Properties

Label 27.1.786...199.1
Degree $27$
Signature $[1, 13]$
Discriminant $-7.864\times 10^{40}$
Root discriminant $32.71$
Ramified prime $1399$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $D_{27}$ (as 27T8)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^27 - x^26 + 6*x^25 - 31*x^24 + 97*x^23 - 269*x^22 + 599*x^21 - 994*x^20 + 1307*x^19 - 1298*x^18 + 592*x^17 + 817*x^16 - 2461*x^15 + 4001*x^14 - 4903*x^13 + 4315*x^12 - 2279*x^11 - 512*x^10 + 3872*x^9 - 6814*x^8 + 7826*x^7 - 7115*x^6 + 5689*x^5 - 3842*x^4 + 1951*x^3 - 627*x^2 + 101*x - 1)
 
gp: K = bnfinit(x^27 - x^26 + 6*x^25 - 31*x^24 + 97*x^23 - 269*x^22 + 599*x^21 - 994*x^20 + 1307*x^19 - 1298*x^18 + 592*x^17 + 817*x^16 - 2461*x^15 + 4001*x^14 - 4903*x^13 + 4315*x^12 - 2279*x^11 - 512*x^10 + 3872*x^9 - 6814*x^8 + 7826*x^7 - 7115*x^6 + 5689*x^5 - 3842*x^4 + 1951*x^3 - 627*x^2 + 101*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 101, -627, 1951, -3842, 5689, -7115, 7826, -6814, 3872, -512, -2279, 4315, -4903, 4001, -2461, 817, 592, -1298, 1307, -994, 599, -269, 97, -31, 6, -1, 1]);
 

\( x^{27} - x^{26} + 6 x^{25} - 31 x^{24} + 97 x^{23} - 269 x^{22} + 599 x^{21} - 994 x^{20} + 1307 x^{19} - 1298 x^{18} + 592 x^{17} + 817 x^{16} - 2461 x^{15} + 4001 x^{14} - 4903 x^{13} + 4315 x^{12} - 2279 x^{11} - 512 x^{10} + 3872 x^{9} - 6814 x^{8} + 7826 x^{7} - 7115 x^{6} + 5689 x^{5} - 3842 x^{4} + 1951 x^{3} - 627 x^{2} + 101 x - 1 \)

