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Properties

Label 16.8.98340377973...9249.5
Degree $16$
Signature $[8, 4]$
Discriminant $13^{10}\cdot 61^{10}$
Root discriminant $64.87$
Ramified primes $13, 61$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_2\times OD_{16}).C_2$ (as 16T123)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-19825, 27755, 86423, 452112, -145654, -336095, -15842, 80410, 18636, -14445, 1380, 1068, -551, 48, 8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 8*x^14 + 48*x^13 - 551*x^12 + 1068*x^11 + 1380*x^10 - 14445*x^9 + 18636*x^8 + 80410*x^7 - 15842*x^6 - 336095*x^5 - 145654*x^4 + 452112*x^3 + 86423*x^2 + 27755*x - 19825)
 
gp: K = bnfinit(x^16 - 4*x^15 + 8*x^14 + 48*x^13 - 551*x^12 + 1068*x^11 + 1380*x^10 - 14445*x^9 + 18636*x^8 + 80410*x^7 - 15842*x^6 - 336095*x^5 - 145654*x^4 + 452112*x^3 + 86423*x^2 + 27755*x - 19825, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 8 x^{14} + 48 x^{13} - 551 x^{12} + 1068 x^{11} + 1380 x^{10} - 14445 x^{9} + 18636 x^{8} + 80410 x^{7} - 15842 x^{6} - 336095 x^{5} - 145654 x^{4} + 452112 x^{3} + 86423 x^{2} + 27755 x - 19825 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(98340377973019428085802419249=13^{10}\cdot 61^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.87$
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:  $13, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{39} a^{13} + \frac{4}{39} a^{12} - \frac{1}{3} a^{10} - \frac{4}{13} a^{9} + \frac{10}{39} a^{8} - \frac{10}{39} a^{7} - \frac{19}{39} a^{6} - \frac{1}{13} a^{5} + \frac{11}{39} a^{4} - \frac{17}{39} a^{3} - \frac{8}{39} a^{2} + \frac{1}{3}$, $\frac{1}{3705} a^{14} + \frac{7}{3705} a^{13} + \frac{122}{741} a^{12} + \frac{101}{285} a^{11} + \frac{9}{1235} a^{10} + \frac{8}{19} a^{9} - \frac{42}{247} a^{8} + \frac{115}{741} a^{7} - \frac{1789}{3705} a^{6} + \frac{1796}{3705} a^{5} - \frac{1271}{3705} a^{4} + \frac{483}{1235} a^{3} - \frac{140}{741} a^{2} + \frac{43}{95} a + \frac{8}{57}$, $\frac{1}{178106483249920003858088742281950467678105} a^{15} - \frac{15949066503404551050524404811505686093}{178106483249920003858088742281950467678105} a^{14} - \frac{28250446800052240372282080153844216436}{35621296649984000771617748456390093535621} a^{13} - \frac{20901303782931418669714038968169575904557}{178106483249920003858088742281950467678105} a^{12} + \frac{28781720798516161686012444896107524047269}{59368827749973334619362914093983489226035} a^{11} + \frac{13732465555816829588163258829965372599590}{35621296649984000771617748456390093535621} a^{10} + \frac{670100447984400324068132409693344951938}{1874805086841263198506197287178425975559} a^{9} + \frac{5944942502234807598092060856400938188036}{35621296649984000771617748456390093535621} a^{8} + \frac{4850695764105263017986368718685208997892}{13700498711532307989083749406303882129085} a^{7} + \frac{10177483652956821821866780932378817447812}{59368827749973334619362914093983489226035} a^{6} - \frac{57734319701944743406432113895681851209876}{178106483249920003858088742281950467678105} a^{5} + \frac{78970471409390378643508582796407947380129}{178106483249920003858088742281950467678105} a^{4} + \frac{8916201228053928129217277186167477672798}{35621296649984000771617748456390093535621} a^{3} + \frac{1145214513677501000543560785119242286286}{3124675144735438664176995478630709959265} a^{2} - \frac{69813679661967383787087947975844450236}{913366580768820532605583293753592141939} a - \frac{934760451180295942825617544282644586229}{2740099742306461597816749881260776425817}$

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:  $11$
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:  \( 383985012.359 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times OD_{16}).C_2$ (as 16T123):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $(C_2\times OD_{16}).C_2$
Character table for $(C_2\times OD_{16}).C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{61}) \), \(\Q(\sqrt{793}) \), 4.4.10309.1 x2, 4.4.48373.1 x2, \(\Q(\sqrt{13}, \sqrt{61})\), 8.8.395451064801.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
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}$
61Data not computed