Properties

Label 27.1.800...063.1
Degree $27$
Signature $[1, 13]$
Discriminant $-8.002\times 10^{38}$
Root discriminant $27.60$
Ramified prime $983$
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 - 9*x^26 + 41*x^25 - 116*x^24 + 234*x^23 - 347*x^22 + 380*x^21 - 274*x^20 + 118*x^19 - 121*x^18 + 474*x^17 - 1071*x^16 + 1556*x^15 - 1531*x^14 + 964*x^13 - 256*x^12 + 122*x^11 - 700*x^10 + 1761*x^9 - 2620*x^8 + 2840*x^7 - 2483*x^6 + 1678*x^5 - 860*x^4 + 463*x^3 - 60*x^2 + 44*x - 1)
 
gp: K = bnfinit(x^27 - 9*x^26 + 41*x^25 - 116*x^24 + 234*x^23 - 347*x^22 + 380*x^21 - 274*x^20 + 118*x^19 - 121*x^18 + 474*x^17 - 1071*x^16 + 1556*x^15 - 1531*x^14 + 964*x^13 - 256*x^12 + 122*x^11 - 700*x^10 + 1761*x^9 - 2620*x^8 + 2840*x^7 - 2483*x^6 + 1678*x^5 - 860*x^4 + 463*x^3 - 60*x^2 + 44*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 44, -60, 463, -860, 1678, -2483, 2840, -2620, 1761, -700, 122, -256, 964, -1531, 1556, -1071, 474, -121, 118, -274, 380, -347, 234, -116, 41, -9, 1]);
 

\( x^{27} - 9 x^{26} + 41 x^{25} - 116 x^{24} + 234 x^{23} - 347 x^{22} + 380 x^{21} - 274 x^{20} + 118 x^{19} - 121 x^{18} + 474 x^{17} - 1071 x^{16} + 1556 x^{15} - 1531 x^{14} + 964 x^{13} - 256 x^{12} + 122 x^{11} - 700 x^{10} + 1761 x^{9} - 2620 x^{8} + 2840 x^{7} - 2483 x^{6} + 1678 x^{5} - 860 x^{4} + 463 x^{3} - 60 x^{2} + 44 x - 1 \)

