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Properties

Label 16.0.14362567555...2129.1
Degree $16$
Signature $[0, 8]$
Discriminant $13^{6}\cdot 29^{14}$
Root discriminant $49.81$
Ramified primes $13, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4\wr C_2$ (as 16T28)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![30416, 84280, 19400, -76748, 51395, 83143, 13076, -8982, 2583, 3586, 95, -319, 53, 31, 5, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 5*x^14 + 31*x^13 + 53*x^12 - 319*x^11 + 95*x^10 + 3586*x^9 + 2583*x^8 - 8982*x^7 + 13076*x^6 + 83143*x^5 + 51395*x^4 - 76748*x^3 + 19400*x^2 + 84280*x + 30416)
 
gp: K = bnfinit(x^16 - 6*x^15 + 5*x^14 + 31*x^13 + 53*x^12 - 319*x^11 + 95*x^10 + 3586*x^9 + 2583*x^8 - 8982*x^7 + 13076*x^6 + 83143*x^5 + 51395*x^4 - 76748*x^3 + 19400*x^2 + 84280*x + 30416, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 5 x^{14} + 31 x^{13} + 53 x^{12} - 319 x^{11} + 95 x^{10} + 3586 x^{9} + 2583 x^{8} - 8982 x^{7} + 13076 x^{6} + 83143 x^{5} + 51395 x^{4} - 76748 x^{3} + 19400 x^{2} + 84280 x + 30416 \)

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:  \(1436256755503642932525262129=13^{6}\cdot 29^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.81$
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:  $13, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{644} a^{14} + \frac{25}{161} a^{13} - \frac{1}{28} a^{12} - \frac{31}{644} a^{11} + \frac{33}{644} a^{10} + \frac{3}{92} a^{9} + \frac{1}{644} a^{8} - \frac{107}{322} a^{7} - \frac{139}{644} a^{6} + \frac{109}{322} a^{5} + \frac{19}{161} a^{4} + \frac{281}{644} a^{3} + \frac{9}{28} a^{2} - \frac{153}{322} a - \frac{62}{161}$, $\frac{1}{2885717420007986227454928101502993112} a^{15} + \frac{362285443818533618571938806303369}{721429355001996556863732025375748278} a^{14} + \frac{306274224548826220259420047855394065}{2885717420007986227454928101502993112} a^{13} - \frac{473194706479661465953007892973226423}{2885717420007986227454928101502993112} a^{12} + \frac{578091424282637221751774663246657363}{2885717420007986227454928101502993112} a^{11} - \frac{145978184381162915331728987082969421}{2885717420007986227454928101502993112} a^{10} + \frac{680620036880195391487645319385886929}{2885717420007986227454928101502993112} a^{9} - \frac{62080930487241222799686112978605248}{360714677500998278431866012687874139} a^{8} + \frac{27943839165873523011046971902406133}{125465974782955922932822960934912744} a^{7} + \frac{97413237836159228954845016124437704}{360714677500998278431866012687874139} a^{6} + \frac{42574467838533800414976506729837732}{360714677500998278431866012687874139} a^{5} - \frac{795810056453365969678137933738205809}{2885717420007986227454928101502993112} a^{4} + \frac{103034672076556913751083970527788983}{412245345715426603922132585928999016} a^{3} - \frac{43057504216855883484845798232287177}{206122672857713301961066292964499508} a^{2} - \frac{23799676242055259134980325713816601}{360714677500998278431866012687874139} a - \frac{61789881894757543398089588318174130}{360714677500998278431866012687874139}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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:  \( -1 \) (order $2$)
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 22252130.9747 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\wr C_2$ (as 16T28):

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

\(\Q(\sqrt{29}) \), 4.4.317057.1, 4.0.24389.1, 4.0.10933.1, 8.0.100525141249.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{16}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$13$13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
29Data not computed