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:  \(-78637606867438430727852801672920631138199\)\(\medspace = -\,1399^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $32.71$
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:  $1399$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{13} - \frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{8} + \frac{1}{9} a^{7} - \frac{2}{9} a^{6} - \frac{1}{9} a^{4} + \frac{2}{9} a^{3} + \frac{1}{9} a - \frac{2}{9}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{13} - \frac{1}{9} a^{12} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} - \frac{1}{9} a^{6} + \frac{2}{9} a^{5} + \frac{1}{9} a^{3} - \frac{2}{9} a^{2} - \frac{1}{9}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{8} - \frac{2}{9}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{9} - \frac{2}{9} a$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{10} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{19} + \frac{1}{9} a^{11} - \frac{2}{9} a^{3}$, $\frac{1}{9} a^{20} + \frac{1}{9} a^{12} - \frac{2}{9} a^{4}$, $\frac{1}{81} a^{21} + \frac{1}{27} a^{20} + \frac{1}{27} a^{19} - \frac{1}{81} a^{18} + \frac{1}{27} a^{17} + \frac{4}{81} a^{15} - \frac{1}{27} a^{14} + \frac{1}{9} a^{13} + \frac{11}{81} a^{12} + \frac{1}{27} a^{11} + \frac{4}{27} a^{10} + \frac{1}{81} a^{9} + \frac{1}{27} a^{8} - \frac{1}{27} a^{7} + \frac{29}{81} a^{6} - \frac{4}{9} a^{5} - \frac{5}{27} a^{4} - \frac{5}{81} a^{3} - \frac{4}{27} a^{2} + \frac{8}{27} a + \frac{23}{81}$, $\frac{1}{1053} a^{22} - \frac{2}{351} a^{21} + \frac{16}{351} a^{20} - \frac{37}{1053} a^{19} + \frac{4}{351} a^{18} + \frac{1}{39} a^{17} - \frac{32}{1053} a^{16} - \frac{4}{351} a^{15} - \frac{4}{117} a^{14} - \frac{88}{1053} a^{13} + \frac{19}{351} a^{12} + \frac{25}{351} a^{11} - \frac{89}{1053} a^{10} - \frac{38}{351} a^{9} - \frac{28}{351} a^{8} - \frac{43}{1053} a^{7} - \frac{35}{117} a^{6} - \frac{158}{351} a^{5} + \frac{301}{1053} a^{4} - \frac{175}{351} a^{3} - \frac{19}{351} a^{2} - \frac{373}{1053} a - \frac{50}{117}$, $\frac{1}{1053} a^{23} - \frac{1}{1053} a^{21} - \frac{22}{1053} a^{20} - \frac{5}{351} a^{19} - \frac{5}{1053} a^{18} - \frac{2}{81} a^{17} + \frac{10}{351} a^{16} - \frac{43}{1053} a^{15} - \frac{31}{1053} a^{14} - \frac{40}{351} a^{13} - \frac{77}{1053} a^{12} - \frac{29}{1053} a^{11} + \frac{5}{351} a^{10} - \frac{79}{1053} a^{9} - \frac{118}{1053} a^{8} + \frac{17}{351} a^{7} - \frac{518}{1053} a^{6} + \frac{265}{1053} a^{5} + \frac{34}{117} a^{4} - \frac{217}{1053} a^{3} + \frac{11}{81} a^{2} + \frac{92}{351} a - \frac{191}{1053}$, $\frac{1}{1053} a^{24} - \frac{2}{1053} a^{21} - \frac{2}{351} a^{20} + \frac{4}{117} a^{19} - \frac{40}{1053} a^{18} + \frac{2}{117} a^{17} + \frac{14}{351} a^{16} - \frac{56}{1053} a^{15} + \frac{23}{351} a^{13} - \frac{37}{1053} a^{12} - \frac{22}{351} a^{11} + \frac{16}{117} a^{10} - \frac{89}{1053} a^{9} + \frac{5}{117} a^{8} - \frac{2}{39} a^{7} + \frac{2}{1053} a^{6} + \frac{139}{351} a^{5} - \frac{34}{117} a^{4} + \frac{190}{1053} a^{3} + \frac{164}{351} a^{2} - \frac{175}{351} a - \frac{437}{1053}$, $\frac{1}{9499113} a^{25} + \frac{2179}{9499113} a^{24} - \frac{770}{3166371} a^{23} - \frac{142}{351819} a^{22} + \frac{22108}{9499113} a^{21} + \frac{5609}{3166371} a^{20} + \frac{163921}{3166371} a^{19} + \frac{7163}{730701} a^{18} + \frac{54868}{1055457} a^{17} - \frac{164327}{3166371} a^{16} + \frac{13225}{306423} a^{15} + \frac{1615}{102141} a^{14} - \frac{142777}{1055457} a^{13} - \frac{1384609}{9499113} a^{12} + \frac{43298}{3166371} a^{11} + \frac{367999}{3166371} a^{10} + \frac{1261300}{9499113} a^{9} + \frac{26657}{1055457} a^{8} + \frac{258833}{3166371} a^{7} - \frac{2493631}{9499113} a^{6} + \frac{1159861}{3166371} a^{5} + \frac{456713}{1055457} a^{4} + \frac{4384573}{9499113} a^{3} + \frac{111658}{3166371} a^{2} + \frac{701534}{9499113} a - \frac{589700}{9499113}$, $\frac{1}{3787305852213} a^{26} - \frac{149548}{3787305852213} a^{25} + \frac{791939755}{3787305852213} a^{24} + \frac{160117921}{420811761357} a^{23} + \frac{67194607}{3787305852213} a^{22} + \frac{561529180}{291331219401} a^{21} + \frac{4954417159}{97110406467} a^{20} + \frac{172818018944}{3787305852213} a^{19} - \frac{126182037067}{3787305852213} a^{18} + \frac{17211615944}{420811761357} a^{17} - \frac{6233087738}{291331219401} a^{16} + \frac{5425039036}{122171156523} a^{15} - \frac{5801079697}{1262435284071} a^{14} - \frac{236014530691}{3787305852213} a^{13} + \frac{11339074948}{80580975579} a^{12} - \frac{12584457790}{1262435284071} a^{11} - \frac{629115830345}{3787305852213} a^{10} + \frac{331516908106}{3787305852213} a^{9} + \frac{41728899608}{420811761357} a^{8} - \frac{318454670992}{3787305852213} a^{7} + \frac{258088011470}{3787305852213} a^{6} + \frac{63213724585}{140270587119} a^{5} + \frac{829334261707}{3787305852213} a^{4} + \frac{638339062513}{3787305852213} a^{3} + \frac{1149502756733}{3787305852213} a^{2} + \frac{553883555774}{1262435284071} a - \frac{1471759457537}{3787305852213}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 6401978481.787342 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{13}\cdot 6401978481.787342 \cdot 2}{2\sqrt{78637606867438430727852801672920631138199}}\approx 1.08609383919128$ (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

3.1.1399.1, 9.1.3830635754401.1

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 $27$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{3}$ $27$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ $27$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{9}$ $27$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{9}$

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
1399Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.1399.2t1.a.a$1$ $ 1399 $ \(\Q(\sqrt{-1399}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.1399.3t2.a.a$2$ $ 1399 $ 3.1.1399.1 $S_3$ (as 3T2) $1$ $0$
* 2.1399.9t3.a.b$2$ $ 1399 $ 9.1.3830635754401.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.1399.9t3.a.c$2$ $ 1399 $ 9.1.3830635754401.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.1399.9t3.a.a$2$ $ 1399 $ 9.1.3830635754401.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.1399.27t8.a.g$2$ $ 1399 $ 27.1.78637606867438430727852801672920631138199.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1399.27t8.a.i$2$ $ 1399 $ 27.1.78637606867438430727852801672920631138199.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1399.27t8.a.d$2$ $ 1399 $ 27.1.78637606867438430727852801672920631138199.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1399.27t8.a.f$2$ $ 1399 $ 27.1.78637606867438430727852801672920631138199.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1399.27t8.a.h$2$ $ 1399 $ 27.1.78637606867438430727852801672920631138199.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1399.27t8.a.b$2$ $ 1399 $ 27.1.78637606867438430727852801672920631138199.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1399.27t8.a.c$2$ $ 1399 $ 27.1.78637606867438430727852801672920631138199.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1399.27t8.a.e$2$ $ 1399 $ 27.1.78637606867438430727852801672920631138199.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1399.27t8.a.a$2$ $ 1399 $ 27.1.78637606867438430727852801672920631138199.1 $D_{27}$ (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 *.