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Properties

Label 16.0.23584074680...5057.5
Degree $16$
Signature $[0, 8]$
Discriminant $17^{13}\cdot 47^{8}$
Root discriminant $68.52$
Ramified primes $17, 47$
Class number $80$ (GRH)
Class group $[4, 20]$ (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![5564881, -6444788, 7132469, -6214060, 3521479, -1832404, 739945, -165022, 27319, -3842, -223, -590, 184, 86, -22, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 22*x^14 + 86*x^13 + 184*x^12 - 590*x^11 - 223*x^10 - 3842*x^9 + 27319*x^8 - 165022*x^7 + 739945*x^6 - 1832404*x^5 + 3521479*x^4 - 6214060*x^3 + 7132469*x^2 - 6444788*x + 5564881)
 
gp: K = bnfinit(x^16 - 4*x^15 - 22*x^14 + 86*x^13 + 184*x^12 - 590*x^11 - 223*x^10 - 3842*x^9 + 27319*x^8 - 165022*x^7 + 739945*x^6 - 1832404*x^5 + 3521479*x^4 - 6214060*x^3 + 7132469*x^2 - 6444788*x + 5564881, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 22 x^{14} + 86 x^{13} + 184 x^{12} - 590 x^{11} - 223 x^{10} - 3842 x^{9} + 27319 x^{8} - 165022 x^{7} + 739945 x^{6} - 1832404 x^{5} + 3521479 x^{4} - 6214060 x^{3} + 7132469 x^{2} - 6444788 x + 5564881 \)

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:  \(235840746805304140638847775057=17^{13}\cdot 47^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.52$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{12} + \frac{1}{6} a^{10} - \frac{1}{6} a^{8} - \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{6} a^{13} + \frac{1}{6} a^{11} - \frac{1}{6} a^{9} - \frac{1}{6} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{6} a^{2} - \frac{1}{3} a$, $\frac{1}{149142} a^{14} - \frac{3623}{74571} a^{13} + \frac{11035}{149142} a^{12} - \frac{14012}{74571} a^{11} - \frac{4741}{21306} a^{10} - \frac{17998}{74571} a^{9} + \frac{10795}{74571} a^{8} - \frac{1733}{21306} a^{7} + \frac{18855}{49714} a^{6} - \frac{2551}{149142} a^{5} - \frac{58981}{149142} a^{4} + \frac{1082}{24857} a^{3} + \frac{22159}{49714} a^{2} + \frac{70667}{149142} a + \frac{1267}{7102}$, $\frac{1}{17135676606428671199631382718073555094758} a^{15} + \frac{6148847187598768229050041266034544}{8567838303214335599815691359036777547379} a^{14} + \frac{99871580756603103150799958768535239449}{17135676606428671199631382718073555094758} a^{13} + \frac{231557493264147380093055312681176430864}{2855946101071445199938563786345592515793} a^{12} - \frac{526892340961342173329516054270221029817}{17135676606428671199631382718073555094758} a^{11} + \frac{4283271973540947001420988780672661324575}{17135676606428671199631382718073555094758} a^{10} - \frac{1969478899168147802933445490632971004707}{8567838303214335599815691359036777547379} a^{9} + \frac{481442491335926344972045805197352778129}{2855946101071445199938563786345592515793} a^{8} + \frac{647548036008769161666533909815030524493}{2855946101071445199938563786345592515793} a^{7} - \frac{3410404862456530425290682085736361972379}{8567838303214335599815691359036777547379} a^{6} - \frac{786797168954735088842053576838722986073}{17135676606428671199631382718073555094758} a^{5} + \frac{190990151626661896487295891993477740161}{407992300153063599991223398049370359399} a^{4} + \frac{2122196040834552754328298227284380638597}{17135676606428671199631382718073555094758} a^{3} - \frac{2114652688038074134294508262492760499023}{5711892202142890399877127572691185031586} a^{2} - \frac{862661271148375639560780180677795142259}{8567838303214335599815691359036777547379} a - \frac{3218450233153368056783916247891477187}{7263957866226651631891217769424991562}$

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

Class group and class number

$C_{4}\times C_{20}$, which has order $80$ (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:  \( 3348223.78149 \) (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

\(\Q(\sqrt{-47}) \), 4.0.37553.1, 8.0.6928449225617.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$
17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$
$47$47.8.4.1$x^{8} + 172302 x^{4} - 103823 x^{2} + 7421994801$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
47.8.4.1$x^{8} + 172302 x^{4} - 103823 x^{2} + 7421994801$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$