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Properties

Label 16.4.30854674434...0897.9
Degree $16$
Signature $[4, 6]$
Discriminant $17^{15}\cdot 47^{6}$
Root discriminant $60.34$
Ramified primes $17, 47$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $(C_2\times C_8).D_4$ (as 16T306)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2904383, 766748, -2665081, 4289566, -3337665, 1254102, -100557, -393476, 288795, -142950, 51042, -13386, 3268, -488, 87, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 87*x^14 - 488*x^13 + 3268*x^12 - 13386*x^11 + 51042*x^10 - 142950*x^9 + 288795*x^8 - 393476*x^7 - 100557*x^6 + 1254102*x^5 - 3337665*x^4 + 4289566*x^3 - 2665081*x^2 + 766748*x + 2904383)
 
gp: K = bnfinit(x^16 - 6*x^15 + 87*x^14 - 488*x^13 + 3268*x^12 - 13386*x^11 + 51042*x^10 - 142950*x^9 + 288795*x^8 - 393476*x^7 - 100557*x^6 + 1254102*x^5 - 3337665*x^4 + 4289566*x^3 - 2665081*x^2 + 766748*x + 2904383, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 87 x^{14} - 488 x^{13} + 3268 x^{12} - 13386 x^{11} + 51042 x^{10} - 142950 x^{9} + 288795 x^{8} - 393476 x^{7} - 100557 x^{6} + 1254102 x^{5} - 3337665 x^{4} + 4289566 x^{3} - 2665081 x^{2} + 766748 x + 2904383 \)

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:  \(30854674434917562989871890897=17^{15}\cdot 47^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.34$
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:  $17, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{134} a^{14} - \frac{5}{134} a^{13} + \frac{15}{134} a^{12} - \frac{2}{67} a^{11} - \frac{19}{134} a^{10} - \frac{5}{134} a^{9} - \frac{17}{134} a^{8} + \frac{11}{134} a^{7} + \frac{18}{67} a^{6} - \frac{8}{67} a^{5} - \frac{3}{67} a^{4} + \frac{57}{134} a^{3} + \frac{32}{67} a^{2} - \frac{45}{134} a$, $\frac{1}{21392949964027742802226053417385542207735094720862} a^{15} - \frac{36486283269873808100525319257158155130844105348}{10696474982013871401113026708692771103867547360431} a^{14} + \frac{48213982927163387734983030392960375711706289264}{10696474982013871401113026708692771103867547360431} a^{13} + \frac{2418668400620979177093133066272666454322869744295}{10696474982013871401113026708692771103867547360431} a^{12} + \frac{1377534986635436538044838841798471478282127485945}{10696474982013871401113026708692771103867547360431} a^{11} + \frac{2298385768187812238632512539519794981898151544431}{10696474982013871401113026708692771103867547360431} a^{10} + \frac{2558029576980003609189374848213820135106885040649}{10696474982013871401113026708692771103867547360431} a^{9} - \frac{827024382907866901165267256742404775658757933264}{10696474982013871401113026708692771103867547360431} a^{8} + \frac{4180905760917233050899685601344701235740095710475}{21392949964027742802226053417385542207735094720862} a^{7} + \frac{1015031378779169744233404310642970754327062719707}{21392949964027742802226053417385542207735094720862} a^{6} - \frac{27357856555852714517015053113452854091985172457}{21392949964027742802226053417385542207735094720862} a^{5} - \frac{1136094770853714998747264432733760367657147053443}{21392949964027742802226053417385542207735094720862} a^{4} + \frac{3392445872398201145692113608375310363536816634809}{10696474982013871401113026708692771103867547360431} a^{3} - \frac{2686044947377331837605444899716114874515583072611}{10696474982013871401113026708692771103867547360431} a^{2} + \frac{2078184699483412981896865227952343562079696458244}{10696474982013871401113026708692771103867547360431} a - \frac{132109672100105644416023737719347868575444212011}{319297760657130489585463483841575256831867085386}$

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

Class group and class number

$C_{2}\times C_{2}$, 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:  $9$
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:  \( 14299681.4226 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_8).D_4$ (as 16T306):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $(C_2\times C_8).D_4$
Character table for $(C_2\times C_8).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
17Data not computed
$47$47.4.2.2$x^{4} - 47 x^{2} + 28717$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
47.4.0.1$x^{4} - x + 39$$1$$4$$0$$C_4$$[\ ]^{4}$
47.4.2.2$x^{4} - 47 x^{2} + 28717$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
47.4.2.2$x^{4} - 47 x^{2} + 28717$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$