Properties

Label 27.1.128...351.1
Degree $27$
Signature $[1, 13]$
Discriminant $-1.287\times 10^{45}$
Root discriminant $46.85$
Ramified primes $13, 227$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{27}$ (as 27T8)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 4*x^26 + 28*x^25 - 42*x^24 + 71*x^23 + 267*x^22 - 142*x^21 + 385*x^20 + 2210*x^19 + 2329*x^18 - 3315*x^17 + 15117*x^16 + 19014*x^15 - 23157*x^14 + 44479*x^13 + 77004*x^12 - 41903*x^11 + 19829*x^10 + 208952*x^9 - 35302*x^8 - 53453*x^7 + 258838*x^6 - 35701*x^5 + 60990*x^4 - 5393*x^3 + 76227*x^2 + 19031*x + 2209)
 
gp: K = bnfinit(x^27 - 4*x^26 + 28*x^25 - 42*x^24 + 71*x^23 + 267*x^22 - 142*x^21 + 385*x^20 + 2210*x^19 + 2329*x^18 - 3315*x^17 + 15117*x^16 + 19014*x^15 - 23157*x^14 + 44479*x^13 + 77004*x^12 - 41903*x^11 + 19829*x^10 + 208952*x^9 - 35302*x^8 - 53453*x^7 + 258838*x^6 - 35701*x^5 + 60990*x^4 - 5393*x^3 + 76227*x^2 + 19031*x + 2209, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2209, 19031, 76227, -5393, 60990, -35701, 258838, -53453, -35302, 208952, 19829, -41903, 77004, 44479, -23157, 19014, 15117, -3315, 2329, 2210, 385, -142, 267, 71, -42, 28, -4, 1]);
 

\(x^{27} - 4 x^{26} + 28 x^{25} - 42 x^{24} + 71 x^{23} + 267 x^{22} - 142 x^{21} + 385 x^{20} + 2210 x^{19} + 2329 x^{18} - 3315 x^{17} + 15117 x^{16} + 19014 x^{15} - 23157 x^{14} + 44479 x^{13} + 77004 x^{12} - 41903 x^{11} + 19829 x^{10} + 208952 x^{9} - 35302 x^{8} - 53453 x^{7} + 258838 x^{6} - 35701 x^{5} + 60990 x^{4} - 5393 x^{3} + 76227 x^{2} + 19031 x + 2209\)  Toggle raw display

