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Properties

Label 16.4.19048273177...4809.1
Degree $16$
Signature $[4, 6]$
Discriminant $17^{12}\cdot 83^{6}$
Root discriminant $43.90$
Ramified primes $17, 83$
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![137, 1008, -1836, 2038, 1347, -8832, 15111, -16704, 12826, -6942, 2589, -718, 145, -46, 19, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 19*x^14 - 46*x^13 + 145*x^12 - 718*x^11 + 2589*x^10 - 6942*x^9 + 12826*x^8 - 16704*x^7 + 15111*x^6 - 8832*x^5 + 1347*x^4 + 2038*x^3 - 1836*x^2 + 1008*x + 137)
 
gp: K = bnfinit(x^16 - 6*x^15 + 19*x^14 - 46*x^13 + 145*x^12 - 718*x^11 + 2589*x^10 - 6942*x^9 + 12826*x^8 - 16704*x^7 + 15111*x^6 - 8832*x^5 + 1347*x^4 + 2038*x^3 - 1836*x^2 + 1008*x + 137, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 19 x^{14} - 46 x^{13} + 145 x^{12} - 718 x^{11} + 2589 x^{10} - 6942 x^{9} + 12826 x^{8} - 16704 x^{7} + 15111 x^{6} - 8832 x^{5} + 1347 x^{4} + 2038 x^{3} - 1836 x^{2} + 1008 x + 137 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(190482731772980153578634809=17^{12}\cdot 83^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.90$
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:  $17, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{28} a^{13} + \frac{1}{14} a^{12} + \frac{1}{14} a^{11} - \frac{3}{14} a^{10} - \frac{3}{14} a^{9} + \frac{1}{7} a^{8} - \frac{3}{28} a^{7} + \frac{3}{14} a^{6} + \frac{5}{28} a^{5} - \frac{5}{14} a^{4} - \frac{1}{28} a^{3} + \frac{1}{7} a^{2} + \frac{1}{28} a - \frac{5}{14}$, $\frac{1}{784} a^{14} - \frac{3}{784} a^{13} - \frac{29}{784} a^{12} - \frac{1}{49} a^{11} + \frac{13}{196} a^{10} + \frac{31}{392} a^{9} - \frac{37}{784} a^{8} - \frac{9}{112} a^{7} - \frac{2}{49} a^{6} - \frac{33}{112} a^{5} + \frac{27}{56} a^{4} - \frac{257}{784} a^{3} + \frac{169}{392} a^{2} - \frac{253}{784} a + \frac{295}{784}$, $\frac{1}{26085776341849507770032} a^{15} - \frac{354392729004907893}{2006598180142269828464} a^{14} + \frac{52895013134795399567}{3726539477407072538576} a^{13} + \frac{701468471129699725499}{13042888170924753885016} a^{12} + \frac{130934372102322146181}{6521444085462376942508} a^{11} - \frac{36721818461282210157}{1003299090071134914232} a^{10} - \frac{5149139340209328677497}{26085776341849507770032} a^{9} + \frac{3222229109713828953959}{26085776341849507770032} a^{8} - \frac{276273093629087160657}{13042888170924753885016} a^{7} + \frac{220865679590655076237}{2006598180142269828464} a^{6} + \frac{92284043179477422610}{232908717337942033661} a^{5} - \frac{10309890234753371769365}{26085776341849507770032} a^{4} - \frac{1746837596970209431837}{6521444085462376942508} a^{3} - \frac{8862403255306033665873}{26085776341849507770032} a^{2} - \frac{1172852550662438865311}{26085776341849507770032} a - \frac{2515516361324750064833}{13042888170924753885016}$

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:  $9$
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:  \( 8455107.16318 \) (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

\(\Q(\sqrt{17}) \), 4.2.23987.1, 4.4.4913.1, 4.2.407779.1, 8.4.166283712841.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
17Data not computed
$83$83.4.2.2$x^{4} - 83 x^{2} + 13778$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
83.4.0.1$x^{4} - x + 22$$1$$4$$0$$C_4$$[\ ]^{4}$
83.8.4.1$x^{8} + 303116 x^{4} - 571787 x^{2} + 22969827364$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$