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Properties

Label 16.0.14030845564...1601.1
Degree $16$
Signature $[0, 8]$
Discriminant $13^{4}\cdot 53^{12}$
Root discriminant $37.30$
Ramified primes $13, 53$
Class number $1$ (GRH)
Class group Trivial (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![169, 0, -313, 0, 430, 0, -704, 0, 465, 0, -52, 0, 82, 0, 19, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 19*x^14 + 82*x^12 - 52*x^10 + 465*x^8 - 704*x^6 + 430*x^4 - 313*x^2 + 169)
 
gp: K = bnfinit(x^16 + 19*x^14 + 82*x^12 - 52*x^10 + 465*x^8 - 704*x^6 + 430*x^4 - 313*x^2 + 169, 1)
 

Normalized defining polynomial

\( x^{16} + 19 x^{14} + 82 x^{12} - 52 x^{10} + 465 x^{8} - 704 x^{6} + 430 x^{4} - 313 x^{2} + 169 \)

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:  \(14030845564476355702701601=13^{4}\cdot 53^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.30$
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:  $13, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{6} a^{8} - \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{6} a^{2} - \frac{1}{3}$, $\frac{1}{6} a^{9} - \frac{1}{6} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{2} a + \frac{1}{6}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} + \frac{1}{6} a$, $\frac{1}{7284} a^{12} - \frac{1}{3642} a^{10} + \frac{391}{7284} a^{8} - \frac{299}{7284} a^{6} + \frac{627}{2428} a^{4} - \frac{1}{2} a^{3} + \frac{644}{1821} a^{2} - \frac{611}{2428}$, $\frac{1}{94692} a^{13} + \frac{809}{15782} a^{11} - \frac{3251}{94692} a^{9} + \frac{9413}{94692} a^{7} - \frac{29683}{94692} a^{5} - \frac{1}{2} a^{4} - \frac{2201}{15782} a^{3} - \frac{1}{2} a^{2} + \frac{35801}{94692} a - \frac{1}{2}$, $\frac{1}{189384} a^{14} - \frac{1}{189384} a^{13} + \frac{5}{189384} a^{12} - \frac{809}{31564} a^{11} - \frac{9335}{189384} a^{10} - \frac{4177}{63128} a^{9} - \frac{1061}{23673} a^{8} + \frac{2123}{63128} a^{7} + \frac{3919}{23673} a^{6} - \frac{17663}{189384} a^{5} + \frac{35167}{189384} a^{4} + \frac{22385}{94692} a^{3} - \frac{1075}{63128} a^{2} - \frac{4237}{189384} a + \frac{227}{14568}$, $\frac{1}{189384} a^{15} - \frac{1}{14568} a^{12} - \frac{157}{14568} a^{11} + \frac{1}{7284} a^{10} - \frac{8015}{189384} a^{9} - \frac{391}{14568} a^{8} - \frac{31495}{189384} a^{7} - \frac{3343}{14568} a^{6} + \frac{18277}{47346} a^{5} - \frac{1841}{4856} a^{4} + \frac{20935}{63128} a^{3} + \frac{533}{7284} a^{2} - \frac{3039}{31564} a - \frac{603}{4856}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 998291.945421 \) (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

\(\Q(\sqrt{53}) \), 4.0.148877.1, 8.0.288136694677.1 x2, 8.0.3745777030801.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
$53$53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$