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Properties

Label 16.0.10188112612...1921.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 61^{8}$
Root discriminant $17.80$
Ramified primes $3, 61$
Class number $1$
Class group Trivial
Galois group $C_2^2.SD_{16}$ (as 16T163)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, -210, 711, -1254, 1471, -1167, 675, -174, -159, 234, -123, -3, 52, -45, 21, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 21*x^14 - 45*x^13 + 52*x^12 - 3*x^11 - 123*x^10 + 234*x^9 - 159*x^8 - 174*x^7 + 675*x^6 - 1167*x^5 + 1471*x^4 - 1254*x^3 + 711*x^2 - 210*x + 25)
 
gp: K = bnfinit(x^16 - 6*x^15 + 21*x^14 - 45*x^13 + 52*x^12 - 3*x^11 - 123*x^10 + 234*x^9 - 159*x^8 - 174*x^7 + 675*x^6 - 1167*x^5 + 1471*x^4 - 1254*x^3 + 711*x^2 - 210*x + 25, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 21 x^{14} - 45 x^{13} + 52 x^{12} - 3 x^{11} - 123 x^{10} + 234 x^{9} - 159 x^{8} - 174 x^{7} + 675 x^{6} - 1167 x^{5} + 1471 x^{4} - 1254 x^{3} + 711 x^{2} - 210 x + 25 \)

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:  \(101881126126588011921=3^{12}\cdot 61^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.80$
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:  $3, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{2}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{8} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{5} a^{11} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{2} - \frac{2}{5} a$, $\frac{1}{25} a^{13} - \frac{1}{25} a^{12} - \frac{1}{25} a^{11} + \frac{2}{25} a^{10} - \frac{1}{25} a^{9} - \frac{12}{25} a^{8} - \frac{1}{25} a^{7} + \frac{8}{25} a^{6} - \frac{2}{25} a^{5} - \frac{1}{5} a^{4} - \frac{6}{25} a^{3} + \frac{3}{25} a^{2} + \frac{8}{25} a + \frac{2}{5}$, $\frac{1}{125} a^{14} - \frac{1}{125} a^{13} - \frac{6}{125} a^{12} + \frac{7}{125} a^{11} + \frac{9}{125} a^{10} + \frac{3}{125} a^{9} + \frac{39}{125} a^{8} + \frac{53}{125} a^{7} + \frac{53}{125} a^{6} - \frac{7}{25} a^{5} - \frac{11}{125} a^{4} - \frac{22}{125} a^{3} + \frac{3}{125} a^{2} - \frac{3}{25} a$, $\frac{1}{690535696375} a^{15} + \frac{144248939}{690535696375} a^{14} - \frac{190752511}{690535696375} a^{13} - \frac{33904397318}{690535696375} a^{12} - \frac{66335209421}{690535696375} a^{11} + \frac{27994263758}{690535696375} a^{10} + \frac{39345102099}{690535696375} a^{9} + \frac{52881301968}{690535696375} a^{8} + \frac{14678173334}{98647956625} a^{7} + \frac{47311903208}{138107139275} a^{6} - \frac{176604891781}{690535696375} a^{5} + \frac{264958287663}{690535696375} a^{4} + \frac{21835483709}{98647956625} a^{3} + \frac{61693032482}{138107139275} a^{2} - \frac{67703079449}{138107139275} a - \frac{4434219021}{27621427855}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{11044739}{181005425} a^{15} - \frac{64874476}{181005425} a^{14} + \frac{222676171}{181005425} a^{13} - \frac{461895273}{181005425} a^{12} + \frac{493679124}{181005425} a^{11} + \frac{14964404}{36201085} a^{10} - \frac{1387533673}{181005425} a^{9} + \frac{2381321113}{181005425} a^{8} - \frac{1304482692}{181005425} a^{7} - \frac{2315223247}{181005425} a^{6} + \frac{1441806551}{36201085} a^{5} - \frac{11690075911}{181005425} a^{4} + \frac{14085911108}{181005425} a^{3} - \frac{11015166882}{181005425} a^{2} + \frac{5236595919}{181005425} a - \frac{164272439}{36201085} \) (order $6$)
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:  \( 10313.9158685 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.SD_{16}$ (as 16T163):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 19 conjugacy class representatives for $C_2^2.SD_{16}$
Character table for $C_2^2.SD_{16}$

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.549.1, 8.0.18385461.1, 8.0.165469149.1, 8.0.10093618089.2

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

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$3$3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$61$61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.8.6.1$x^{8} - 61 x^{4} + 59536$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$