17954520*x + 3418472, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3418472, 17954520, 50090044, 94764543, 137815936, 164525402, 167922122, 149538616, 116977534, 80603954, 48978608, 26430338, 12463122, 5099608, 1828206, 637440, 237408, 112334, 61822, 34184, 15834, 6094, 1648, 342, 62, 16, -2, 1]);
 

\( x^{27} - 2 x^{26} + 16 x^{25} + 62 x^{24} + 342 x^{23} + 1648 x^{22} + 6094 x^{21} + 15834 x^{20} + 34184 x^{19} + 61822 x^{18} + 112334 x^{17} + 237408 x^{16} + 637440 x^{15} + 1828206 x^{14} + 5099608 x^{13} + 12463122 x^{12} + 26430338 x^{11} + 48978608 x^{10} + 80603954 x^{9} + 116977534 x^{8} + 149538616 x^{7} + 167922122 x^{6} + 164525402 x^{5} + 137815936 x^{4} + 94764543 x^{3} + 50090044 x^{2} + 17954520 x + 3418472 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $27$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[1, 13]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-19977770457280786686246139734781554667936251692291\)\(\medspace = -\,7^{18}\cdot 419^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $66.98$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $7, 419$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{5}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{9} - \frac{1}{8} a^{6} + \frac{1}{8} a^{3}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{10} - \frac{1}{8} a^{7} + \frac{1}{8} a^{4}$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{11} - \frac{1}{8} a^{8} + \frac{1}{8} a^{5}$, $\frac{1}{32} a^{15} - \frac{1}{8} a^{11} + \frac{1}{16} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{8} a^{5} + \frac{13}{32} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{32} a^{16} + \frac{1}{16} a^{10} - \frac{1}{4} a^{7} - \frac{3}{32} a^{4} + \frac{1}{4} a$, $\frac{1}{32} a^{17} + \frac{1}{16} a^{11} - \frac{3}{32} a^{5}$, $\frac{1}{64} a^{18} - \frac{1}{64} a^{15} - \frac{1}{16} a^{14} + \frac{1}{32} a^{12} - \frac{1}{8} a^{10} + \frac{3}{32} a^{9} - \frac{1}{8} a^{8} + \frac{1}{8} a^{7} - \frac{11}{64} a^{6} + \frac{1}{8} a^{4} - \frac{5}{64} a^{3} + \frac{3}{16} a^{2} - \frac{1}{8} a + \frac{1}{8}$, $\frac{1}{64} a^{19} - \frac{1}{64} a^{16} + \frac{1}{32} a^{13} - \frac{1}{8} a^{11} + \frac{3}{32} a^{10} - \frac{1}{8} a^{8} - \frac{11}{64} a^{7} + \frac{1}{8} a^{5} - \frac{5}{64} a^{4} - \frac{1}{2} a^{3} + \frac{1}{8} a^{2} + \frac{1}{8} a - \frac{1}{2}$, $\frac{1}{64} a^{20} - \frac{1}{64} a^{17} + \frac{1}{32} a^{14} + \frac{3}{32} a^{11} + \frac{5}{64} a^{8} - \frac{5}{64} a^{5} - \frac{1}{8} a^{2}$, $\frac{1}{128} a^{21} + \frac{1}{128} a^{15} - \frac{1}{32} a^{14} - \frac{1}{16} a^{13} - \frac{1}{16} a^{12} - \frac{1}{8} a^{11} + \frac{11}{128} a^{9} + \frac{1}{16} a^{8} + \frac{1}{8} a^{7} + \frac{1}{8} a^{5} + \frac{51}{128} a^{3} - \frac{1}{32} a^{2} - \frac{1}{16} a - \frac{7}{16}$, $\frac{1}{256} a^{22} - \frac{1}{256} a^{21} - \frac{1}{64} a^{17} - \frac{3}{256} a^{16} + \frac{3}{256} a^{15} + \frac{3}{64} a^{14} + \frac{1}{32} a^{12} + \frac{3}{32} a^{11} - \frac{29}{256} a^{10} + \frac{29}{256} a^{9} - \frac{1}{32} a^{8} - \frac{3}{16} a^{7} - \frac{1}{4} a^{6} - \frac{5}{64} a^{5} - \frac{33}{256} a^{4} - \frac{95}{256} a^{3} - \frac{1}{64} a^{2} + \frac{7}{16} a + \frac{15}{32}$, $\frac{1}{4352} a^{23} + \frac{1}{4352} a^{22} - \frac{5}{2176} a^{21} + \frac{3}{544} a^{20} - \frac{7}{1088} a^{19} + \frac{3}{544} a^{18} - \frac{67}{4352} a^{17} + \frac{1}{256} a^{16} - \frac{1}{128} a^{15} - \frac{9}{544} a^{14} - \frac{15}{272} a^{13} + \frac{3}{136} a^{12} - \frac{189}{4352} a^{11} - \frac{117}{4352} a^{10} - \frac{239}{2176} a^{9} + \frac{39}{544} a^{8} + \frac{269}{1088} a^{7} + \frac{79}{544} a^{6} - \frac{577}{4352} a^{5} - \frac{733}{4352} a^{4} + \frac{581}{2176} a^{3} + \frac{231}{544} a^{2} - \frac{253}{544} a - \frac{87}{272}$, $\frac{1}{134912} a^{24} + \frac{13}{134912} a^{23} + \frac{35}{67456} a^{22} - \frac{235}{67456} a^{21} - \frac{95}{16864} a^{20} - \frac{39}{16864} a^{19} + \frac{29}{7936} a^{18} - \frac{1399}{134912} a^{17} + \frac{23}{3968} a^{16} + \frac{117}{67456} a^{15} + \frac{135}{8432} a^{14} + \frac{455}{8432} a^{13} + \frac{691}{134912} a^{12} - \frac{15577}{134912} a^{11} + \frac{1609}{67456} a^{10} - \frac{7217}{67456} a^{9} + \frac{2039}{16864} a^{8} - \frac{1911}{16864} a^{7} - \frac{239}{4352} a^{6} - \frac{17789}{134912} a^{5} - \frac{1267}{67456} a^{4} - \frac{825}{67456} a^{3} - \frac{3271}{8432} a^{2} + \frac{303}{1054} a + \frac{1761}{8432}$, $\frac{1}{539648} a^{25} + \frac{1}{539648} a^{24} + \frac{19}{269824} a^{23} + \frac{395}{539648} a^{22} - \frac{1103}{539648} a^{21} + \frac{311}{134912} a^{20} + \frac{45}{31744} a^{19} - \frac{123}{539648} a^{18} + \frac{415}{269824} a^{17} + \frac{863}{539648} a^{16} - \frac{3283}{539648} a^{15} + \frac{8391}{134912} a^{14} - \frac{23677}{539648} a^{13} - \frac{33045}{539648} a^{12} - \frac{9399}{269824} a^{11} + \frac{769}{31744} a^{10} + \frac{6403}{539648} a^{9} + \frac{5925}{134912} a^{8} - \frac{53105}{539648} a^{7} + \frac{103111}{539648} a^{6} - \frac{1979}{269824} a^{5} - \frac{51067}{539648} a^{4} + \frac{204831}{539648} a^{3} - \frac{3347}{7936} a^{2} + \frac{1255}{33728} a + \frac{32601}{67456}$, $\frac{1}{18255996954386381970253586769864394354559177863283014741479831000064} a^{26} + \frac{7890763687351371537457862927170324033346669217171864066332865}{9127998477193190985126793384932197177279588931641507370739915500032} a^{25} + \frac{34249733947455797235946119289248362168328290803796214678211431}{18255996954386381970253586769864394354559177863283014741479831000064} a^{24} - \frac{1734504584278516565693697787286563397180621459512742900339782379}{18255996954386381970253586769864394354559177863283014741479831000064} a^{23} + \frac{22046365039173917865540554428479638183973039665155136765529817}{2281999619298297746281698346233049294319897232910376842684978875008} a^{22} + \frac{48360198116681080457109028030260701149298967661155386382568522925}{18255996954386381970253586769864394354559177863283014741479831000064} a^{21} + \frac{99551735796253658585467789060625948593755121711140796691277365209}{18255996954386381970253586769864394354559177863283014741479831000064} a^{20} - \frac{35236695865022458530291914082237370341775133448284026194996660983}{9127998477193190985126793384932197177279588931641507370739915500032} a^{19} + \frac{77897082301786914094533814328937195593896538991223053274646853667}{18255996954386381970253586769864394354559177863283014741479831000064} a^{18} + \frac{245002854648605788775537987910807191983565545572907784534450290513}{18255996954386381970253586769864394354559177863283014741479831000064} a^{17} - \frac{19799177161705970628366141260522992197918377085904319399879591789}{2281999619298297746281698346233049294319897232910376842684978875008} a^{16} + \frac{11185157356092295425756635700450353501173696759328897977848471593}{18255996954386381970253586769864394354559177863283014741479831000064} a^{15} - \frac{168085732586088680026453751132758731251308024248725848542182748705}{18255996954386381970253586769864394354559177863283014741479831000064} a^{14} + \frac{405649254481841965834421789892830895245472630899687089779836643095}{9127998477193190985126793384932197177279588931641507370739915500032} a^{13} + \frac{8876641960182249920309486510763349379044464946845267230910484013}{1073882173787434233544328633521434962032892815487236161263519470592} a^{12} - \frac{1104840388245157320986904170246035013681217611491002979898771964017}{18255996954386381970253586769864394354559177863283014741479831000064} a^{11} + \frac{11169217583117905016465235212996000411287155605178985559855650515}{134235271723429279193041079190179370254111601935904520157939933824} a^{10} + \frac{738603314655675498415470332187161674310871388716977895718041016951}{18255996954386381970253586769864394354559177863283014741479831000064} a^{9} + \frac{2268479672559347009323662350445911949252755386934631738620914071907}{18255996954386381970253586769864394354559177863283014741479831000064} a^{8} - \frac{2265570274189874582295264404810744698372929284897555859672172834413}{9127998477193190985126793384932197177279588931641507370739915500032} a^{7} - \frac{2818022974148983600626246322162400338452846310572746515803652301647}{18255996954386381970253586769864394354559177863283014741479831000064} a^{6} + \frac{3786502078758917964993733252295597777899663744778516308285096389643}{18255996954386381970253586769864394354559177863283014741479831000064} a^{5} + \frac{78485024021221517039638323204203583618659081347296491595982178177}{2281999619298297746281698346233049294319897232910376842684978875008} a^{4} - \frac{1746491598996815684695521933880204043906581109463305658790178633165}{18255996954386381970253586769864394354559177863283014741479831000064} a^{3} - \frac{115875730189312570849401450819247814086984700307651483388141874119}{4563999238596595492563396692466098588639794465820753685369957750016} a^{2} + \frac{585787995369731399439545123947101573540982361046017030570048803635}{2281999619298297746281698346233049294319897232910376842684978875008} a + \frac{691166909515204696291588592073567606900634960874700382003276419433}{2281999619298297746281698346233049294319897232910376842684978875008}$

