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Properties

Label 16.0.26436931289...625.13
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 101^{8}$
Root discriminant $33.60$
Ramified primes $5, 101$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T157)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4481, -31174, 99078, -187268, 224283, -165310, 69244, -17579, 8038, -4162, 161, 276, 53, 11, -8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 8*x^14 + 11*x^13 + 53*x^12 + 276*x^11 + 161*x^10 - 4162*x^9 + 8038*x^8 - 17579*x^7 + 69244*x^6 - 165310*x^5 + 224283*x^4 - 187268*x^3 + 99078*x^2 - 31174*x + 4481)
 
gp: K = bnfinit(x^16 - 4*x^15 - 8*x^14 + 11*x^13 + 53*x^12 + 276*x^11 + 161*x^10 - 4162*x^9 + 8038*x^8 - 17579*x^7 + 69244*x^6 - 165310*x^5 + 224283*x^4 - 187268*x^3 + 99078*x^2 - 31174*x + 4481, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 8 x^{14} + 11 x^{13} + 53 x^{12} + 276 x^{11} + 161 x^{10} - 4162 x^{9} + 8038 x^{8} - 17579 x^{7} + 69244 x^{6} - 165310 x^{5} + 224283 x^{4} - 187268 x^{3} + 99078 x^{2} - 31174 x + 4481 \)

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:  \(2643693128974804931640625=5^{12}\cdot 101^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.60$
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:  $5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{451} a^{14} - \frac{219}{451} a^{13} + \frac{59}{451} a^{12} + \frac{115}{451} a^{11} + \frac{172}{451} a^{10} - \frac{204}{451} a^{9} + \frac{59}{451} a^{8} + \frac{95}{451} a^{7} - \frac{171}{451} a^{6} - \frac{213}{451} a^{5} + \frac{135}{451} a^{4} - \frac{26}{451} a^{3} - \frac{193}{451} a^{2} + \frac{158}{451} a + \frac{67}{451}$, $\frac{1}{40866345881373407406872150630021} a^{15} - \frac{8078616773910818777788329780}{40866345881373407406872150630021} a^{14} + \frac{1774202173389496161692487774176}{3715122352852127946079286420911} a^{13} - \frac{575595641991450666840992747518}{3715122352852127946079286420911} a^{12} - \frac{15734815300917091996207124253645}{40866345881373407406872150630021} a^{11} - \frac{16603040430633969488710405242152}{40866345881373407406872150630021} a^{10} + \frac{8221088108589381591297104923479}{40866345881373407406872150630021} a^{9} + \frac{7587578667464810829262559063306}{40866345881373407406872150630021} a^{8} - \frac{16694061148303899243931420196066}{40866345881373407406872150630021} a^{7} + \frac{1363527114749700716214626704651}{40866345881373407406872150630021} a^{6} + \frac{13267092637750419948294138097210}{40866345881373407406872150630021} a^{5} - \frac{4796522683887289984443133900894}{40866345881373407406872150630021} a^{4} - \frac{5967431839416277029071862484665}{40866345881373407406872150630021} a^{3} + \frac{18430798840753342398287080791429}{40866345881373407406872150630021} a^{2} + \frac{5726841594386033411380434404375}{40866345881373407406872150630021} a - \frac{9082015605565897354382158538406}{40866345881373407406872150630021}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  \( \frac{515547728719676720412478929}{150798324285510728438642622251} a^{15} - \frac{143939081866973915191428972}{13708938571410066221694783841} a^{14} - \frac{5441130799674299294827856989}{150798324285510728438642622251} a^{13} + \frac{246753100771927072614009140}{150798324285510728438642622251} a^{12} + \frac{25628444397211843931057187959}{150798324285510728438642622251} a^{11} + \frac{164948872002328199003923312028}{150798324285510728438642622251} a^{10} + \frac{243849735295167742736438545153}{150798324285510728438642622251} a^{9} - \frac{1863511595119265304513288839993}{150798324285510728438642622251} a^{8} + \frac{2522221099431816888204701676539}{150798324285510728438642622251} a^{7} - \frac{7224117204767819086131476669509}{150798324285510728438642622251} a^{6} + \frac{29340141790808627056810843480890}{150798324285510728438642622251} a^{5} - \frac{59807892528715995574712514519186}{150798324285510728438642622251} a^{4} + \frac{67600125961883772389131144946551}{150798324285510728438642622251} a^{3} - \frac{45955723429481862535843099240108}{150798324285510728438642622251} a^{2} + \frac{18718433438262287316236993144394}{150798324285510728438642622251} a - \frac{3615857000030092635887875040591}{150798324285510728438642622251} \) (order $10$)
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:  \( 318080.298133 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T157):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.16098453125.4 x2, 8.0.159390625.2

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

Sibling fields

Degree 8 siblings: data not computed
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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$101$101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$