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Properties

Label 16.0.81094542259...5856.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{8}\cdot 659^{4}$
Root discriminant $17.55$
Ramified primes $2, 3, 659$
Class number $2$
Class group $[2]$
Galois group $C_2\times C_2^2:S_4:C_2$ (as 16T724)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![169, -234, 173, -90, 45, 96, -212, 174, -61, 0, 46, -84, 84, -48, 20, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 20*x^14 - 48*x^13 + 84*x^12 - 84*x^11 + 46*x^10 - 61*x^8 + 174*x^7 - 212*x^6 + 96*x^5 + 45*x^4 - 90*x^3 + 173*x^2 - 234*x + 169)
 
gp: K = bnfinit(x^16 - 6*x^15 + 20*x^14 - 48*x^13 + 84*x^12 - 84*x^11 + 46*x^10 - 61*x^8 + 174*x^7 - 212*x^6 + 96*x^5 + 45*x^4 - 90*x^3 + 173*x^2 - 234*x + 169, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 20 x^{14} - 48 x^{13} + 84 x^{12} - 84 x^{11} + 46 x^{10} - 61 x^{8} + 174 x^{7} - 212 x^{6} + 96 x^{5} + 45 x^{4} - 90 x^{3} + 173 x^{2} - 234 x + 169 \)

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:  \(81094542259068665856=2^{16}\cdot 3^{8}\cdot 659^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.55$
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:  $2, 3, 659$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{7531} a^{14} + \frac{118}{443} a^{13} + \frac{654}{7531} a^{12} + \frac{220}{443} a^{11} + \frac{3745}{7531} a^{10} - \frac{3590}{7531} a^{9} - \frac{2361}{7531} a^{8} - \frac{2400}{7531} a^{7} + \frac{3722}{7531} a^{6} - \frac{3}{443} a^{5} - \frac{3447}{7531} a^{4} + \frac{953}{7531} a^{3} + \frac{3250}{7531} a^{2} + \frac{2517}{7531} a - \frac{1234}{7531}$, $\frac{1}{940364207879473} a^{15} - \frac{2513645660}{940364207879473} a^{14} - \frac{50953599405871}{940364207879473} a^{13} - \frac{71577939655712}{940364207879473} a^{12} + \frac{281066668433986}{940364207879473} a^{11} + \frac{433604011614807}{940364207879473} a^{10} - \frac{154549072397115}{940364207879473} a^{9} - \frac{238234945376}{4255041664613} a^{8} + \frac{4248117178127}{940364207879473} a^{7} + \frac{367385176864893}{940364207879473} a^{6} - \frac{336620772200665}{940364207879473} a^{5} + \frac{250862595568434}{940364207879473} a^{4} - \frac{389699901164984}{940364207879473} a^{3} + \frac{216634620481293}{940364207879473} a^{2} - \frac{14252187249273}{55315541639969} a + \frac{35162230920877}{72335708298421}$

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:  \( \frac{74856358879}{55315541639969} a^{15} - \frac{18270677950}{55315541639969} a^{14} - \frac{669595321712}{55315541639969} a^{13} + \frac{2990165899161}{55315541639969} a^{12} - \frac{7802203826321}{55315541639969} a^{11} + \frac{13894167234852}{55315541639969} a^{10} - \frac{5011407785623}{55315541639969} a^{9} - \frac{598988483101}{4255041664613} a^{8} + \frac{20366581713503}{55315541639969} a^{7} - \frac{5798425268907}{55315541639969} a^{6} + \frac{15968984254627}{55315541639969} a^{5} - \frac{13203115593497}{55315541639969} a^{4} - \frac{36613726496888}{55315541639969} a^{3} + \frac{56421462217721}{55315541639969} a^{2} - \frac{14614994916536}{55315541639969} a - \frac{238759973912}{4255041664613} \) (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:  \( 6048.2036157 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^2:S_4:C_2$ (as 16T724):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 384
The 28 conjugacy class representatives for $C_2\times C_2^2:S_4:C_2$
Character table for $C_2\times C_2^2:S_4:C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{12})\), 8.0.9005250816.1

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

Sibling fields

Degree 12 siblings: data not computed
Degree 16 siblings: data not computed
Degree 24 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
659Data not computed