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Properties

Label 27.1.813...999.1
Degree $27$
Signature $[1, 13]$
Discriminant $-8.139\times 10^{42}$
Root discriminant $38.84$
Ramified prime $1999$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $D_{27}$ (as 27T8)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 11*x^26 + 70*x^25 - 273*x^24 + 723*x^23 - 1456*x^22 + 2649*x^21 - 4775*x^20 + 8022*x^19 - 11719*x^18 + 15552*x^17 - 20687*x^16 + 27099*x^15 - 31222*x^14 + 31020*x^13 - 30638*x^12 + 32802*x^11 - 31588*x^10 + 22446*x^9 - 12521*x^8 + 9384*x^7 - 8740*x^6 + 4644*x^5 - 254*x^4 - 460*x^3 - 287*x^2 + 245*x + 1)
 
gp: K = bnfinit(x^27 - 11*x^26 + 70*x^25 - 273*x^24 + 723*x^23 - 1456*x^22 + 2649*x^21 - 4775*x^20 + 8022*x^19 - 11719*x^18 + 15552*x^17 - 20687*x^16 + 27099*x^15 - 31222*x^14 + 31020*x^13 - 30638*x^12 + 32802*x^11 - 31588*x^10 + 22446*x^9 - 12521*x^8 + 9384*x^7 - 8740*x^6 + 4644*x^5 - 254*x^4 - 460*x^3 - 287*x^2 + 245*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 245, -287, -460, -254, 4644, -8740, 9384, -12521, 22446, -31588, 32802, -30638, 31020, -31222, 27099, -20687, 15552, -11719, 8022, -4775, 2649, -1456, 723, -273, 70, -11, 1]);
 

\( x^{27} - 11 x^{26} + 70 x^{25} - 273 x^{24} + 723 x^{23} - 1456 x^{22} + 2649 x^{21} - 4775 x^{20} + 8022 x^{19} - 11719 x^{18} + 15552 x^{17} - 20687 x^{16} + 27099 x^{15} - 31222 x^{14} + 31020 x^{13} - 30638 x^{12} + 32802 x^{11} - 31588 x^{10} + 22446 x^{9} - 12521 x^{8} + 9384 x^{7} - 8740 x^{6} + 4644 x^{5} - 254 x^{4} - 460 x^{3} - 287 x^{2} + 245 x + 1 \)

