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Properties

Label 16.0.33657152197...4416.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{46}\cdot 3^{14}$
Root discriminant $19.18$
Ramified primes $2, 3$
Class number $2$
Class group $[2]$
Galois group $C_8:C_2^2$ (as 16T38)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![49, 168, 340, 504, 556, 432, 316, 288, 214, 72, -20, -24, 4, 0, 4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 4*x^14 + 4*x^12 - 24*x^11 - 20*x^10 + 72*x^9 + 214*x^8 + 288*x^7 + 316*x^6 + 432*x^5 + 556*x^4 + 504*x^3 + 340*x^2 + 168*x + 49)
 
gp: K = bnfinit(x^16 + 4*x^14 + 4*x^12 - 24*x^11 - 20*x^10 + 72*x^9 + 214*x^8 + 288*x^7 + 316*x^6 + 432*x^5 + 556*x^4 + 504*x^3 + 340*x^2 + 168*x + 49, 1)
 

Normalized defining polynomial

\( x^{16} + 4 x^{14} + 4 x^{12} - 24 x^{11} - 20 x^{10} + 72 x^{9} + 214 x^{8} + 288 x^{7} + 316 x^{6} + 432 x^{5} + 556 x^{4} + 504 x^{3} + 340 x^{2} + 168 x + 49 \)

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:  \(336571521970697404416=2^{46}\cdot 3^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.18$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{4} a^{4} + \frac{1}{4}$, $\frac{1}{308} a^{13} + \frac{37}{308} a^{12} + \frac{19}{154} a^{11} + \frac{19}{154} a^{10} - \frac{47}{308} a^{9} - \frac{1}{44} a^{8} + \frac{17}{77} a^{7} - \frac{4}{77} a^{6} + \frac{3}{308} a^{5} + \frac{3}{308} a^{4} + \frac{73}{154} a^{3} - \frac{41}{154} a^{2} + \frac{131}{308} a - \frac{7}{44}$, $\frac{1}{308} a^{14} - \frac{1}{14} a^{12} + \frac{9}{154} a^{11} - \frac{67}{308} a^{10} + \frac{19}{154} a^{9} - \frac{29}{154} a^{8} - \frac{17}{77} a^{7} - \frac{3}{44} a^{6} - \frac{27}{77} a^{5} - \frac{3}{22} a^{4} - \frac{47}{154} a^{3} - \frac{69}{308} a^{2} - \frac{61}{154} a + \frac{3}{22}$, $\frac{1}{314270959156} a^{15} + \frac{1984369}{22447925654} a^{14} - \frac{171553441}{314270959156} a^{13} - \frac{1663026313}{22447925654} a^{12} + \frac{16600823579}{314270959156} a^{11} + \frac{5319323045}{78567739789} a^{10} - \frac{13233688649}{314270959156} a^{9} + \frac{12991436273}{78567739789} a^{8} + \frac{9225276547}{314270959156} a^{7} + \frac{7796059784}{78567739789} a^{6} - \frac{4335985751}{314270959156} a^{5} - \frac{1198030000}{2534443219} a^{4} - \frac{140156018251}{314270959156} a^{3} - \frac{2549856195}{22447925654} a^{2} + \frac{123250056101}{314270959156} a + \frac{1960689191}{22447925654}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( 7938.46011493 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8:C_2^2$ (as 16T38):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_8:C_2^2$
Character table for $C_8:C_2^2$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-2}) \), 4.2.1728.1, 4.2.6912.1, \(\Q(\sqrt{-2}, \sqrt{3})\), 8.2.2293235712.2, 8.2.2293235712.1, 8.0.191102976.4

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 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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
$3$3.8.7.2$x^{8} - 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
3.8.7.2$x^{8} - 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$