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:  \(-1287062756823986424273208421285283554009333351\)\(\medspace = -\,13^{13}\cdot 227^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $46.85$
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:  $13, 227$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13} a^{12} + \frac{1}{13} a^{9} - \frac{6}{13} a^{6} + \frac{5}{13} a^{3} - \frac{1}{13}$, $\frac{1}{13} a^{13} + \frac{1}{13} a^{10} - \frac{6}{13} a^{7} + \frac{5}{13} a^{4} - \frac{1}{13} a$, $\frac{1}{13} a^{14} + \frac{1}{13} a^{11} - \frac{6}{13} a^{8} + \frac{5}{13} a^{5} - \frac{1}{13} a^{2}$, $\frac{1}{13} a^{15} + \frac{6}{13} a^{9} - \frac{2}{13} a^{6} - \frac{6}{13} a^{3} + \frac{1}{13}$, $\frac{1}{13} a^{16} + \frac{6}{13} a^{10} - \frac{2}{13} a^{7} - \frac{6}{13} a^{4} + \frac{1}{13} a$, $\frac{1}{13} a^{17} + \frac{6}{13} a^{11} - \frac{2}{13} a^{8} - \frac{6}{13} a^{5} + \frac{1}{13} a^{2}$, $\frac{1}{13} a^{18} + \frac{5}{13} a^{9} + \frac{4}{13} a^{6} - \frac{3}{13} a^{3} + \frac{6}{13}$, $\frac{1}{13} a^{19} + \frac{5}{13} a^{10} + \frac{4}{13} a^{7} - \frac{3}{13} a^{4} + \frac{6}{13} a$, $\frac{1}{91} a^{20} + \frac{2}{91} a^{19} - \frac{2}{91} a^{18} - \frac{2}{91} a^{17} - \frac{1}{91} a^{16} + \frac{3}{91} a^{15} - \frac{1}{91} a^{14} - \frac{2}{91} a^{13} - \frac{3}{91} a^{12} + \frac{18}{91} a^{11} + \frac{41}{91} a^{10} - \frac{8}{91} a^{9} - \frac{12}{91} a^{8} + \frac{5}{13} a^{7} - \frac{9}{91} a^{6} + \frac{30}{91} a^{5} - \frac{10}{91} a^{4} - \frac{2}{13} a^{3} + \frac{18}{91} a^{2} + \frac{2}{7} a - \frac{32}{91}$, $\frac{1}{13013} a^{21} - \frac{10}{1859} a^{20} + \frac{50}{13013} a^{19} - \frac{194}{13013} a^{18} + \frac{311}{13013} a^{17} - \frac{471}{13013} a^{16} - \frac{61}{1859} a^{15} - \frac{34}{1859} a^{14} + \frac{456}{13013} a^{13} + \frac{17}{13013} a^{12} - \frac{5833}{13013} a^{11} + \frac{1975}{13013} a^{10} - \frac{2502}{13013} a^{9} + \frac{3321}{13013} a^{8} + \frac{1006}{13013} a^{7} + \frac{4150}{13013} a^{6} + \frac{106}{1859} a^{5} + \frac{4241}{13013} a^{4} + \frac{753}{13013} a^{3} - \frac{6345}{13013} a^{2} + \frac{267}{1859} a + \frac{1751}{13013}$, $\frac{1}{13013} a^{22} + \frac{12}{13013} a^{20} + \frac{17}{13013} a^{19} + \frac{30}{13013} a^{18} - \frac{437}{13013} a^{17} - \frac{17}{1001} a^{16} + \frac{474}{13013} a^{15} - \frac{45}{13013} a^{14} + \frac{191}{13013} a^{13} - \frac{30}{1859} a^{12} - \frac{5506}{13013} a^{11} - \frac{750}{1859} a^{10} + \frac{6502}{13013} a^{9} - \frac{5048}{13013} a^{8} + \frac{1497}{13013} a^{7} - \frac{1765}{13013} a^{6} + \frac{263}{1859} a^{5} + \frac{1756}{13013} a^{4} - \frac{62}{1183} a^{3} + \frac{2592}{13013} a^{2} - \frac{896}{1859} a - \frac{1983}{13013}$, $\frac{1}{32831187389} a^{23} + \frac{38078}{2525475953} a^{22} + \frac{263255}{32831187389} a^{21} + \frac{13352125}{2525475953} a^{20} + \frac{138742609}{32831187389} a^{19} - \frac{11719439}{4690169627} a^{18} + \frac{812060659}{32831187389} a^{17} - \frac{1031178641}{32831187389} a^{16} - \frac{832232967}{32831187389} a^{15} + \frac{567648079}{32831187389} a^{14} - \frac{101919060}{32831187389} a^{13} - \frac{694048000}{32831187389} a^{12} + \frac{262516510}{2525475953} a^{11} + \frac{15794305718}{32831187389} a^{10} + \frac{14796286543}{32831187389} a^{9} + \frac{234234939}{1931246317} a^{8} + \frac{4791672685}{32831187389} a^{7} + \frac{658026669}{2525475953} a^{6} - \frac{7454500618}{32831187389} a^{5} - \frac{799295}{25081121} a^{4} + \frac{2217150401}{4690169627} a^{3} - \frac{1261962947}{4690169627} a^{2} - \frac{972239964}{2984653399} a + \frac{4744434294}{32831187389}$, $\frac{1}{32831187389} a^{24} + \frac{690231}{32831187389} a^{22} + \frac{151654}{32831187389} a^{21} - \frac{166775}{64248899} a^{20} + \frac{422606453}{32831187389} a^{19} - \frac{222635368}{32831187389} a^{18} - \frac{796535078}{32831187389} a^{17} + \frac{1086552017}{32831187389} a^{16} - \frac{389182806}{32831187389} a^{15} + \frac{375307630}{32831187389} a^{14} + \frac{123101492}{4690169627} a^{13} - \frac{1170284438}{32831187389} a^{12} - \frac{87556412}{1727957231} a^{11} - \frac{5022633202}{32831187389} a^{10} + \frac{10797141997}{32831187389} a^{9} + \frac{317546521}{32831187389} a^{8} - \frac{4351310079}{32831187389} a^{7} + \frac{7741336298}{32831187389} a^{6} + \frac{4320077516}{32831187389} a^{5} + \frac{1167687818}{2984653399} a^{4} - \frac{13482856159}{32831187389} a^{3} - \frac{15554742529}{32831187389} a^{2} + \frac{3126599635}{32831187389} a - \frac{9178792284}{32831187389}$, $\frac{1}{523033646294159} a^{25} + \frac{18}{74719092327737} a^{24} + \frac{193}{523033646294159} a^{23} + \frac{16601461330}{523033646294159} a^{22} + \frac{4244798319}{523033646294159} a^{21} - \frac{822304632991}{523033646294159} a^{20} + \frac{7056405286235}{523033646294159} a^{19} + \frac{13248912255927}{523033646294159} a^{18} - \frac{2541159895485}{523033646294159} a^{17} + \frac{6164195203032}{523033646294159} a^{16} + \frac{16860351078640}{523033646294159} a^{15} + \frac{1141904505874}{47548513299469} a^{14} - \frac{16574990243257}{523033646294159} a^{13} - \frac{1564029979894}{74719092327737} a^{12} - \frac{1771508114132}{27528086647061} a^{11} - \frac{23360182650449}{523033646294159} a^{10} - \frac{1400742565977}{4888164918637} a^{9} + \frac{103184134443274}{523033646294159} a^{8} - \frac{542936003431}{3932583806723} a^{7} - \frac{18384955880722}{74719092327737} a^{6} + \frac{30208588644495}{523033646294159} a^{5} + \frac{257883409950519}{523033646294159} a^{4} - \frac{229668110010253}{523033646294159} a^{3} + \frac{187960243724587}{523033646294159} a^{2} - \frac{220610074041613}{523033646294159} a + \frac{6090731561546}{40233357407243}$, $\frac{1}{3379320388706561299} a^{26} - \frac{5}{10462292225097713} a^{25} + \frac{657032}{3379320388706561299} a^{24} + \frac{34574003}{3379320388706561299} a^{23} - \frac{74177854409982}{3379320388706561299} a^{22} + \frac{107367355540727}{3379320388706561299} a^{21} + \frac{1307487188814}{4208369101751633} a^{20} - \frac{47236015878639570}{3379320388706561299} a^{19} + \frac{21210462270947552}{3379320388706561299} a^{18} - \frac{2077912598311538}{177858967826661121} a^{17} - \frac{6753238529253737}{177858967826661121} a^{16} + \frac{10667455458595031}{3379320388706561299} a^{15} - \frac{8728765614449423}{259947722208197023} a^{14} + \frac{5837062928583954}{307210944427869209} a^{13} + \frac{9027276069613541}{307210944427869209} a^{12} + \frac{64654691866060966}{3379320388706561299} a^{11} - \frac{751544366123693477}{3379320388706561299} a^{10} - \frac{54144319316417591}{177858967826661121} a^{9} + \frac{1195617125811139561}{3379320388706561299} a^{8} - \frac{921052561218035217}{3379320388706561299} a^{7} + \frac{52102795801780452}{177858967826661121} a^{6} + \frac{68594112031913014}{3379320388706561299} a^{5} - \frac{215770993557652626}{3379320388706561299} a^{4} + \frac{23209391718075866}{307210944427869209} a^{3} + \frac{77298324589153181}{3379320388706561299} a^{2} + \frac{290937407124098386}{3379320388706561299} a - \frac{153582814682476854}{3379320388706561299}$  Toggle raw display