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Class group and class number

$C_{3}$, which has order $3$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $13$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 951820331630620.2 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{13}\cdot 951820331630620.2 \cdot 3}{2\sqrt{19977770457280786686246139734781554667936251692291}}\approx 15.1964103173312$ (assuming GRH)

Galois group

$D_{27}$ (as 27T8):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 54
The 15 conjugacy class representatives for $D_{27}$
Character table for $D_{27}$

Intermediate fields

3.1.419.1, 9.1.30821664721.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ $27$ $27$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ $27$ $27$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ $27$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ $27$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
7Data not computed
419Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.419.2t1.a.a$1$ $ 419 $ \(\Q(\sqrt{-419}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.419.3t2.a.a$2$ $ 419 $ 3.1.419.1 $S_3$ (as 3T2) $1$ $0$
* 2.419.9t3.a.c$2$ $ 419 $ 9.1.30821664721.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.419.9t3.a.a$2$ $ 419 $ 9.1.30821664721.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.419.9t3.a.b$2$ $ 419 $ 9.1.30821664721.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.20531.27t8.a.h$2$ $ 7^{2} \cdot 419 $ 27.1.19977770457280786686246139734781554667936251692291.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.20531.27t8.a.a$2$ $ 7^{2} \cdot 419 $ 27.1.19977770457280786686246139734781554667936251692291.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.20531.27t8.a.g$2$ $ 7^{2} \cdot 419 $ 27.1.19977770457280786686246139734781554667936251692291.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.20531.27t8.a.b$2$ $ 7^{2} \cdot 419 $ 27.1.19977770457280786686246139734781554667936251692291.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.20531.27t8.a.i$2$ $ 7^{2} \cdot 419 $ 27.1.19977770457280786686246139734781554667936251692291.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.20531.27t8.a.d$2$ $ 7^{2} \cdot 419 $ 27.1.19977770457280786686246139734781554667936251692291.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.20531.27t8.a.f$2$ $ 7^{2} \cdot 419 $ 27.1.19977770457280786686246139734781554667936251692291.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.20531.27t8.a.c$2$ $ 7^{2} \cdot 419 $ 27.1.19977770457280786686246139734781554667936251692291.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.20531.27t8.a.e$2$ $ 7^{2} \cdot 419 $ 27.1.19977770457280786686246139734781554667936251692291.1 $D_{27}$ (as 27T8) $1$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.