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Properties

Label 16.0.16821502541...4417.1
Degree $16$
Signature $[0, 8]$
Discriminant $17^{13}\cdot 19^{8}$
Root discriminant $43.56$
Ramified primes $17, 19$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T289)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![353, 2080, 3524, -2019, -3789, 2213, 2425, -1031, -323, 826, 305, 49, -29, -21, 17, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 17*x^14 - 21*x^13 - 29*x^12 + 49*x^11 + 305*x^10 + 826*x^9 - 323*x^8 - 1031*x^7 + 2425*x^6 + 2213*x^5 - 3789*x^4 - 2019*x^3 + 3524*x^2 + 2080*x + 353)
 
gp: K = bnfinit(x^16 - 4*x^15 + 17*x^14 - 21*x^13 - 29*x^12 + 49*x^11 + 305*x^10 + 826*x^9 - 323*x^8 - 1031*x^7 + 2425*x^6 + 2213*x^5 - 3789*x^4 - 2019*x^3 + 3524*x^2 + 2080*x + 353, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 17 x^{14} - 21 x^{13} - 29 x^{12} + 49 x^{11} + 305 x^{10} + 826 x^{9} - 323 x^{8} - 1031 x^{7} + 2425 x^{6} + 2213 x^{5} - 3789 x^{4} - 2019 x^{3} + 3524 x^{2} + 2080 x + 353 \)

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:  \(168215025416361753462674417=17^{13}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.56$
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:  $17, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{566542029546354743628803766221} a^{15} + \frac{138760494746341220106227166335}{566542029546354743628803766221} a^{14} + \frac{153572390229303266459218383018}{566542029546354743628803766221} a^{13} + \frac{270296387748073330765567887746}{566542029546354743628803766221} a^{12} - \frac{41533958530980485256386802112}{566542029546354743628803766221} a^{11} - \frac{125844623021659101477417590367}{566542029546354743628803766221} a^{10} - \frac{214654449029819633407878180307}{566542029546354743628803766221} a^{9} - \frac{192149118216441091782234238914}{566542029546354743628803766221} a^{8} + \frac{1177309417201177828245013903}{24632262154189336679513207227} a^{7} - \frac{237999965739711491700305590579}{566542029546354743628803766221} a^{6} - \frac{115574204972560336074904710728}{566542029546354743628803766221} a^{5} + \frac{10481675875975077824125955360}{566542029546354743628803766221} a^{4} + \frac{25139279053187065498854187884}{566542029546354743628803766221} a^{3} + \frac{216480184801412526321892373728}{566542029546354743628803766221} a^{2} - \frac{278546730790146608740892274837}{566542029546354743628803766221} a + \frac{38395748996403494135250340484}{566542029546354743628803766221}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 479856.088485 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T289):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 44 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-19}) \), 4.0.6137.1, 8.0.185037184097.1

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

Sibling fields

Degree 16 siblings: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$17$17.8.7.6$x^{8} + 37179$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.6.4$x^{8} + 136 x^{4} + 7803$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$
19Data not computed