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Properties

Label 27.1.154...079.1
Degree $27$
Signature $[1, 13]$
Discriminant $-1.543\times 10^{42}$
Root discriminant $36.52$
Ramified prime $1759$
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 - 6*x^26 + 31*x^25 - 80*x^24 + 117*x^23 - 51*x^22 - 243*x^21 + 807*x^20 - 1350*x^19 + 966*x^18 + 1935*x^17 - 9345*x^16 + 23103*x^15 - 43548*x^14 + 68751*x^13 - 94575*x^12 + 115269*x^11 - 125268*x^10 + 121785*x^9 - 105567*x^8 + 81027*x^7 - 54357*x^6 + 31122*x^5 - 14664*x^4 + 5393*x^3 - 1350*x^2 + 179*x - 1)
 
gp: K = bnfinit(x^27 - 6*x^26 + 31*x^25 - 80*x^24 + 117*x^23 - 51*x^22 - 243*x^21 + 807*x^20 - 1350*x^19 + 966*x^18 + 1935*x^17 - 9345*x^16 + 23103*x^15 - 43548*x^14 + 68751*x^13 - 94575*x^12 + 115269*x^11 - 125268*x^10 + 121785*x^9 - 105567*x^8 + 81027*x^7 - 54357*x^6 + 31122*x^5 - 14664*x^4 + 5393*x^3 - 1350*x^2 + 179*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 179, -1350, 5393, -14664, 31122, -54357, 81027, -105567, 121785, -125268, 115269, -94575, 68751, -43548, 23103, -9345, 1935, 966, -1350, 807, -243, -51, 117, -80, 31, -6, 1]);
 

\( x^{27} - 6 x^{26} + 31 x^{25} - 80 x^{24} + 117 x^{23} - 51 x^{22} - 243 x^{21} + 807 x^{20} - 1350 x^{19} + 966 x^{18} + 1935 x^{17} - 9345 x^{16} + 23103 x^{15} - 43548 x^{14} + 68751 x^{13} - 94575 x^{12} + 115269 x^{11} - 125268 x^{10} + 121785 x^{9} - 105567 x^{8} + 81027 x^{7} - 54357 x^{6} + 31122 x^{5} - 14664 x^{4} + 5393 x^{3} - 1350 x^{2} + 179 x - 1 \)

