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Properties

Label 16.0.48003408671...0689.8
Degree $16$
Signature $[0, 8]$
Discriminant $17^{10}\cdot 47^{8}$
Root discriminant $40.28$
Ramified primes $17, 47$
Class number $20$ (GRH)
Class group $[20]$ (GRH)
Galois group $C_4\wr C_2$ (as 16T42)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16336, -3816, 5492, 1602, 3101, 3352, 336, 2056, -131, 222, 75, -74, 81, -56, 21, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 21*x^14 - 56*x^13 + 81*x^12 - 74*x^11 + 75*x^10 + 222*x^9 - 131*x^8 + 2056*x^7 + 336*x^6 + 3352*x^5 + 3101*x^4 + 1602*x^3 + 5492*x^2 - 3816*x + 16336)
 
gp: K = bnfinit(x^16 - 6*x^15 + 21*x^14 - 56*x^13 + 81*x^12 - 74*x^11 + 75*x^10 + 222*x^9 - 131*x^8 + 2056*x^7 + 336*x^6 + 3352*x^5 + 3101*x^4 + 1602*x^3 + 5492*x^2 - 3816*x + 16336, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 21 x^{14} - 56 x^{13} + 81 x^{12} - 74 x^{11} + 75 x^{10} + 222 x^{9} - 131 x^{8} + 2056 x^{7} + 336 x^{6} + 3352 x^{5} + 3101 x^{4} + 1602 x^{3} + 5492 x^{2} - 3816 x + 16336 \)

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:  \(48003408671952806969030689=17^{10}\cdot 47^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.28$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{14} a^{11} - \frac{3}{14} a^{10} + \frac{1}{7} a^{8} + \frac{1}{14} a^{7} + \frac{3}{14} a^{6} - \frac{5}{14} a^{5} + \frac{3}{7} a^{4} + \frac{3}{7} a^{3} + \frac{1}{14} a^{2} + \frac{2}{7} a - \frac{2}{7}$, $\frac{1}{28} a^{12} - \frac{1}{14} a^{10} + \frac{1}{14} a^{9} - \frac{1}{4} a^{8} + \frac{3}{14} a^{7} + \frac{1}{7} a^{6} - \frac{1}{14} a^{5} - \frac{11}{28} a^{4} - \frac{1}{14} a^{3} + \frac{1}{4} a^{2} + \frac{2}{7} a - \frac{3}{7}$, $\frac{1}{84} a^{13} + \frac{1}{42} a^{11} - \frac{5}{42} a^{10} + \frac{1}{12} a^{9} - \frac{1}{6} a^{8} - \frac{5}{21} a^{7} - \frac{3}{14} a^{6} + \frac{25}{84} a^{5} + \frac{2}{21} a^{4} + \frac{17}{84} a^{3} - \frac{5}{14} a^{2} + \frac{8}{21} a - \frac{4}{21}$, $\frac{1}{168} a^{14} - \frac{1}{168} a^{13} + \frac{1}{84} a^{12} - \frac{19}{168} a^{10} - \frac{1}{8} a^{9} - \frac{1}{7} a^{8} - \frac{1}{6} a^{7} + \frac{37}{168} a^{6} - \frac{5}{24} a^{5} - \frac{15}{56} a^{4} - \frac{59}{168} a^{3} + \frac{4}{21} a^{2} - \frac{4}{21}$, $\frac{1}{2122608577430068114363704} a^{15} + \frac{2984879498122037021953}{1061304288715034057181852} a^{14} + \frac{1593610898610170967019}{2122608577430068114363704} a^{13} + \frac{306383038092925348865}{58961349373057447621214} a^{12} - \frac{703966272353434870329}{33692199641747112926408} a^{11} - \frac{111606828216177333530815}{1061304288715034057181852} a^{10} + \frac{181691174460588749678153}{2122608577430068114363704} a^{9} + \frac{39974335101183589426715}{1061304288715034057181852} a^{8} - \frac{54165949730065043083121}{707536192476689371454568} a^{7} + \frac{98896115924879022837145}{530652144357517028590926} a^{6} + \frac{453970870500847603959961}{1061304288715034057181852} a^{5} + \frac{868895370436098600409}{2915671122843500157093} a^{4} - \frac{61082943665123086996593}{235845397492229790484856} a^{3} + \frac{46852542925083059965051}{353768096238344685727284} a^{2} - \frac{5066939587284542851064}{20409697859904501099651} a - \frac{99419100724797459684646}{265326072178758514295463}$

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

Class group and class number

$C_{20}$, which has order $20$ (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:  \( 145848.015244 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\wr C_2$ (as 16T42):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_4\wr C_2$
Character table for $C_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{-47}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-799}) \), 4.0.37553.1 x2, 4.2.13583.1 x2, \(\Q(\sqrt{17}, \sqrt{-47})\), 8.0.6928449225617.2, 8.0.23973872753.2, 8.0.407555836801.2

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$17$17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
$47$47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$