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Properties

Label 16.0.68157975826...473.12
Degree $16$
Signature $[0, 8]$
Discriminant $17^{15}\cdot 47^{8}$
Root discriminant $97.63$
Ramified primes $17, 47$
Class number $128$ (GRH)
Class group $[8, 16]$ (GRH)
Galois group $D_{16}$ (as 16T56)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19185895, -6159188, 16674731, -2814377, 5768725, -229641, 1078359, 53505, 156705, -1561, 17093, -1163, 1075, -131, 53, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 53*x^14 - 131*x^13 + 1075*x^12 - 1163*x^11 + 17093*x^10 - 1561*x^9 + 156705*x^8 + 53505*x^7 + 1078359*x^6 - 229641*x^5 + 5768725*x^4 - 2814377*x^3 + 16674731*x^2 - 6159188*x + 19185895)
 
gp: K = bnfinit(x^16 - 6*x^15 + 53*x^14 - 131*x^13 + 1075*x^12 - 1163*x^11 + 17093*x^10 - 1561*x^9 + 156705*x^8 + 53505*x^7 + 1078359*x^6 - 229641*x^5 + 5768725*x^4 - 2814377*x^3 + 16674731*x^2 - 6159188*x + 19185895, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 53 x^{14} - 131 x^{13} + 1075 x^{12} - 1163 x^{11} + 17093 x^{10} - 1561 x^{9} + 156705 x^{8} + 53505 x^{7} + 1078359 x^{6} - 229641 x^{5} + 5768725 x^{4} - 2814377 x^{3} + 16674731 x^{2} - 6159188 x + 19185895 \)

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:  \(68157975826732896644627006991473=17^{15}\cdot 47^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $97.63$
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:  $17, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{10} a^{14} + \frac{3}{10} a^{13} - \frac{3}{10} a^{12} + \frac{3}{10} a^{11} + \frac{1}{10} a^{10} - \frac{3}{10} a^{9} + \frac{3}{10} a^{8} - \frac{1}{2} a^{7} + \frac{1}{10} a^{6} - \frac{1}{10} a^{5} - \frac{3}{10} a^{4} - \frac{1}{2} a^{3} - \frac{1}{10} a^{2} - \frac{1}{10} a - \frac{1}{2}$, $\frac{1}{6146374507611745796492889666386790152103245730970510} a^{15} - \frac{531054771747790516346044309831951486264919025821}{91736932949429041738699845766967017195570831805530} a^{14} + \frac{738153558476232797502054457137511409300559276259407}{6146374507611745796492889666386790152103245730970510} a^{13} + \frac{821087198065629483011122606724836213822199522063473}{6146374507611745796492889666386790152103245730970510} a^{12} - \frac{1032690541299977205563793706004373231353309777624389}{6146374507611745796492889666386790152103245730970510} a^{11} - \frac{2382201532450583147518992663070183865618019783562563}{6146374507611745796492889666386790152103245730970510} a^{10} + \frac{823297276598028844338353951281992115171891884695813}{6146374507611745796492889666386790152103245730970510} a^{9} - \frac{436795418460678541444267817240287131446358517048061}{1229274901522349159298577933277358030420649146194102} a^{8} + \frac{321175496244263369511729411223218372196479313010911}{6146374507611745796492889666386790152103245730970510} a^{7} - \frac{79902938490755318638904945790136337831959580972019}{323493395137460305078573140336146850110697143735290} a^{6} - \frac{952798277768062320182874053514391377589956638121493}{6146374507611745796492889666386790152103245730970510} a^{5} + \frac{41978949508947527162465561696901007555027916949987}{1229274901522349159298577933277358030420649146194102} a^{4} - \frac{573988432847595528956726664772494700741744072738991}{6146374507611745796492889666386790152103245730970510} a^{3} - \frac{323295148696161650782538522074469860314921139963121}{6146374507611745796492889666386790152103245730970510} a^{2} - \frac{11679480736696715811858589817847689184252692696901}{64698679027492061015714628067229370022139428747058} a - \frac{128351097543313042103608857584693165906564692172672}{614637450761174579649288966638679015210324573097051}$

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

Class group and class number

$C_{8}\times C_{16}$, which has order $128$ (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:  \( 33651486.7879 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{16}$ (as 16T56):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $D_{16}$
Character table for $D_{16}$

Intermediate fields

\(\Q(\sqrt{-799}) \), 4.0.10852817.1, 8.0.2002321826203313.1

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

Sibling fields

Galois closure: data not computed
Degree 16 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
17Data not computed
$47$47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$