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Properties

Label 16.0.95712680702...4569.1
Degree $16$
Signature $[0, 8]$
Discriminant $13^{14}\cdot 79^{6}$
Root discriminant $48.56$
Ramified primes $13, 79$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^3.C_4$ (as 16T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![145557, 186138, 227610, 85257, 28884, -6908, 3038, -3549, 5870, -1404, 1509, -603, 177, -82, 19, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 19*x^14 - 82*x^13 + 177*x^12 - 603*x^11 + 1509*x^10 - 1404*x^9 + 5870*x^8 - 3549*x^7 + 3038*x^6 - 6908*x^5 + 28884*x^4 + 85257*x^3 + 227610*x^2 + 186138*x + 145557)
 
gp: K = bnfinit(x^16 - 3*x^15 + 19*x^14 - 82*x^13 + 177*x^12 - 603*x^11 + 1509*x^10 - 1404*x^9 + 5870*x^8 - 3549*x^7 + 3038*x^6 - 6908*x^5 + 28884*x^4 + 85257*x^3 + 227610*x^2 + 186138*x + 145557, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 19 x^{14} - 82 x^{13} + 177 x^{12} - 603 x^{11} + 1509 x^{10} - 1404 x^{9} + 5870 x^{8} - 3549 x^{7} + 3038 x^{6} - 6908 x^{5} + 28884 x^{4} + 85257 x^{3} + 227610 x^{2} + 186138 x + 145557 \)

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:  \(957126807028111655268824569=13^{14}\cdot 79^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.56$
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:  $13, 79$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{2}{9} a^{4} + \frac{1}{9} a^{3} - \frac{2}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{1}{3} a^{7} + \frac{2}{9} a^{5} + \frac{1}{3} a^{4} - \frac{4}{9} a^{3} + \frac{4}{9} a^{2}$, $\frac{1}{27} a^{14} + \frac{1}{27} a^{12} + \frac{2}{27} a^{11} + \frac{1}{9} a^{10} - \frac{1}{9} a^{8} - \frac{1}{3} a^{7} + \frac{11}{27} a^{6} - \frac{2}{9} a^{5} - \frac{13}{27} a^{4} - \frac{8}{27} a^{3} - \frac{4}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{2303585470741836910930129229639949} a^{15} + \frac{945973081324329364440353514842}{85317980397845811515930712208887} a^{14} + \frac{28849542632076709692871615002349}{2303585470741836910930129229639949} a^{13} + \frac{50907368139721004148463824122387}{2303585470741836910930129229639949} a^{12} - \frac{81714870698637997620929958891059}{767861823580612303643376409879983} a^{11} + \frac{1284733436820212493249590799301}{9479775599760645723992301356543} a^{10} - \frac{16070186213299832048689530976795}{767861823580612303643376409879983} a^{9} + \frac{33032024969146751218963546511807}{255953941193537434547792136626661} a^{8} + \frac{14340106472563123494701914180829}{2303585470741836910930129229639949} a^{7} - \frac{203326304660063945054593548624785}{767861823580612303643376409879983} a^{6} + \frac{697477876695292663903691349060557}{2303585470741836910930129229639949} a^{5} - \frac{690765759624858074583471358827236}{2303585470741836910930129229639949} a^{4} - \frac{26719570322148709398615298191658}{767861823580612303643376409879983} a^{3} + \frac{126193385772973836277933941544993}{255953941193537434547792136626661} a^{2} + \frac{20722224897306098792543855913379}{85317980397845811515930712208887} a - \frac{2440333127575239424735480886144}{28439326799281937171976904069629}$

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:  \( 47106468.997 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{13}) \), 4.0.2197.1, 4.2.173563.1, 4.2.13351.1, 8.0.391613494597.1 x2, 8.0.30124114969.2

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 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
13Data not computed
79Data not computed