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:  \(-1543319746516623033280478216838436483146079\)\(\medspace = -\,1759^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $36.52$
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:  $1759$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{6}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{7}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{8} - \frac{2}{9}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{9} - \frac{2}{9} a$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{10} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{19} + \frac{1}{9} a^{11} - \frac{2}{9} a^{3}$, $\frac{1}{9} a^{20} + \frac{1}{9} a^{12} - \frac{2}{9} a^{4}$, $\frac{1}{9} a^{21} + \frac{1}{9} a^{13} - \frac{2}{9} a^{5}$, $\frac{1}{27} a^{22} + \frac{1}{27} a^{21} - \frac{1}{27} a^{20} - \frac{1}{27} a^{18} - \frac{1}{27} a^{17} + \frac{1}{27} a^{16} + \frac{1}{27} a^{14} + \frac{1}{27} a^{13} - \frac{1}{27} a^{12} - \frac{1}{27} a^{10} - \frac{1}{27} a^{9} + \frac{1}{27} a^{8} - \frac{2}{27} a^{6} - \frac{2}{27} a^{5} + \frac{2}{27} a^{4} + \frac{2}{27} a^{2} + \frac{2}{27} a - \frac{2}{27}$, $\frac{1}{27} a^{23} + \frac{1}{27} a^{21} + \frac{1}{27} a^{20} - \frac{1}{27} a^{19} - \frac{1}{27} a^{17} - \frac{1}{27} a^{16} + \frac{1}{27} a^{15} + \frac{1}{27} a^{13} + \frac{1}{27} a^{12} - \frac{1}{27} a^{11} - \frac{1}{27} a^{9} - \frac{1}{27} a^{8} - \frac{2}{27} a^{7} - \frac{2}{27} a^{5} - \frac{2}{27} a^{4} + \frac{2}{27} a^{3} + \frac{2}{27} a + \frac{2}{27}$, $\frac{1}{27} a^{24} - \frac{1}{9} a^{8} + \frac{2}{27}$, $\frac{1}{202797} a^{25} + \frac{254}{22533} a^{24} + \frac{220}{67599} a^{23} + \frac{442}{28971} a^{22} - \frac{1472}{28971} a^{21} + \frac{7313}{202797} a^{20} + \frac{3403}{67599} a^{19} + \frac{10271}{202797} a^{18} + \frac{9236}{202797} a^{17} + \frac{235}{202797} a^{16} + \frac{3856}{67599} a^{15} + \frac{11230}{202797} a^{14} + \frac{766}{5481} a^{13} - \frac{30865}{202797} a^{12} + \frac{6691}{67599} a^{11} - \frac{8143}{202797} a^{10} + \frac{19178}{202797} a^{9} + \frac{31636}{202797} a^{8} + \frac{32230}{67599} a^{7} - \frac{2675}{28971} a^{6} - \frac{20306}{202797} a^{5} - \frac{72379}{202797} a^{4} - \frac{20714}{67599} a^{3} + \frac{5501}{28971} a^{2} + \frac{14401}{28971} a + \frac{17494}{202797}$, $\frac{1}{3401596152173459846238129} a^{26} + \frac{4307896242397707175}{3401596152173459846238129} a^{25} - \frac{58800317973104262888547}{3401596152173459846238129} a^{24} + \frac{696690136732004143805}{1133865384057819948746043} a^{23} - \frac{609424976164941145585}{161980769151117135535149} a^{22} - \frac{46066059511880976971}{1725822502371111033099} a^{21} + \frac{10230492152795716098671}{377955128019273316248681} a^{20} - \frac{44249908117598173524424}{1133865384057819948746043} a^{19} + \frac{4564622200880778041351}{161980769151117135535149} a^{18} - \frac{20032557056116754684054}{377955128019273316248681} a^{17} + \frac{9722229141917537957788}{377955128019273316248681} a^{16} + \frac{984162272193725786324}{7509042278528608932093} a^{15} + \frac{448355665696345932485}{7509042278528608932093} a^{14} - \frac{113341777598021175587}{3669467262323041905327} a^{13} - \frac{3831422452649316745766}{53993589717039045178383} a^{12} - \frac{2925478470162800351686}{1133865384057819948746043} a^{11} + \frac{135067254605921093428537}{1133865384057819948746043} a^{10} - \frac{152996910428730473766967}{1133865384057819948746043} a^{9} + \frac{86822969295656201363557}{1133865384057819948746043} a^{8} - \frac{205318045204383381101590}{1133865384057819948746043} a^{7} - \frac{347912314047464599446505}{1133865384057819948746043} a^{6} - \frac{178338945657990561998836}{377955128019273316248681} a^{5} + \frac{37701538126997227113758}{125985042673091105416227} a^{4} + \frac{227420428785178017308372}{1133865384057819948746043} a^{3} + \frac{91694467203047109886865}{485942307453351406605447} a^{2} + \frac{222647107579038322344767}{3401596152173459846238129} a + \frac{897274803829459144634932}{3401596152173459846238129}$

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:  \( 71844122831.24568 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{13}\cdot 71844122831.24568 \cdot 1}{2\sqrt{1543319746516623033280478216838436483146079}}\approx 1.37562981139276$ (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

3.1.1759.1, 9.1.9573337234561.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ $27$ $27$ $27$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $27$ $27$ $27$ $27$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
1759Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.1759.2t1.a.a$1$ $ 1759 $ \(\Q(\sqrt{-1759}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.1759.3t2.a.a$2$ $ 1759 $ 3.1.1759.1 $S_3$ (as 3T2) $1$ $0$
* 2.1759.9t3.a.a$2$ $ 1759 $ 9.1.9573337234561.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.1759.9t3.a.b$2$ $ 1759 $ 9.1.9573337234561.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.1759.9t3.a.c$2$ $ 1759 $ 9.1.9573337234561.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.1759.27t8.a.h$2$ $ 1759 $ 27.1.1543319746516623033280478216838436483146079.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1759.27t8.a.e$2$ $ 1759 $ 27.1.1543319746516623033280478216838436483146079.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1759.27t8.a.d$2$ $ 1759 $ 27.1.1543319746516623033280478216838436483146079.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1759.27t8.a.g$2$ $ 1759 $ 27.1.1543319746516623033280478216838436483146079.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1759.27t8.a.a$2$ $ 1759 $ 27.1.1543319746516623033280478216838436483146079.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1759.27t8.a.b$2$ $ 1759 $ 27.1.1543319746516623033280478216838436483146079.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1759.27t8.a.f$2$ $ 1759 $ 27.1.1543319746516623033280478216838436483146079.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1759.27t8.a.c$2$ $ 1759 $ 27.1.1543319746516623033280478216838436483146079.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1759.27t8.a.i$2$ $ 1759 $ 27.1.1543319746516623033280478216838436483146079.1 $D_{27}$ (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 *.