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Properties

Label 16.0.20924832437...5449.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 13^{14}$
Root discriminant $21.51$
Ramified primes $3, 13$
Class number $4$
Class group $[4]$
Galois group $C_4\wr C_2$ (as 16T42)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 14, -12, -8, -3, 29, -36, 298, -36, 29, -3, -8, -12, 14, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 14*x^14 - 12*x^13 - 8*x^12 - 3*x^11 + 29*x^10 - 36*x^9 + 298*x^8 - 36*x^7 + 29*x^6 - 3*x^5 - 8*x^4 - 12*x^3 + 14*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^16 - 6*x^15 + 14*x^14 - 12*x^13 - 8*x^12 - 3*x^11 + 29*x^10 - 36*x^9 + 298*x^8 - 36*x^7 + 29*x^6 - 3*x^5 - 8*x^4 - 12*x^3 + 14*x^2 - 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 14 x^{14} - 12 x^{13} - 8 x^{12} - 3 x^{11} + 29 x^{10} - 36 x^{9} + 298 x^{8} - 36 x^{7} + 29 x^{6} - 3 x^{5} - 8 x^{4} - 12 x^{3} + 14 x^{2} - 6 x + 1 \)

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:  \(2092483243792415845449=3^{12}\cdot 13^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.51$
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:  $3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{10} + \frac{1}{3} a^{7} + \frac{1}{6} a^{6} + \frac{1}{6} a^{5} + \frac{1}{3} a^{4} - \frac{1}{6} a + \frac{1}{6}$, $\frac{1}{30} a^{12} - \frac{1}{15} a^{11} + \frac{1}{10} a^{10} + \frac{2}{15} a^{8} - \frac{13}{30} a^{7} + \frac{2}{5} a^{6} + \frac{7}{30} a^{5} - \frac{1}{5} a^{4} + \frac{13}{30} a^{2} + \frac{4}{15} a + \frac{11}{30}$, $\frac{1}{240} a^{13} - \frac{1}{60} a^{12} + \frac{1}{20} a^{11} + \frac{29}{240} a^{10} - \frac{13}{120} a^{9} + \frac{19}{240} a^{8} + \frac{9}{20} a^{7} + \frac{9}{20} a^{6} - \frac{5}{16} a^{5} - \frac{59}{120} a^{4} + \frac{43}{240} a^{3} - \frac{17}{60} a^{2} - \frac{5}{12} a + \frac{103}{240}$, $\frac{1}{24240} a^{14} - \frac{23}{12120} a^{13} - \frac{67}{6060} a^{12} - \frac{659}{24240} a^{11} - \frac{617}{6060} a^{10} + \frac{637}{8080} a^{9} - \frac{187}{4040} a^{8} - \frac{437}{2020} a^{7} - \frac{9707}{24240} a^{6} + \frac{151}{1515} a^{5} - \frac{953}{24240} a^{4} - \frac{699}{4040} a^{3} - \frac{67}{6060} a^{2} - \frac{9641}{24240} a - \frac{5807}{12120}$, $\frac{1}{533280} a^{15} - \frac{7}{533280} a^{14} - \frac{749}{533280} a^{13} + \frac{11}{9696} a^{12} + \frac{10211}{533280} a^{11} + \frac{73}{1111} a^{10} - \frac{377}{177760} a^{9} - \frac{12743}{533280} a^{8} + \frac{94837}{533280} a^{7} - \frac{127697}{533280} a^{6} + \frac{60713}{133320} a^{5} + \frac{51459}{177760} a^{4} + \frac{839}{9696} a^{3} + \frac{54647}{533280} a^{2} - \frac{14603}{177760} a - \frac{34099}{533280}$

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

Class group and class number

$C_{4}$, which has order $4$

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:  \( -\frac{103363}{106656} a^{15} + \frac{590837}{106656} a^{14} - \frac{426453}{35552} a^{13} + \frac{79651}{9696} a^{12} + \frac{359957}{35552} a^{11} + \frac{102219}{17776} a^{10} - \frac{2836067}{106656} a^{9} + \frac{2943811}{106656} a^{8} - \frac{9982349}{35552} a^{7} - \frac{4803461}{106656} a^{6} - \frac{729183}{17776} a^{5} - \frac{387933}{35552} a^{4} - \frac{1391}{9696} a^{3} + \frac{1254227}{106656} a^{2} - \frac{371527}{35552} a + \frac{361343}{106656} \) (order $6$)
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:  \( 22778.0314327 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\wr C_2$ (as 16T42):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_4\wr C_2$
Character table for $C_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{-39}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-3}) \), 4.0.19773.1 x2, 4.2.6591.1 x2, \(\Q(\sqrt{-3}, \sqrt{13})\), 8.0.45743668893.2, 8.0.45743668893.1, 8.0.390971529.2

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$3$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$13$13.8.7.1$x^{8} - 13$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
13.8.7.1$x^{8} - 13$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$