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Properties

Label 16.0.79377267773...0761.5
Degree $16$
Signature $[0, 8]$
Discriminant $13^{14}\cdot 17^{10}$
Root discriminant $55.43$
Ramified primes $13, 17$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $(C_2\times C_4).D_4$ (as 16T121)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5517801, -479196, 2699541, -881433, 999786, -96505, 192731, -34044, 12753, -1192, 2537, -839, 297, -28, 13, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 13*x^14 - 28*x^13 + 297*x^12 - 839*x^11 + 2537*x^10 - 1192*x^9 + 12753*x^8 - 34044*x^7 + 192731*x^6 - 96505*x^5 + 999786*x^4 - 881433*x^3 + 2699541*x^2 - 479196*x + 5517801)
 
gp: K = bnfinit(x^16 - 4*x^15 + 13*x^14 - 28*x^13 + 297*x^12 - 839*x^11 + 2537*x^10 - 1192*x^9 + 12753*x^8 - 34044*x^7 + 192731*x^6 - 96505*x^5 + 999786*x^4 - 881433*x^3 + 2699541*x^2 - 479196*x + 5517801, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 13 x^{14} - 28 x^{13} + 297 x^{12} - 839 x^{11} + 2537 x^{10} - 1192 x^{9} + 12753 x^{8} - 34044 x^{7} + 192731 x^{6} - 96505 x^{5} + 999786 x^{4} - 881433 x^{3} + 2699541 x^{2} - 479196 x + 5517801 \)

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:  \(7937726777341695855516080761=13^{14}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.43$
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:  $13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{11} + \frac{1}{3} a^{10} - \frac{1}{9} a^{9} + \frac{4}{9} a^{8} - \frac{1}{3} a^{7} - \frac{4}{9} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{4}{9} a^{2}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{11} + \frac{2}{9} a^{10} - \frac{4}{9} a^{9} - \frac{4}{9} a^{8} - \frac{4}{9} a^{7} - \frac{2}{9} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{4}{9} a^{3} + \frac{4}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{2349} a^{14} + \frac{71}{2349} a^{13} + \frac{118}{2349} a^{12} - \frac{313}{2349} a^{11} - \frac{157}{783} a^{10} - \frac{407}{2349} a^{9} - \frac{322}{2349} a^{8} + \frac{497}{2349} a^{7} + \frac{146}{783} a^{6} + \frac{385}{783} a^{5} + \frac{608}{2349} a^{4} + \frac{773}{2349} a^{3} + \frac{50}{261} a^{2} - \frac{53}{261} a$, $\frac{1}{72144187884824724824739048651571716307298787} a^{15} - \frac{1406168239397618815998117288072278999977}{72144187884824724824739048651571716307298787} a^{14} - \frac{3444805565323590901717529859465439695397925}{72144187884824724824739048651571716307298787} a^{13} + \frac{2626631864659537703164739715419447007210515}{72144187884824724824739048651571716307298787} a^{12} - \frac{724879265322025364289164184657162201537751}{24048062628274908274913016217190572102432929} a^{11} - \frac{16706109688360765774217922018288377534259755}{72144187884824724824739048651571716307298787} a^{10} - \frac{15394544707259373724160145296764498323003742}{72144187884824724824739048651571716307298787} a^{9} - \frac{27521976043699229847526751542405327813867579}{72144187884824724824739048651571716307298787} a^{8} + \frac{10993682570435711360412000268322897624568794}{24048062628274908274913016217190572102432929} a^{7} + \frac{864580646339085746779207620656592566886595}{24048062628274908274913016217190572102432929} a^{6} + \frac{25091797818991322608841788705715952391341458}{72144187884824724824739048651571716307298787} a^{5} + \frac{133584790906523763005516527694424562555798}{2487730616718093959473760298330059183010303} a^{4} + \frac{266811144376063832307251589415381819175590}{890668986232404010181963563599650818608627} a^{3} - \frac{3067158035865789979627492312235940717463499}{8016020876091636091637672072396857367477643} a^{2} + \frac{329154746951060397602642740790747599822366}{890668986232404010181963563599650818608627} a + \frac{623686402383864839261830876332892533883}{3412524851465149464298710971646171718807}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 79554197.7611 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_4).D_4$ (as 16T121):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $(C_2\times C_4).D_4$
Character table for $(C_2\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 4.4.37349.1, 4.0.2197.1, 4.0.2873.1, 8.4.89093921102069.2, 8.4.89093921102069.1, 8.0.1394947801.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
13Data not computed
$17$17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.8.6.2$x^{8} + 85 x^{4} + 2601$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$