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Properties

Label 16.0.82163008108...3649.1
Degree $16$
Signature $[0, 8]$
Discriminant $17^{14}\cdot 47^{4}$
Root discriminant $31.24$
Ramified primes $17, 47$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T158)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![68, 0, 527, -1054, 102, 1700, -2465, 1190, 851, -1708, 1303, -566, 155, -56, 28, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 28*x^14 - 56*x^13 + 155*x^12 - 566*x^11 + 1303*x^10 - 1708*x^9 + 851*x^8 + 1190*x^7 - 2465*x^6 + 1700*x^5 + 102*x^4 - 1054*x^3 + 527*x^2 + 68)
 
gp: K = bnfinit(x^16 - 8*x^15 + 28*x^14 - 56*x^13 + 155*x^12 - 566*x^11 + 1303*x^10 - 1708*x^9 + 851*x^8 + 1190*x^7 - 2465*x^6 + 1700*x^5 + 102*x^4 - 1054*x^3 + 527*x^2 + 68, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 28 x^{14} - 56 x^{13} + 155 x^{12} - 566 x^{11} + 1303 x^{10} - 1708 x^{9} + 851 x^{8} + 1190 x^{7} - 2465 x^{6} + 1700 x^{5} + 102 x^{4} - 1054 x^{3} + 527 x^{2} + 68 \)

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:  \(821630081083204084623649=17^{14}\cdot 47^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.24$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{7} - \frac{1}{2} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{4} - \frac{3}{8} a^{3} + \frac{1}{8} a^{2} - \frac{1}{2}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{8} - \frac{1}{8} a^{5} - \frac{1}{2} a^{3} + \frac{1}{8} a^{2} - \frac{1}{2}$, $\frac{1}{3520} a^{12} - \frac{3}{1760} a^{11} + \frac{3}{704} a^{10} - \frac{1}{176} a^{9} + \frac{23}{320} a^{8} - \frac{39}{1760} a^{7} - \frac{331}{3520} a^{6} + \frac{101}{440} a^{5} - \frac{733}{3520} a^{4} - \frac{789}{1760} a^{3} - \frac{971}{3520} a^{2} + \frac{1}{4} a - \frac{49}{880}$, $\frac{1}{3520} a^{13} - \frac{21}{3520} a^{11} + \frac{7}{352} a^{10} + \frac{133}{3520} a^{9} - \frac{1}{11} a^{8} + \frac{81}{3520} a^{7} - \frac{149}{1760} a^{6} - \frac{57}{704} a^{5} + \frac{23}{440} a^{4} + \frac{91}{320} a^{3} - \frac{273}{1760} a^{2} - \frac{49}{880} a - \frac{147}{440}$, $\frac{1}{7040} a^{14} - \frac{1}{7040} a^{13} - \frac{7}{1408} a^{11} + \frac{189}{3520} a^{10} + \frac{7}{7040} a^{9} + \frac{217}{3520} a^{8} + \frac{623}{7040} a^{7} + \frac{51}{3520} a^{6} - \frac{163}{7040} a^{5} + \frac{79}{440} a^{4} + \frac{279}{1408} a^{3} + \frac{2839}{7040} a^{2} - \frac{49}{352} a - \frac{59}{352}$, $\frac{1}{17959040} a^{15} + \frac{317}{4489760} a^{14} - \frac{1889}{17959040} a^{13} + \frac{5}{326528} a^{12} - \frac{801997}{17959040} a^{11} + \frac{181339}{17959040} a^{10} + \frac{54627}{1632640} a^{9} + \frac{1404599}{17959040} a^{8} + \frac{1196769}{17959040} a^{7} - \frac{316051}{3591808} a^{6} - \frac{1776003}{17959040} a^{5} - \frac{637579}{17959040} a^{4} + \frac{4287687}{8979520} a^{3} - \frac{4927579}{17959040} a^{2} - \frac{85717}{224488} a + \frac{264787}{897952}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 696158.290051 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T158):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.2.230911.1, 4.4.4913.1, 4.2.13583.1, 8.0.906438128657.1 x2, 8.2.19285917631.1 x2, 8.4.53319889921.1

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

Sibling fields

Degree 8 siblings: data not computed
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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$17$17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$
$47$$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$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}$