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Properties

Label 16.0.22745398789...6704.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{28}\cdot 3^{14}\cdot 11^{6}$
Root discriminant $21.62$
Ramified primes $2, 3, 11$
Class number $2$
Class group $[2]$
Galois group $D_4.C_2^3.C_2$ (as 16T264)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1261, -5822, 13170, -19006, 19172, -13854, 7006, -2318, 546, -338, 358, -210, 44, 14, -6, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 6*x^14 + 14*x^13 + 44*x^12 - 210*x^11 + 358*x^10 - 338*x^9 + 546*x^8 - 2318*x^7 + 7006*x^6 - 13854*x^5 + 19172*x^4 - 19006*x^3 + 13170*x^2 - 5822*x + 1261)
 
gp: K = bnfinit(x^16 - 2*x^15 - 6*x^14 + 14*x^13 + 44*x^12 - 210*x^11 + 358*x^10 - 338*x^9 + 546*x^8 - 2318*x^7 + 7006*x^6 - 13854*x^5 + 19172*x^4 - 19006*x^3 + 13170*x^2 - 5822*x + 1261, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 6 x^{14} + 14 x^{13} + 44 x^{12} - 210 x^{11} + 358 x^{10} - 338 x^{9} + 546 x^{8} - 2318 x^{7} + 7006 x^{6} - 13854 x^{5} + 19172 x^{4} - 19006 x^{3} + 13170 x^{2} - 5822 x + 1261 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2274539878974650056704=2^{28}\cdot 3^{14}\cdot 11^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.62$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{8} + \frac{1}{6} a^{7} - \frac{1}{6} a^{6} + \frac{1}{6} a^{5} - \frac{1}{6} a^{4} + \frac{1}{6} a^{3} - \frac{1}{6} a^{2} + \frac{1}{6} a + \frac{1}{12}$, $\frac{1}{12} a^{9} + \frac{1}{4} a + \frac{1}{3}$, $\frac{1}{12} a^{10} + \frac{1}{4} a^{2} + \frac{1}{3} a$, $\frac{1}{12} a^{11} + \frac{1}{4} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{48} a^{12} - \frac{1}{24} a^{11} - \frac{1}{24} a^{10} - \frac{1}{24} a^{9} - \frac{1}{48} a^{8} - \frac{1}{6} a^{7} - \frac{1}{12} a^{6} - \frac{1}{6} a^{5} - \frac{1}{48} a^{4} - \frac{11}{24} a^{3} - \frac{3}{8} a^{2} - \frac{11}{24} a - \frac{5}{16}$, $\frac{1}{48} a^{13} - \frac{1}{24} a^{11} - \frac{1}{24} a^{10} - \frac{1}{48} a^{9} - \frac{1}{24} a^{8} - \frac{1}{12} a^{7} - \frac{1}{6} a^{6} - \frac{1}{48} a^{5} + \frac{1}{6} a^{4} + \frac{7}{24} a^{3} - \frac{11}{24} a^{2} - \frac{5}{16} a - \frac{1}{8}$, $\frac{1}{96} a^{14} - \frac{1}{96} a^{12} + \frac{1}{96} a^{10} - \frac{1}{24} a^{9} - \frac{1}{96} a^{8} - \frac{1}{12} a^{7} - \frac{13}{96} a^{6} - \frac{1}{6} a^{5} + \frac{5}{96} a^{4} + \frac{1}{4} a^{3} - \frac{13}{96} a^{2} + \frac{5}{24} a + \frac{37}{96}$, $\frac{1}{405435168} a^{15} + \frac{384275}{135145056} a^{14} - \frac{27541}{45048352} a^{13} - \frac{2747869}{405435168} a^{12} - \frac{189949}{135145056} a^{11} - \frac{3927865}{135145056} a^{10} - \frac{510991}{17627616} a^{9} - \frac{1097711}{135145056} a^{8} + \frac{5387827}{45048352} a^{7} + \frac{28828843}{405435168} a^{6} - \frac{25625621}{135145056} a^{5} - \frac{1260959}{45048352} a^{4} - \frac{139026397}{405435168} a^{3} + \frac{9711639}{45048352} a^{2} - \frac{11414881}{135145056} a + \frac{1243513}{4179744}$

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

Class group and class number

$C_{2}$, which has order $2$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{15650831}{405435168} a^{15} - \frac{901837}{33786264} a^{14} - \frac{11273591}{45048352} a^{13} + \frac{22361653}{101358792} a^{12} + \frac{84233255}{45048352} a^{11} - \frac{8069622}{1407761} a^{10} + \frac{125988631}{17627616} a^{9} - \frac{167008019}{33786264} a^{8} + \frac{2066714287}{135145056} a^{7} - \frac{7098736801}{101358792} a^{6} + \frac{24955775833}{135145056} a^{5} - \frac{10651560637}{33786264} a^{4} + \frac{151028632093}{405435168} a^{3} - \frac{1291274965}{4223283} a^{2} + \frac{21790174777}{135145056} a - \frac{45355825}{1044936} \) (order $12$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 52295.3343635 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4.C_2^3.C_2$ (as 16T264):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $D_4.C_2^3.C_2$
Character table for $D_4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), 4.0.4752.1, 4.4.4752.1, \(\Q(\zeta_{12})\), 8.0.22581504.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.8.16.4$x^{8} + 6 x^{6} + 6 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 20$$4$$2$$16$$D_4$$[2, 3]^{2}$
2.8.12.13$x^{8} + 12 x^{4} + 16$$4$$2$$12$$D_4$$[2, 2]^{2}$
3Data not computed
$11$11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.8.6.1$x^{8} + 143 x^{4} + 5929$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$