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Properties

Label 16.4.64236691634...6801.1
Degree $16$
Signature $[4, 6]$
Discriminant $3^{8}\cdot 7^{8}\cdot 19^{8}$
Root discriminant $19.97$
Ramified primes $3, 7, 19$
Class number $1$
Class group Trivial
Galois group $C_2\times D_8$ (as 16T29)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, -21, -8, 88, -84, 2, -46, 186, -44, -218, 136, 90, -106, 14, 18, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 18*x^14 + 14*x^13 - 106*x^12 + 90*x^11 + 136*x^10 - 218*x^9 - 44*x^8 + 186*x^7 - 46*x^6 + 2*x^5 - 84*x^4 + 88*x^3 - 8*x^2 - 21*x + 9)
 
gp: K = bnfinit(x^16 - 8*x^15 + 18*x^14 + 14*x^13 - 106*x^12 + 90*x^11 + 136*x^10 - 218*x^9 - 44*x^8 + 186*x^7 - 46*x^6 + 2*x^5 - 84*x^4 + 88*x^3 - 8*x^2 - 21*x + 9, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 18 x^{14} + 14 x^{13} - 106 x^{12} + 90 x^{11} + 136 x^{10} - 218 x^{9} - 44 x^{8} + 186 x^{7} - 46 x^{6} + 2 x^{5} - 84 x^{4} + 88 x^{3} - 8 x^{2} - 21 x + 9 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(642366916348420476801=3^{8}\cdot 7^{8}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.97$
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:  $3, 7, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{7} a^{10} + \frac{2}{7} a^{9} - \frac{3}{7} a^{8} + \frac{3}{7} a^{6} - \frac{2}{7} a^{5} - \frac{3}{7} a^{4} + \frac{3}{7} a^{3} + \frac{2}{7} a^{2} - \frac{3}{7} a - \frac{3}{7}$, $\frac{1}{7} a^{11} - \frac{1}{7} a^{8} + \frac{3}{7} a^{7} - \frac{1}{7} a^{6} + \frac{1}{7} a^{5} + \frac{2}{7} a^{4} + \frac{3}{7} a^{3} + \frac{3}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{12} - \frac{1}{7} a^{9} + \frac{3}{7} a^{8} - \frac{1}{7} a^{7} + \frac{1}{7} a^{6} + \frac{2}{7} a^{5} + \frac{3}{7} a^{4} + \frac{3}{7} a^{2} - \frac{1}{7} a$, $\frac{1}{49} a^{13} - \frac{3}{49} a^{12} + \frac{2}{49} a^{11} + \frac{1}{49} a^{10} + \frac{17}{49} a^{9} + \frac{17}{49} a^{8} + \frac{17}{49} a^{7} + \frac{24}{49} a^{6} - \frac{12}{49} a^{5} - \frac{11}{49} a^{4} - \frac{20}{49} a^{3} - \frac{13}{49} a^{2} + \frac{17}{49} a + \frac{6}{49}$, $\frac{1}{1617} a^{14} - \frac{1}{231} a^{13} + \frac{2}{33} a^{12} - \frac{5}{231} a^{11} + \frac{23}{539} a^{10} - \frac{269}{539} a^{9} - \frac{674}{1617} a^{8} + \frac{278}{1617} a^{7} + \frac{90}{539} a^{6} - \frac{256}{539} a^{5} - \frac{193}{1617} a^{4} + \frac{445}{1617} a^{3} + \frac{433}{1617} a^{2} - \frac{496}{1617} a + \frac{223}{539}$, $\frac{1}{95403} a^{15} + \frac{2}{8673} a^{14} + \frac{108}{31801} a^{13} + \frac{6371}{95403} a^{12} - \frac{302}{8673} a^{11} - \frac{1153}{31801} a^{10} + \frac{29647}{95403} a^{9} - \frac{42005}{95403} a^{8} - \frac{44204}{95403} a^{7} - \frac{328}{2891} a^{6} + \frac{39146}{95403} a^{5} - \frac{23731}{95403} a^{4} + \frac{6745}{31801} a^{3} + \frac{7639}{95403} a^{2} + \frac{42781}{95403} a - \frac{1068}{31801}$

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

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
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:  \( 25672.5485932 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_8$ (as 16T29):

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 $C_2\times D_8$
Character table for $C_2\times D_8$

Intermediate fields

\(\Q(\sqrt{21}) \), \(\Q(\sqrt{133}) \), \(\Q(\sqrt{57}) \), 4.2.53067.1, 4.2.53067.2, \(\Q(\sqrt{21}, \sqrt{57})\), 8.2.8448319467.1, 8.2.8448319467.2, 8.4.25344958401.1

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 16 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$19$19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$