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Properties

Label 16.0.14215874243...8544.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{14}\cdot 11^{6}$
Root discriminant $18.18$
Ramified primes $2, 3, 11$
Class number $2$
Class group $[2]$
Galois group $C_2\times D_4:D_4$ (as 16T265)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![169, -260, 71, 84, -31, -16, -14, -48, 106, -72, 94, -104, 29, 12, -1, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - x^14 + 12*x^13 + 29*x^12 - 104*x^11 + 94*x^10 - 72*x^9 + 106*x^8 - 48*x^7 - 14*x^6 - 16*x^5 - 31*x^4 + 84*x^3 + 71*x^2 - 260*x + 169)
 
gp: K = bnfinit(x^16 - 4*x^15 - x^14 + 12*x^13 + 29*x^12 - 104*x^11 + 94*x^10 - 72*x^9 + 106*x^8 - 48*x^7 - 14*x^6 - 16*x^5 - 31*x^4 + 84*x^3 + 71*x^2 - 260*x + 169, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - x^{14} + 12 x^{13} + 29 x^{12} - 104 x^{11} + 94 x^{10} - 72 x^{9} + 106 x^{8} - 48 x^{7} - 14 x^{6} - 16 x^{5} - 31 x^{4} + 84 x^{3} + 71 x^{2} - 260 x + 169 \)

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:  \(142158742435915628544=2^{24}\cdot 3^{14}\cdot 11^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.18$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{11} + \frac{1}{8} a^{9} + \frac{1}{8} a^{7} - \frac{1}{8} a^{5} + \frac{1}{4} a^{4} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2} + \frac{1}{8} a + \frac{1}{8}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} + \frac{1}{8} a^{9} - \frac{1}{8} a^{8} + \frac{1}{8} a^{7} + \frac{1}{8} a^{6} + \frac{1}{8} a^{5} - \frac{3}{8} a^{4} - \frac{1}{8} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{1}{8}$, $\frac{1}{1144} a^{14} - \frac{7}{572} a^{13} + \frac{6}{143} a^{12} - \frac{1}{22} a^{11} - \frac{101}{1144} a^{10} - \frac{41}{572} a^{9} + \frac{95}{1144} a^{8} + \frac{37}{286} a^{7} - \frac{9}{1144} a^{6} - \frac{135}{572} a^{5} - \frac{213}{1144} a^{4} - \frac{25}{143} a^{3} - \frac{49}{572} a^{2} - \frac{23}{143} a + \frac{19}{88}$, $\frac{1}{119046797584} a^{15} + \frac{7035565}{119046797584} a^{14} - \frac{1882295959}{59523398792} a^{13} + \frac{946879547}{59523398792} a^{12} + \frac{1705276277}{119046797584} a^{11} + \frac{13419049967}{119046797584} a^{10} + \frac{6698909175}{119046797584} a^{9} - \frac{18679450063}{119046797584} a^{8} + \frac{4694025365}{119046797584} a^{7} - \frac{6460802429}{119046797584} a^{6} - \frac{26588221065}{119046797584} a^{5} - \frac{15375602259}{119046797584} a^{4} - \frac{848680451}{29761699396} a^{3} + \frac{7998100475}{59523398792} a^{2} - \frac{17644325163}{119046797584} a - \frac{3110793337}{9157445968}$

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:  \( -\frac{449580717}{29761699396} a^{15} + \frac{2676141367}{59523398792} a^{14} + \frac{927347309}{14880849698} a^{13} - \frac{3485037077}{29761699396} a^{12} - \frac{4301569575}{7440424849} a^{11} + \frac{58141006675}{59523398792} a^{10} - \frac{9189638451}{29761699396} a^{9} + \frac{49551563419}{59523398792} a^{8} - \frac{8595113035}{7440424849} a^{7} - \frac{24610209925}{59523398792} a^{6} - \frac{14765407}{2289361492} a^{5} + \frac{18389788487}{59523398792} a^{4} + \frac{38789424581}{29761699396} a^{3} - \frac{4730753172}{7440424849} a^{2} - \frac{43884689993}{29761699396} a + \frac{9957608941}{4578722984} \) (order $12$)
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:  \( 19384.644979 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_4:D_4$ (as 16T265):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_2\times D_4:D_4$
Character table for $C_2\times D_4:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), 4.4.4752.1, 4.0.4752.1, \(\Q(\zeta_{12})\), 8.0.11923034112.10, 8.0.11923034112.2, 8.0.22581504.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

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 R R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
3Data not computed
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$