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Properties

Label 16.0.39948709956...7376.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 47^{8}$
Root discriminant $19.39$
Ramified primes $2, 47$
Class number $5$
Class group $[5]$
Galois group $D_{8}$ (as 16T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, -162, -189, 630, -119, -848, 1142, -932, 834, -532, 334, -160, 81, -26, 11, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 11*x^14 - 26*x^13 + 81*x^12 - 160*x^11 + 334*x^10 - 532*x^9 + 834*x^8 - 932*x^7 + 1142*x^6 - 848*x^5 - 119*x^4 + 630*x^3 - 189*x^2 - 162*x + 81)
 
gp: K = bnfinit(x^16 - 2*x^15 + 11*x^14 - 26*x^13 + 81*x^12 - 160*x^11 + 334*x^10 - 532*x^9 + 834*x^8 - 932*x^7 + 1142*x^6 - 848*x^5 - 119*x^4 + 630*x^3 - 189*x^2 - 162*x + 81, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 11 x^{14} - 26 x^{13} + 81 x^{12} - 160 x^{11} + 334 x^{10} - 532 x^{9} + 834 x^{8} - 932 x^{7} + 1142 x^{6} - 848 x^{5} - 119 x^{4} + 630 x^{3} - 189 x^{2} - 162 x + 81 \)

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:  \(399487099562283237376=2^{24}\cdot 47^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.39$
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:  $2, 47$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{6} a^{8} - \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{6} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{3} + \frac{1}{3} a$, $\frac{1}{12} a^{10} - \frac{1}{12} a^{8} - \frac{1}{6} a^{7} - \frac{1}{12} a^{6} - \frac{1}{6} a^{5} + \frac{5}{12} a^{4} - \frac{1}{6} a^{3} - \frac{5}{12} a^{2} + \frac{1}{3} a + \frac{1}{4}$, $\frac{1}{12} a^{11} - \frac{1}{12} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{12} a^{3} - \frac{1}{2} a^{2} + \frac{1}{12} a - \frac{1}{2}$, $\frac{1}{12} a^{12} - \frac{1}{3} a^{4} + \frac{1}{4}$, $\frac{1}{12} a^{13} + \frac{1}{6} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{252} a^{14} - \frac{2}{63} a^{13} - \frac{1}{36} a^{12} - \frac{1}{126} a^{11} + \frac{1}{28} a^{10} + \frac{1}{126} a^{9} + \frac{1}{252} a^{8} + \frac{11}{63} a^{7} - \frac{1}{12} a^{6} + \frac{1}{9} a^{5} - \frac{55}{252} a^{4} + \frac{1}{9} a^{3} + \frac{22}{63} a^{2} + \frac{1}{14} a + \frac{5}{14}$, $\frac{1}{22436008259112} a^{15} - \frac{3886058657}{3205144037016} a^{14} + \frac{295615400377}{11218004129556} a^{13} - \frac{458745297109}{11218004129556} a^{12} - \frac{7870319275}{1068381345672} a^{11} - \frac{580895075239}{22436008259112} a^{10} - \frac{1032246189545}{22436008259112} a^{9} + \frac{617027565227}{22436008259112} a^{8} - \frac{148778539205}{830963268856} a^{7} + \frac{684799879255}{3205144037016} a^{6} + \frac{347868477671}{22436008259112} a^{5} + \frac{4156357816927}{22436008259112} a^{4} + \frac{1333109595794}{2804501032389} a^{3} + \frac{423551874791}{1869667354926} a^{2} - \frac{1113450688723}{2492889806568} a - \frac{242800772661}{830963268856}$

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

Class group and class number

$C_{5}$, which has order $5$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3314.20332109 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_8$ (as 16T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $D_{8}$
Character table for $D_{8}$

Intermediate fields

\(\Q(\sqrt{-94}) \), \(\Q(\sqrt{-47}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{2}, \sqrt{-47})\), 4.0.17672.1 x2, 4.2.3008.1 x2, 8.0.19987173376.1, 8.0.2498396672.1 x4, 8.2.425259008.1 x4

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
$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}$