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Properties

Label 16.0.34993857500...7721.1
Degree $16$
Signature $[0, 8]$
Discriminant $41^{9}\cdot 83^{12}$
Root discriminant $222.07$
Ramified primes $41, 83$
Class number $30$ (GRH)
Class group $[30]$ (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![27286024997, 3583260107, 112953200, 1264951561, 1624056654, 661782541, 165000919, 35576212, 7676074, -170920, -430991, -34753, 9857, 822, -146, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 146*x^14 + 822*x^13 + 9857*x^12 - 34753*x^11 - 430991*x^10 - 170920*x^9 + 7676074*x^8 + 35576212*x^7 + 165000919*x^6 + 661782541*x^5 + 1624056654*x^4 + 1264951561*x^3 + 112953200*x^2 + 3583260107*x + 27286024997)
 
gp: K = bnfinit(x^16 - 6*x^15 - 146*x^14 + 822*x^13 + 9857*x^12 - 34753*x^11 - 430991*x^10 - 170920*x^9 + 7676074*x^8 + 35576212*x^7 + 165000919*x^6 + 661782541*x^5 + 1624056654*x^4 + 1264951561*x^3 + 112953200*x^2 + 3583260107*x + 27286024997, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 146 x^{14} + 822 x^{13} + 9857 x^{12} - 34753 x^{11} - 430991 x^{10} - 170920 x^{9} + 7676074 x^{8} + 35576212 x^{7} + 165000919 x^{6} + 661782541 x^{5} + 1624056654 x^{4} + 1264951561 x^{3} + 112953200 x^{2} + 3583260107 x + 27286024997 \)

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:  \(34993857500868339818743792665125437721=41^{9}\cdot 83^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $222.07$
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}{374591988342080742109519157365456500866758618291948386124253948254038497260237} a^{15} + \frac{38205417857407293569122037634848418554155570094470110721093857294573158370612}{374591988342080742109519157365456500866758618291948386124253948254038497260237} a^{14} + \frac{58672490941178898127393049931677624825323728726580608502333131814804797622729}{374591988342080742109519157365456500866758618291948386124253948254038497260237} a^{13} + \frac{180810330171292840757211734646086118353857986643975968133832661499481959882223}{374591988342080742109519157365456500866758618291948386124253948254038497260237} a^{12} + \frac{12763701790922914933313181962281406774361929774273270102161651058385747238915}{34053817122007340191774468851405136442432601662904398738568540750367136114567} a^{11} - \frac{26853742119011424277110378710108079918749840751235599674157679929226893110032}{374591988342080742109519157365456500866758618291948386124253948254038497260237} a^{10} + \frac{151654758346341837656705852901988671025844642544160141260038441956964839486353}{374591988342080742109519157365456500866758618291948386124253948254038497260237} a^{9} - \frac{1958046537927397084835100326361769871252331159214541809460907831602425873632}{34053817122007340191774468851405136442432601662904398738568540750367136114567} a^{8} + \frac{8676595579263635798955154579816325576349381994732971822123269038936003926106}{53513141191725820301359879623636642980965516898849769446321992607719785322891} a^{7} - \frac{4284444978987343926163734268516561585090040692221965389943619015165077982456}{53513141191725820301359879623636642980965516898849769446321992607719785322891} a^{6} - \frac{10409775543258947747391582059647267900967040063881398101087293185872396871222}{374591988342080742109519157365456500866758618291948386124253948254038497260237} a^{5} - \frac{7728650271910140876234282051256056113204799113486903893710210234944315795642}{53513141191725820301359879623636642980965516898849769446321992607719785322891} a^{4} - \frac{82429276660478127992639038594585960658186851641833684777976631188682815890633}{374591988342080742109519157365456500866758618291948386124253948254038497260237} a^{3} + \frac{36547492429555200940577531677383282700419070124332328661573818626606735516527}{374591988342080742109519157365456500866758618291948386124253948254038497260237} a^{2} + \frac{151493996237902805071778242599931588942527317390464603095451325598927744624982}{374591988342080742109519157365456500866758618291948386124253948254038497260237} a + \frac{1160039383983158201177538915136053512158769368086342313651210131139866618307}{4864831017429620027396352693057876634633228808986342676938362964338162302081}$

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

Class group and class number

$C_{30}$, which has order $30$ (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:  \( 58262717024.6 \) (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

\(\Q(\sqrt{-83}) \), 4.0.282449.1, 8.0.3270874941641.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$41$41.4.2.2$x^{4} - 41 x^{2} + 20172$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.8.7.4$x^{8} - 1912896$$8$$1$$7$$C_8$$[\ ]_{8}$
$83$83.8.6.2$x^{8} + 249 x^{4} + 27556$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
83.8.6.2$x^{8} + 249 x^{4} + 27556$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$