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:  \(-800194812883861824535483445924579987063\)\(\medspace = -\,983^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $27.60$
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:  $983$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{5} a^{17} + \frac{2}{5} a^{16} - \frac{2}{5} a^{15} - \frac{2}{5} a^{14} + \frac{2}{5} a^{12} + \frac{2}{5} a^{11} - \frac{1}{5} a^{9} - \frac{2}{5} a^{7} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{2} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{18} - \frac{1}{5} a^{16} + \frac{2}{5} a^{15} - \frac{1}{5} a^{14} + \frac{2}{5} a^{13} - \frac{2}{5} a^{12} + \frac{1}{5} a^{11} - \frac{1}{5} a^{10} + \frac{2}{5} a^{9} - \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{19} - \frac{1}{5} a^{16} + \frac{2}{5} a^{15} - \frac{2}{5} a^{13} - \frac{2}{5} a^{12} + \frac{1}{5} a^{11} + \frac{2}{5} a^{10} + \frac{2}{5} a^{9} - \frac{1}{5} a^{8} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5}$, $\frac{1}{5} a^{20} - \frac{1}{5} a^{16} - \frac{2}{5} a^{15} + \frac{1}{5} a^{14} - \frac{2}{5} a^{13} - \frac{2}{5} a^{12} - \frac{1}{5} a^{11} + \frac{2}{5} a^{10} - \frac{2}{5} a^{9} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{21} - \frac{1}{5} a^{15} + \frac{1}{5} a^{14} - \frac{2}{5} a^{13} + \frac{1}{5} a^{12} - \frac{1}{5} a^{11} - \frac{2}{5} a^{10} - \frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{85} a^{22} + \frac{6}{85} a^{21} + \frac{7}{85} a^{20} - \frac{6}{85} a^{19} - \frac{6}{85} a^{17} - \frac{14}{85} a^{16} - \frac{14}{85} a^{15} - \frac{42}{85} a^{14} + \frac{12}{85} a^{13} - \frac{19}{85} a^{12} - \frac{28}{85} a^{11} - \frac{6}{85} a^{10} - \frac{4}{17} a^{9} + \frac{23}{85} a^{8} - \frac{13}{85} a^{7} + \frac{37}{85} a^{6} - \frac{3}{17} a^{5} - \frac{1}{17} a^{4} - \frac{37}{85} a^{3} - \frac{24}{85} a^{2} + \frac{21}{85} a + \frac{32}{85}$, $\frac{1}{1105} a^{23} - \frac{2}{1105} a^{22} + \frac{19}{221} a^{21} + \frac{6}{1105} a^{20} - \frac{54}{1105} a^{19} - \frac{23}{1105} a^{18} + \frac{6}{65} a^{17} - \frac{79}{221} a^{16} - \frac{236}{1105} a^{15} + \frac{433}{1105} a^{14} + \frac{72}{1105} a^{13} - \frac{97}{1105} a^{12} - \frac{394}{1105} a^{11} - \frac{42}{221} a^{10} + \frac{108}{221} a^{9} + \frac{66}{221} a^{8} + \frac{37}{85} a^{7} - \frac{4}{17} a^{6} - \frac{293}{1105} a^{5} + \frac{326}{1105} a^{4} - \frac{28}{65} a^{3} - \frac{5}{221} a^{2} + \frac{16}{65} a - \frac{103}{1105}$, $\frac{1}{5525} a^{24} + \frac{2}{5525} a^{23} + \frac{22}{5525} a^{22} - \frac{4}{5525} a^{21} - \frac{43}{5525} a^{20} - \frac{512}{5525} a^{19} - \frac{432}{5525} a^{18} - \frac{4}{85} a^{17} - \frac{2674}{5525} a^{16} - \frac{264}{5525} a^{15} - \frac{991}{5525} a^{14} + \frac{2284}{5525} a^{13} - \frac{652}{5525} a^{12} + \frac{93}{325} a^{11} - \frac{424}{1105} a^{10} + \frac{1359}{5525} a^{9} + \frac{44}{325} a^{8} + \frac{193}{425} a^{7} - \frac{173}{1105} a^{6} - \frac{1418}{5525} a^{5} + \frac{108}{221} a^{4} + \frac{16}{65} a^{3} - \frac{1583}{5525} a^{2} + \frac{1167}{5525} a - \frac{282}{5525}$, $\frac{1}{5525} a^{25} - \frac{2}{5525} a^{23} - \frac{8}{5525} a^{22} + \frac{11}{221} a^{21} - \frac{42}{425} a^{20} - \frac{538}{5525} a^{19} - \frac{41}{5525} a^{18} + \frac{226}{5525} a^{17} - \frac{2486}{5525} a^{16} - \frac{2373}{5525} a^{15} + \frac{87}{425} a^{14} + \frac{436}{1105} a^{13} + \frac{81}{1105} a^{12} + \frac{388}{5525} a^{11} + \frac{2064}{5525} a^{10} + \frac{319}{1105} a^{9} + \frac{1043}{5525} a^{8} + \frac{1072}{5525} a^{7} + \frac{84}{425} a^{6} - \frac{1864}{5525} a^{5} + \frac{98}{1105} a^{4} + \frac{1902}{5525} a^{3} + \frac{2623}{5525} a^{2} + \frac{1889}{5525} a - \frac{691}{5525}$, $\frac{1}{160497705249110837897575} a^{26} - \frac{1143171342020832313}{14590700477191894354325} a^{25} + \frac{12042547297511383639}{160497705249110837897575} a^{24} - \frac{261342174697961794}{2918140095438378870865} a^{23} - \frac{739809879310502280739}{160497705249110837897575} a^{22} + \frac{740811871484393469103}{32099541049822167579515} a^{21} + \frac{3258012399365648422627}{160497705249110837897575} a^{20} + \frac{9418735920655884211396}{160497705249110837897575} a^{19} + \frac{10269350004228255221792}{160497705249110837897575} a^{18} - \frac{12902260304265240580209}{160497705249110837897575} a^{17} - \frac{52956548233273764927189}{160497705249110837897575} a^{16} + \frac{30871640536410862160916}{160497705249110837897575} a^{15} - \frac{46955886372723557565459}{160497705249110837897575} a^{14} + \frac{2406294423568932068119}{160497705249110837897575} a^{13} + \frac{1604046812257698793573}{9441041485241813993975} a^{12} + \frac{36264304417158786945836}{160497705249110837897575} a^{11} - \frac{76577735361570364878322}{160497705249110837897575} a^{10} + \frac{45841298791240909448282}{160497705249110837897575} a^{9} + \frac{23443617690375582435321}{160497705249110837897575} a^{8} - \frac{9721655120203278319656}{32099541049822167579515} a^{7} - \frac{13395596628200100868117}{32099541049822167579515} a^{6} - \frac{4703429698717194610411}{14590700477191894354325} a^{5} + \frac{68282674755647920441022}{160497705249110837897575} a^{4} - \frac{29089499530605548866318}{160497705249110837897575} a^{3} - \frac{26217258769654822472563}{160497705249110837897575} a^{2} + \frac{75812345561325992011044}{160497705249110837897575} a - \frac{16548831501749163795569}{160497705249110837897575}$

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 $2$)
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:  \( 460164457.10213304 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{13}\cdot 460164457.10213304 \cdot 1}{2\sqrt{800194812883861824535483445924579987063}}\approx 0.386948790637645$ (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

3.1.983.1, 9.1.933714431521.1

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 $27$ $27$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ $27$ $27$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ $27$ $27$ $27$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{9}$

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
983Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.983.2t1.a.a$1$ $ 983 $ \(\Q(\sqrt{-983}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.983.3t2.a.a$2$ $ 983 $ 3.1.983.1 $S_3$ (as 3T2) $1$ $0$
* 2.983.9t3.a.c$2$ $ 983 $ 9.1.933714431521.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.983.9t3.a.a$2$ $ 983 $ 9.1.933714431521.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.983.9t3.a.b$2$ $ 983 $ 9.1.933714431521.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.983.27t8.a.d$2$ $ 983 $ 27.1.800194812883861824535483445924579987063.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.983.27t8.a.h$2$ $ 983 $ 27.1.800194812883861824535483445924579987063.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.983.27t8.a.b$2$ $ 983 $ 27.1.800194812883861824535483445924579987063.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.983.27t8.a.c$2$ $ 983 $ 27.1.800194812883861824535483445924579987063.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.983.27t8.a.g$2$ $ 983 $ 27.1.800194812883861824535483445924579987063.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.983.27t8.a.f$2$ $ 983 $ 27.1.800194812883861824535483445924579987063.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.983.27t8.a.e$2$ $ 983 $ 27.1.800194812883861824535483445924579987063.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.983.27t8.a.a$2$ $ 983 $ 27.1.800194812883861824535483445924579987063.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.983.27t8.a.i$2$ $ 983 $ 27.1.800194812883861824535483445924579987063.1 $D_{27}$ (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 *.