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Properties

Label 16.0.16622447092...4761.1
Degree $16$
Signature $[0, 8]$
Discriminant $41^{15}\cdot 83^{12}$
Root discriminant $893.91$
Ramified primes $41, 83$
Class number $256$ (GRH)
Class group $[2, 2, 4, 16]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T260)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![448464248273, 374090699762, 313772956188, 247366851315, 142489068976, 36052562864, 17677601785, -1118934092, 79402250, 48276837, -3124258, -89769, 48712, -3257, -60, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 60*x^14 - 3257*x^13 + 48712*x^12 - 89769*x^11 - 3124258*x^10 + 48276837*x^9 + 79402250*x^8 - 1118934092*x^7 + 17677601785*x^6 + 36052562864*x^5 + 142489068976*x^4 + 247366851315*x^3 + 313772956188*x^2 + 374090699762*x + 448464248273)
 
gp: K = bnfinit(x^16 - 6*x^15 - 60*x^14 - 3257*x^13 + 48712*x^12 - 89769*x^11 - 3124258*x^10 + 48276837*x^9 + 79402250*x^8 - 1118934092*x^7 + 17677601785*x^6 + 36052562864*x^5 + 142489068976*x^4 + 247366851315*x^3 + 313772956188*x^2 + 374090699762*x + 448464248273, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 60 x^{14} - 3257 x^{13} + 48712 x^{12} - 89769 x^{11} - 3124258 x^{10} + 48276837 x^{9} + 79402250 x^{8} - 1118934092 x^{7} + 17677601785 x^{6} + 36052562864 x^{5} + 142489068976 x^{4} + 247366851315 x^{3} + 313772956188 x^{2} + 374090699762 x + 448464248273 \)

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:  \(166224470923824362155644060831037034515503474761=41^{15}\cdot 83^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $893.91$
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:  $41, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{657266631032519224392593613009030149787336919008287316525027886983020736286087895085107121382519} a^{15} + \frac{108128118626278408865308054507207848622431336642965724524532320642815660830048023655682437991417}{657266631032519224392593613009030149787336919008287316525027886983020736286087895085107121382519} a^{14} + \frac{301102564753078151533715476569363619857784251486516342755326989130871108382139949204152171015615}{657266631032519224392593613009030149787336919008287316525027886983020736286087895085107121382519} a^{13} - \frac{3733748918767989938462772748816321927536593877696481054269350755450521600094577234608088520023}{657266631032519224392593613009030149787336919008287316525027886983020736286087895085107121382519} a^{12} + \frac{65096161294002252266243545909806280027225063803881685820544066810172304784337307187407705080067}{657266631032519224392593613009030149787336919008287316525027886983020736286087895085107121382519} a^{11} - \frac{103233620142549440379205894682448920702643673947514541022282664677690510233560103098889726234151}{657266631032519224392593613009030149787336919008287316525027886983020736286087895085107121382519} a^{10} - \frac{79660844981937084831392192528975946762611444872970280110905241288248597363770366821910385774747}{657266631032519224392593613009030149787336919008287316525027886983020736286087895085107121382519} a^{9} - \frac{256990261617344798821245739825785089718217664721551013335835175271413274685493993586790120156934}{657266631032519224392593613009030149787336919008287316525027886983020736286087895085107121382519} a^{8} + \frac{190840670463076233490393055645748253681005997984900920408323551401552859167632038439257862139738}{657266631032519224392593613009030149787336919008287316525027886983020736286087895085107121382519} a^{7} - \frac{137254944941334275853316959128423870307989905932022003959031580327995043739057341550192438883270}{657266631032519224392593613009030149787336919008287316525027886983020736286087895085107121382519} a^{6} + \frac{39648131720877337255495926142482122673145025090855384037102482432043699264694661575726996607980}{657266631032519224392593613009030149787336919008287316525027886983020736286087895085107121382519} a^{5} - \frac{205911571216463063677896828857554859070718214730875373338677894123481677616130953006401293762507}{657266631032519224392593613009030149787336919008287316525027886983020736286087895085107121382519} a^{4} + \frac{176548433872570556542977542980538215719725531249991008097263823132192275889196505183626766577746}{657266631032519224392593613009030149787336919008287316525027886983020736286087895085107121382519} a^{3} - \frac{135987694746321394409959376080354928548682486313470723163630631534677829550761986995100015370533}{657266631032519224392593613009030149787336919008287316525027886983020736286087895085107121382519} a^{2} + \frac{972360258025792761430187662277809687214042686492102323307313557526457408511560102486953856014}{657266631032519224392593613009030149787336919008287316525027886983020736286087895085107121382519} a + \frac{166669170710270500803634733904894792752153256909422980995981153060909149795148876640675967648662}{657266631032519224392593613009030149787336919008287316525027886983020736286087895085107121382519}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{16}$, which has order $256$ (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:  \( 570145056055000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T260):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-3403}) \), 4.0.474796769.1, 8.0.9242710845966413801.1

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

Sibling fields

Degree 16 sibling: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
41Data not computed
$83$83.8.6.2$x^{8} + 249 x^{4} + 27556$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
83.8.6.2$x^{8} + 249 x^{4} + 27556$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$