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Properties

Label 16.4.27231268499...8081.3
Degree $16$
Signature $[4, 6]$
Discriminant $13^{12}\cdot 43^{8}$
Root discriminant $44.89$
Ramified primes $13, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_4:C_4$ (as 16T26)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8567, -39611, 68978, -72878, 3095, 50594, -27441, -5701, 6044, 1048, -1077, -149, 114, 6, 5, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 5*x^14 + 6*x^13 + 114*x^12 - 149*x^11 - 1077*x^10 + 1048*x^9 + 6044*x^8 - 5701*x^7 - 27441*x^6 + 50594*x^5 + 3095*x^4 - 72878*x^3 + 68978*x^2 - 39611*x + 8567)
 
gp: K = bnfinit(x^16 - 6*x^15 + 5*x^14 + 6*x^13 + 114*x^12 - 149*x^11 - 1077*x^10 + 1048*x^9 + 6044*x^8 - 5701*x^7 - 27441*x^6 + 50594*x^5 + 3095*x^4 - 72878*x^3 + 68978*x^2 - 39611*x + 8567, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 5 x^{14} + 6 x^{13} + 114 x^{12} - 149 x^{11} - 1077 x^{10} + 1048 x^{9} + 6044 x^{8} - 5701 x^{7} - 27441 x^{6} + 50594 x^{5} + 3095 x^{4} - 72878 x^{3} + 68978 x^{2} - 39611 x + 8567 \)

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:  \(272312684996154152285848081=13^{12}\cdot 43^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.89$
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:  $13, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} + \frac{1}{9} a^{7} + \frac{1}{9} a^{6} + \frac{4}{9} a^{5} + \frac{1}{3} a^{4} + \frac{2}{9} a^{3} + \frac{2}{9} a^{2} - \frac{4}{9} a + \frac{4}{9}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{11} + \frac{1}{9} a^{10} - \frac{1}{9} a^{9} + \frac{1}{9} a^{8} - \frac{1}{9} a^{7} + \frac{2}{9} a^{6} + \frac{4}{9} a^{5} + \frac{2}{9} a^{4} - \frac{2}{9} a^{3} + \frac{1}{9} a^{2} - \frac{1}{3} a + \frac{1}{9}$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{11} - \frac{1}{9} a^{9} - \frac{1}{9} a^{8} + \frac{1}{9} a^{7} + \frac{1}{3} a^{6} - \frac{2}{9} a^{5} + \frac{4}{9} a^{4} + \frac{2}{9} a^{3} + \frac{4}{9} a^{2} - \frac{1}{9} a - \frac{4}{9}$, $\frac{1}{11248067411723472915928952015793} a^{15} + \frac{128969915834946745081448342585}{11248067411723472915928952015793} a^{14} + \frac{63336518355908454686012468833}{1249785267969274768436550223977} a^{13} - \frac{86765365286584086110792521540}{3749355803907824305309650671931} a^{12} - \frac{86758958260577655591587849429}{1249785267969274768436550223977} a^{11} + \frac{1200544954387692909151712475946}{11248067411723472915928952015793} a^{10} - \frac{1224836794455735812810971807192}{11248067411723472915928952015793} a^{9} + \frac{839387301384984213441972106946}{11248067411723472915928952015793} a^{8} - \frac{21337157457596465885571002197}{3749355803907824305309650671931} a^{7} - \frac{3105068181054125387436068724307}{11248067411723472915928952015793} a^{6} + \frac{1562722524240581532658789503400}{11248067411723472915928952015793} a^{5} + \frac{15324359494817534645347413370}{11248067411723472915928952015793} a^{4} + \frac{1355203685445995718253231842373}{11248067411723472915928952015793} a^{3} - \frac{173585377477084671719729927996}{3749355803907824305309650671931} a^{2} + \frac{2162868831556626042033716451482}{11248067411723472915928952015793} a - \frac{2757993699369547727700927928198}{11248067411723472915928952015793}$

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:  \( 12995615.5812 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4:C_4$ (as 16T26):

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 $D_4:C_4$
Character table for $D_4:C_4$

Intermediate fields

\(\Q(\sqrt{13}) \), 4.4.4062253.1, 4.2.7267.1, 4.2.94471.1, 8.2.383765103163.1, 8.2.2270799427.1, 8.4.16501899436009.1

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 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$13$13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
43Data not computed