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

Class group and class number

Trivial group, which has order $1$ (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 $2$)  Toggle raw display
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:  \( 1184470448230.2556 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{13}\cdot 1184470448230.2556 \cdot 1}{2\sqrt{1287062756823986424273208421285283554009333351}}\approx 0.785348930660551$ (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

3.1.2951.1, 9.1.75836247976801.1

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 $27$ $27$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{9}$ $27$ $27$ $27$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ $27$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
227Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.2951.2t1.a.a$1$ $ 13 \cdot 227 $ \(\Q(\sqrt{-2951}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.2951.3t2.a.a$2$ $ 13 \cdot 227 $ 3.1.2951.1 $S_3$ (as 3T2) $1$ $0$
* 2.2951.9t3.a.b$2$ $ 13 \cdot 227 $ 9.1.75836247976801.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.2951.9t3.a.c$2$ $ 13 \cdot 227 $ 9.1.75836247976801.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.2951.9t3.a.a$2$ $ 13 \cdot 227 $ 9.1.75836247976801.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.2951.27t8.a.c$2$ $ 13 \cdot 227 $ 27.1.1287062756823986424273208421285283554009333351.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2951.27t8.a.b$2$ $ 13 \cdot 227 $ 27.1.1287062756823986424273208421285283554009333351.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2951.27t8.a.e$2$ $ 13 \cdot 227 $ 27.1.1287062756823986424273208421285283554009333351.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2951.27t8.a.i$2$ $ 13 \cdot 227 $ 27.1.1287062756823986424273208421285283554009333351.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2951.27t8.a.f$2$ $ 13 \cdot 227 $ 27.1.1287062756823986424273208421285283554009333351.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2951.27t8.a.a$2$ $ 13 \cdot 227 $ 27.1.1287062756823986424273208421285283554009333351.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2951.27t8.a.h$2$ $ 13 \cdot 227 $ 27.1.1287062756823986424273208421285283554009333351.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2951.27t8.a.g$2$ $ 13 \cdot 227 $ 27.1.1287062756823986424273208421285283554009333351.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2951.27t8.a.d$2$ $ 13 \cdot 227 $ 27.1.1287062756823986424273208421285283554009333351.1 $D_{27}$ (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 *.