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:  \(-8138911451501750747538217172562287688025999\)\(\medspace = -\,1999^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $38.84$
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:  $1999$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{6}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{7}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{8} - \frac{2}{9}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{9} - \frac{2}{9} a$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{10} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{19} + \frac{1}{9} a^{11} - \frac{2}{9} a^{3}$, $\frac{1}{27} a^{20} + \frac{1}{27} a^{19} + \frac{1}{27} a^{18} - \frac{1}{27} a^{17} + \frac{1}{27} a^{16} + \frac{1}{9} a^{15} + \frac{1}{9} a^{14} - \frac{2}{27} a^{12} + \frac{4}{27} a^{11} - \frac{2}{27} a^{10} + \frac{2}{27} a^{9} + \frac{4}{27} a^{8} - \frac{1}{9} a^{7} - \frac{1}{9} a^{6} + \frac{1}{27} a^{4} - \frac{5}{27} a^{3} + \frac{1}{27} a^{2} - \frac{1}{27} a - \frac{5}{27}$, $\frac{1}{27} a^{21} + \frac{1}{27} a^{18} - \frac{1}{27} a^{17} - \frac{1}{27} a^{16} - \frac{1}{9} a^{14} - \frac{2}{27} a^{13} - \frac{1}{9} a^{12} + \frac{1}{9} a^{11} - \frac{2}{27} a^{10} - \frac{1}{27} a^{9} - \frac{1}{27} a^{8} + \frac{1}{9} a^{6} + \frac{1}{27} a^{5} + \frac{1}{9} a^{4} - \frac{1}{9} a^{3} + \frac{1}{27} a^{2} + \frac{2}{27} a + \frac{2}{27}$, $\frac{1}{27} a^{22} + \frac{1}{27} a^{19} - \frac{1}{27} a^{18} - \frac{1}{27} a^{17} - \frac{1}{9} a^{15} - \frac{2}{27} a^{14} - \frac{1}{9} a^{13} + \frac{1}{9} a^{12} - \frac{2}{27} a^{11} - \frac{1}{27} a^{10} - \frac{1}{27} a^{9} + \frac{1}{9} a^{7} + \frac{1}{27} a^{6} + \frac{1}{9} a^{5} - \frac{1}{9} a^{4} + \frac{1}{27} a^{3} + \frac{2}{27} a^{2} + \frac{2}{27} a$, $\frac{1}{27} a^{23} + \frac{1}{27} a^{19} + \frac{1}{27} a^{18} + \frac{1}{27} a^{17} - \frac{1}{27} a^{16} + \frac{4}{27} a^{15} + \frac{1}{9} a^{14} + \frac{1}{9} a^{13} - \frac{2}{27} a^{11} + \frac{4}{27} a^{10} - \frac{2}{27} a^{9} + \frac{2}{27} a^{8} - \frac{5}{27} a^{7} - \frac{1}{9} a^{6} - \frac{1}{9} a^{5} + \frac{1}{27} a^{3} - \frac{5}{27} a^{2} + \frac{1}{27} a - \frac{1}{27}$, $\frac{1}{189} a^{24} - \frac{1}{63} a^{23} - \frac{2}{189} a^{22} + \frac{1}{63} a^{21} + \frac{1}{189} a^{19} - \frac{1}{27} a^{18} - \frac{1}{27} a^{17} - \frac{2}{63} a^{16} + \frac{1}{9} a^{15} + \frac{13}{189} a^{14} - \frac{1}{21} a^{13} - \frac{5}{63} a^{12} - \frac{20}{189} a^{11} + \frac{29}{189} a^{10} + \frac{2}{189} a^{9} + \frac{2}{21} a^{8} - \frac{2}{21} a^{7} - \frac{92}{189} a^{6} - \frac{16}{63} a^{5} + \frac{2}{9} a^{4} + \frac{73}{189} a^{3} - \frac{7}{27} a^{2} - \frac{76}{189} a + \frac{41}{189}$, $\frac{1}{6968997} a^{25} + \frac{907}{409941} a^{24} - \frac{119131}{6968997} a^{23} - \frac{16438}{2322999} a^{22} + \frac{111016}{6968997} a^{21} + \frac{5534}{774333} a^{20} - \frac{117550}{6968997} a^{19} + \frac{5057}{110619} a^{18} + \frac{281212}{6968997} a^{17} + \frac{38770}{774333} a^{16} - \frac{89272}{6968997} a^{15} - \frac{34507}{258111} a^{14} + \frac{994459}{6968997} a^{13} - \frac{45130}{331857} a^{12} - \frac{64678}{6968997} a^{11} - \frac{113357}{2322999} a^{10} - \frac{1043939}{6968997} a^{9} - \frac{5744}{110619} a^{8} + \frac{1188476}{6968997} a^{7} - \frac{4358}{331857} a^{6} - \frac{2482178}{6968997} a^{5} + \frac{147064}{2322999} a^{4} - \frac{60232}{6968997} a^{3} - \frac{575689}{2322999} a^{2} + \frac{43906}{331857} a + \frac{63727}{6968997}$, $\frac{1}{79855436126887669833699303} a^{26} + \frac{104801159591417746}{26618478708962556611233101} a^{25} + \frac{127072061881935984532597}{79855436126887669833699303} a^{24} - \frac{68189693900012324788004}{4697378595699274696099959} a^{23} - \frac{391889938218029037163703}{79855436126887669833699303} a^{22} - \frac{1366807648025127246305939}{79855436126887669833699303} a^{21} + \frac{509293846790868260523068}{79855436126887669833699303} a^{20} + \frac{3644658169847964099826343}{79855436126887669833699303} a^{19} - \frac{2804983123956889631105531}{79855436126887669833699303} a^{18} + \frac{3010924131809467857903754}{79855436126887669833699303} a^{17} - \frac{2236115651416121970414559}{79855436126887669833699303} a^{16} - \frac{8611860194426630851480495}{79855436126887669833699303} a^{15} - \frac{2609933185132288020964031}{79855436126887669833699303} a^{14} - \frac{406653363276452098478081}{79855436126887669833699303} a^{13} - \frac{7810592198905232755225342}{79855436126887669833699303} a^{12} + \frac{4389508766377380161854151}{79855436126887669833699303} a^{11} - \frac{2892691666211877008047808}{79855436126887669833699303} a^{10} + \frac{207272027466504508000153}{1629702778099748363953047} a^{9} - \frac{9942350894977123238864734}{79855436126887669833699303} a^{8} - \frac{4706344577079224194741642}{11407919446698238547671329} a^{7} - \frac{308225689722384267609029}{4202917690888824728089437} a^{6} + \frac{224126230088268066994585}{4202917690888824728089437} a^{5} + \frac{10544844761348223144716288}{79855436126887669833699303} a^{4} + \frac{13799927362403524405532933}{79855436126887669833699303} a^{3} - \frac{1910091649126601849897194}{8872826236320852203744367} a^{2} - \frac{1490760866678323788247544}{4202917690888824728089437} a + \frac{6430148992251618469806301}{79855436126887669833699303}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  \( 35248883953.15641 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{13}\cdot 35248883953.15641 \cdot 5}{2\sqrt{8138911451501750747538217172562287688025999}}\approx 1.46950308511013$ (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

3.1.1999.1, 9.1.15968023992001.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ $27$ $27$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ $27$ $27$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ $27$ $27$

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
1999Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.1999.2t1.a.a$1$ $ 1999 $ \(\Q(\sqrt{-1999}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.1999.3t2.a.a$2$ $ 1999 $ 3.1.1999.1 $S_3$ (as 3T2) $1$ $0$
* 2.1999.9t3.a.c$2$ $ 1999 $ 9.1.15968023992001.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.1999.9t3.a.a$2$ $ 1999 $ 9.1.15968023992001.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.1999.9t3.a.b$2$ $ 1999 $ 9.1.15968023992001.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.1999.27t8.a.i$2$ $ 1999 $ 27.1.8138911451501750747538217172562287688025999.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1999.27t8.a.d$2$ $ 1999 $ 27.1.8138911451501750747538217172562287688025999.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1999.27t8.a.f$2$ $ 1999 $ 27.1.8138911451501750747538217172562287688025999.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1999.27t8.a.g$2$ $ 1999 $ 27.1.8138911451501750747538217172562287688025999.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1999.27t8.a.c$2$ $ 1999 $ 27.1.8138911451501750747538217172562287688025999.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1999.27t8.a.b$2$ $ 1999 $ 27.1.8138911451501750747538217172562287688025999.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1999.27t8.a.e$2$ $ 1999 $ 27.1.8138911451501750747538217172562287688025999.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1999.27t8.a.a$2$ $ 1999 $ 27.1.8138911451501750747538217172562287688025999.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1999.27t8.a.h$2$ $ 1999 $ 27.1.8138911451501750747538217172562287688025999.1 $D_{27}$ (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 *.