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Properties

Label 16.0.23584074680...5057.2
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 16T260)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![62867, 12019, 1483, 150510, 78771, -264952, 98504, 6973, -15947, 5471, 479, -199, 261, -107, 45, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 45*x^14 - 107*x^13 + 261*x^12 - 199*x^11 + 479*x^10 + 5471*x^9 - 15947*x^8 + 6973*x^7 + 98504*x^6 - 264952*x^5 + 78771*x^4 + 150510*x^3 + 1483*x^2 + 12019*x + 62867)
 
gp: K = bnfinit(x^16 - 8*x^15 + 45*x^14 - 107*x^13 + 261*x^12 - 199*x^11 + 479*x^10 + 5471*x^9 - 15947*x^8 + 6973*x^7 + 98504*x^6 - 264952*x^5 + 78771*x^4 + 150510*x^3 + 1483*x^2 + 12019*x + 62867, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 45 x^{14} - 107 x^{13} + 261 x^{12} - 199 x^{11} + 479 x^{10} + 5471 x^{9} - 15947 x^{8} + 6973 x^{7} + 98504 x^{6} - 264952 x^{5} + 78771 x^{4} + 150510 x^{3} + 1483 x^{2} + 12019 x + 62867 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{742} a^{14} - \frac{25}{106} a^{13} - \frac{2}{371} a^{12} + \frac{1}{742} a^{11} + \frac{325}{742} a^{10} + \frac{7}{53} a^{9} - \frac{30}{371} a^{8} - \frac{1}{2} a^{7} + \frac{66}{371} a^{6} - \frac{33}{106} a^{5} - \frac{3}{106} a^{4} - \frac{92}{371} a^{3} - \frac{155}{371} a^{2} + \frac{323}{742} a + \frac{19}{106}$, $\frac{1}{54288164559040025611206776688250122409934} a^{15} - \frac{8534958336904499919716699026156067491}{27144082279520012805603388344125061204967} a^{14} + \frac{10019511553539333967660713369648345956269}{54288164559040025611206776688250122409934} a^{13} + \frac{10566945845516675240368374941160980374321}{54288164559040025611206776688250122409934} a^{12} + \frac{3361346294714685394639367258773032196154}{27144082279520012805603388344125061204967} a^{11} - \frac{10088883126689886404178603528234838218547}{54288164559040025611206776688250122409934} a^{10} + \frac{4847850185696198191902522478030145717435}{27144082279520012805603388344125061204967} a^{9} + \frac{13548569806437061798230016485279204771945}{54288164559040025611206776688250122409934} a^{8} + \frac{6884765725892592289214019421554543014587}{54288164559040025611206776688250122409934} a^{7} - \frac{13602236605751894639329651069117082306495}{54288164559040025611206776688250122409934} a^{6} - \frac{29036647922098102031955704538335102866}{3877726039931430400800484049160723029281} a^{5} + \frac{18448624245468431651021092932360024482361}{54288164559040025611206776688250122409934} a^{4} - \frac{661094458881513662754225925367745925811}{3877726039931430400800484049160723029281} a^{3} + \frac{8294813804646786975095615108565448011401}{54288164559040025611206776688250122409934} a^{2} + \frac{10427419268789980137254976993190174934299}{27144082279520012805603388344125061204967} a - \frac{1298590168313355072023805438384439640527}{7755452079862860801600968098321446058562}$

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:  \( 1688424.44596 \) (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

\(\Q(\sqrt{-47}) \), 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 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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}$