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Properties

Label 16.0.14505315669...641.15
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 53^{8}$
Root discriminant $49.84$
Ramified primes $13, 53$
Class number $16$ (GRH)
Class group $[4, 4]$ (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![44001, 90297, 150733, 19678, 21560, -31552, 13896, 2786, 7236, -1536, -309, -398, 210, 31, 2, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 2*x^14 + 31*x^13 + 210*x^12 - 398*x^11 - 309*x^10 - 1536*x^9 + 7236*x^8 + 2786*x^7 + 13896*x^6 - 31552*x^5 + 21560*x^4 + 19678*x^3 + 150733*x^2 + 90297*x + 44001)
 
gp: K = bnfinit(x^16 - 7*x^15 + 2*x^14 + 31*x^13 + 210*x^12 - 398*x^11 - 309*x^10 - 1536*x^9 + 7236*x^8 + 2786*x^7 + 13896*x^6 - 31552*x^5 + 21560*x^4 + 19678*x^3 + 150733*x^2 + 90297*x + 44001, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 2 x^{14} + 31 x^{13} + 210 x^{12} - 398 x^{11} - 309 x^{10} - 1536 x^{9} + 7236 x^{8} + 2786 x^{7} + 13896 x^{6} - 31552 x^{5} + 21560 x^{4} + 19678 x^{3} + 150733 x^{2} + 90297 x + 44001 \)

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:  \(1450531566903202684958906641=13^{12}\cdot 53^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.84$
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:  $13, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{11} + \frac{1}{9} a^{10} - \frac{1}{9} a^{9} + \frac{1}{9} a^{8} + \frac{1}{9} a^{6} + \frac{4}{9} a^{5} - \frac{2}{9} a^{4} + \frac{1}{9} a^{3} - \frac{2}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{8} + \frac{1}{9} a^{7} - \frac{4}{9} a^{6} + \frac{2}{9} a^{5} - \frac{1}{9} a^{4} - \frac{1}{9} a^{3} + \frac{4}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{1053} a^{14} + \frac{2}{81} a^{13} + \frac{2}{351} a^{12} + \frac{46}{351} a^{11} - \frac{1}{117} a^{10} + \frac{7}{81} a^{9} - \frac{50}{351} a^{8} + \frac{40}{1053} a^{7} + \frac{4}{9} a^{6} - \frac{5}{39} a^{5} + \frac{1}{13} a^{4} - \frac{22}{81} a^{3} - \frac{190}{1053} a^{2} + \frac{2}{117} a + \frac{16}{117}$, $\frac{1}{81700281299765658482327822959863} a^{15} - \frac{7018613565337985219481679537}{27233427099921886160775940986621} a^{14} + \frac{525397478709026281908363313220}{81700281299765658482327822959863} a^{13} + \frac{145969643494157531523058537751}{9077809033307295386925313662207} a^{12} + \frac{3809270478334847218467395811475}{27233427099921886160775940986621} a^{11} - \frac{3469498314898170957753956693930}{81700281299765658482327822959863} a^{10} - \frac{726431484172929945624598298129}{81700281299765658482327822959863} a^{9} + \frac{1844146624385004010754101990063}{81700281299765658482327822959863} a^{8} + \frac{9784415637304143510591167380588}{81700281299765658482327822959863} a^{7} + \frac{4316469099790262159027997791114}{9077809033307295386925313662207} a^{6} - \frac{3198197739007002962685173098457}{9077809033307295386925313662207} a^{5} - \frac{8858274586833692835656668888972}{81700281299765658482327822959863} a^{4} + \frac{7113834521870479347982913890708}{81700281299765658482327822959863} a^{3} - \frac{28182932191333083263343884350387}{81700281299765658482327822959863} a^{2} - \frac{223740794400652388635182540814}{3025936344435765128975104554069} a - \frac{3478529629958390656603289095073}{9077809033307295386925313662207}$

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

Class group and class number

$C_{4}\times C_{4}$, which has order $16$ (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:  \( 17791001.9014 \) (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, 8.0.718600843493.4 x2, 8.0.13558506481.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$13$13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
$53$53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$