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Properties

Label 16.0.47526285764...7681.4
Degree $16$
Signature $[0, 8]$
Discriminant $13^{8}\cdot 17^{12}$
Root discriminant $30.19$
Ramified primes $13, 17$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T172)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![608, -1740, 4616, -11605, 19712, -22373, 19093, -13610, 8297, -4323, 1976, -796, 275, -85, 22, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 22*x^14 - 85*x^13 + 275*x^12 - 796*x^11 + 1976*x^10 - 4323*x^9 + 8297*x^8 - 13610*x^7 + 19093*x^6 - 22373*x^5 + 19712*x^4 - 11605*x^3 + 4616*x^2 - 1740*x + 608)
 
gp: K = bnfinit(x^16 - 4*x^15 + 22*x^14 - 85*x^13 + 275*x^12 - 796*x^11 + 1976*x^10 - 4323*x^9 + 8297*x^8 - 13610*x^7 + 19093*x^6 - 22373*x^5 + 19712*x^4 - 11605*x^3 + 4616*x^2 - 1740*x + 608, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 22 x^{14} - 85 x^{13} + 275 x^{12} - 796 x^{11} + 1976 x^{10} - 4323 x^{9} + 8297 x^{8} - 13610 x^{7} + 19093 x^{6} - 22373 x^{5} + 19712 x^{4} - 11605 x^{3} + 4616 x^{2} - 1740 x + 608 \)

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:  \(475262857646065983187681=13^{8}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.19$
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:  $13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{6} a^{8} + \frac{1}{6} a^{7} - \frac{1}{3} a^{6} + \frac{1}{6} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} + \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} + \frac{1}{6} a^{2} + \frac{1}{6} a$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3} + \frac{1}{6} a^{2}$, $\frac{1}{18} a^{12} - \frac{1}{18} a^{10} - \frac{1}{18} a^{8} - \frac{7}{18} a^{7} - \frac{7}{18} a^{6} - \frac{2}{9} a^{5} - \frac{1}{2} a^{4} - \frac{1}{3} a^{3} + \frac{7}{18} a^{2} - \frac{5}{18} a - \frac{1}{9}$, $\frac{1}{18} a^{13} - \frac{1}{18} a^{11} - \frac{1}{18} a^{9} - \frac{1}{18} a^{8} - \frac{1}{18} a^{7} + \frac{1}{9} a^{6} - \frac{1}{6} a^{5} - \frac{5}{18} a^{3} + \frac{1}{18} a^{2} + \frac{2}{9} a - \frac{1}{3}$, $\frac{1}{72} a^{14} - \frac{1}{72} a^{13} + \frac{1}{72} a^{12} + \frac{1}{18} a^{11} - \frac{1}{24} a^{10} + \frac{1}{24} a^{9} - \frac{5}{72} a^{8} - \frac{4}{9} a^{7} + \frac{23}{72} a^{6} - \frac{17}{72} a^{5} - \frac{1}{9} a^{4} - \frac{5}{24} a^{3} - \frac{1}{72} a^{2} + \frac{17}{36} a + \frac{1}{9}$, $\frac{1}{287073078852024} a^{15} + \frac{632038131871}{287073078852024} a^{14} + \frac{2849336232877}{287073078852024} a^{13} + \frac{43372592037}{7974252190334} a^{12} - \frac{1540299838319}{31897008761336} a^{11} + \frac{83840980265}{95691026284008} a^{10} - \frac{4359629721529}{287073078852024} a^{9} - \frac{69782984861}{71768269713006} a^{8} - \frac{137131276355461}{287073078852024} a^{7} + \frac{3865922196759}{31897008761336} a^{6} + \frac{234756440551}{7974252190334} a^{5} + \frac{110063358895217}{287073078852024} a^{4} + \frac{94909954859875}{287073078852024} a^{3} + \frac{40280851691231}{143536539426012} a^{2} + \frac{1428361816599}{3987126095167} a + \frac{300465592679}{35884134856503}$

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

Class group and class number

$C_{4}$, 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:  $7$
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:  \( 829552.961859 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T172):

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

\(\Q(\sqrt{17}) \), 4.0.3757.1, 4.4.830297.1, 4.0.63869.1, 8.0.40552535777.1, 8.0.239955833.1, 8.0.689393108209.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$17$17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$