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Properties

Label 16.0.23922443742...4089.1
Degree $16$
Signature $[0, 8]$
Discriminant $13^{14}\cdot 157^{4}$
Root discriminant $33.39$
Ramified primes $13, 157$
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![2349, -486, 5463, -1275, 4987, -987, 2161, -2917, 1261, -9, 329, -385, 111, 15, -3, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 3*x^14 + 15*x^13 + 111*x^12 - 385*x^11 + 329*x^10 - 9*x^9 + 1261*x^8 - 2917*x^7 + 2161*x^6 - 987*x^5 + 4987*x^4 - 1275*x^3 + 5463*x^2 - 486*x + 2349)
 
gp: K = bnfinit(x^16 - 4*x^15 - 3*x^14 + 15*x^13 + 111*x^12 - 385*x^11 + 329*x^10 - 9*x^9 + 1261*x^8 - 2917*x^7 + 2161*x^6 - 987*x^5 + 4987*x^4 - 1275*x^3 + 5463*x^2 - 486*x + 2349, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 3 x^{14} + 15 x^{13} + 111 x^{12} - 385 x^{11} + 329 x^{10} - 9 x^{9} + 1261 x^{8} - 2917 x^{7} + 2161 x^{6} - 987 x^{5} + 4987 x^{4} - 1275 x^{3} + 5463 x^{2} - 486 x + 2349 \)

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:  \(2392244374201127641154089=13^{14}\cdot 157^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.39$
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:  $13, 157$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{11} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{9} a^{8} - \frac{2}{9} a^{6} - \frac{4}{9} a^{5} - \frac{2}{9} a^{4} + \frac{2}{9} a^{3} + \frac{1}{9} a^{2}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{8} + \frac{1}{9} a^{7} + \frac{4}{9} a^{6} - \frac{1}{9} a^{5} + \frac{4}{9} a^{4} - \frac{4}{9} a^{3} + \frac{2}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{13770} a^{14} + \frac{167}{4590} a^{13} - \frac{23}{918} a^{12} + \frac{211}{4590} a^{11} + \frac{137}{4590} a^{10} + \frac{1979}{13770} a^{9} + \frac{131}{810} a^{8} + \frac{1243}{13770} a^{7} - \frac{823}{13770} a^{6} - \frac{5033}{13770} a^{5} - \frac{1993}{13770} a^{4} + \frac{389}{13770} a^{3} - \frac{19}{90} a^{2} - \frac{577}{1530} a - \frac{179}{510}$, $\frac{1}{260931922621902597897270} a^{15} + \frac{5852029144813265509}{260931922621902597897270} a^{14} + \frac{201668860740064160363}{5116312208272599958770} a^{13} + \frac{1498045991619330698027}{28992435846878066433030} a^{12} + \frac{442349031816975185713}{5798487169375613286606} a^{11} - \frac{19666621015435498679293}{260931922621902597897270} a^{10} - \frac{11693096443738035116627}{86977307540634199299090} a^{9} - \frac{37867851769741544486641}{260931922621902597897270} a^{8} + \frac{12526454602066972885097}{86977307540634199299090} a^{7} - \frac{426845794960306077449}{86977307540634199299090} a^{6} - \frac{39311678557759876385099}{86977307540634199299090} a^{5} + \frac{23219134154519792979641}{52186384524380519579454} a^{4} + \frac{19700084135978882999527}{52186384524380519579454} a^{3} + \frac{9758623201353901696069}{28992435846878066433030} a^{2} + \frac{7978285539998620748627}{28992435846878066433030} a + \frac{1794502898818426431074}{4832072641146344405505}$

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:  \( 2733055.66648 \) (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

\(\Q(\sqrt{13}) \), 4.0.2197.1, 8.0.9851517169.1 x2, 8.0.118976015041.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
13Data not computed
157Data not computed