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Properties

Label 16.0.90339182788...4624.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 17^{14}\cdot 47^{4}$
Root discriminant $176.70$
Ramified primes $2, 17, 47$
Class number $115488$ (GRH)
Class group $[2, 2, 2, 6, 2406]$ (GRH)
Galois group $C_4.C_2^2:D_4$ (as 16T209)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![602604304, 0, 642666640, 0, 261780824, 0, 52887000, 0, 5779728, 0, 346460, 0, 10982, 0, 170, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 170*x^14 + 10982*x^12 + 346460*x^10 + 5779728*x^8 + 52887000*x^6 + 261780824*x^4 + 642666640*x^2 + 602604304)
 
gp: K = bnfinit(x^16 + 170*x^14 + 10982*x^12 + 346460*x^10 + 5779728*x^8 + 52887000*x^6 + 261780824*x^4 + 642666640*x^2 + 602604304, 1)
 

Normalized defining polynomial

\( x^{16} + 170 x^{14} + 10982 x^{12} + 346460 x^{10} + 5779728 x^{8} + 52887000 x^{6} + 261780824 x^{4} + 642666640 x^{2} + 602604304 \)

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:  \(903391827881520588378160364334874624=2^{40}\cdot 17^{14}\cdot 47^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $176.70$
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:  $2, 17, 47$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{68} a^{8}$, $\frac{1}{68} a^{9}$, $\frac{1}{68} a^{10}$, $\frac{1}{68} a^{11}$, $\frac{1}{14144} a^{12} - \frac{1}{416} a^{10} - \frac{1}{208} a^{8} + \frac{11}{52} a^{6} - \frac{5}{104} a^{4} + \frac{27}{104} a^{2} + \frac{1}{104}$, $\frac{1}{268736} a^{13} + \frac{503}{134368} a^{11} + \frac{21}{3536} a^{9} + \frac{167}{988} a^{7} - \frac{5}{1976} a^{5} - \frac{701}{1976} a^{3} + \frac{729}{1976} a$, $\frac{1}{224363473267979968} a^{14} + \frac{134260952641}{112181736633989984} a^{12} - \frac{12181195181737}{2952150964052368} a^{10} + \frac{17924556694841}{3505679269812187} a^{8} - \frac{2088727490177}{126902417006776} a^{6} + \frac{358868168591235}{1649731421088088} a^{4} + \frac{695459468987117}{1649731421088088} a^{2} - \frac{198006533853}{1142473283302}$, $\frac{1}{4262905992091619392} a^{15} + \frac{134260952641}{2131452996045809696} a^{13} - \frac{272665103774593}{56090868316994992} a^{11} - \frac{1784249621944735}{266431624505726212} a^{9} + \frac{251716106523375}{2411145923128744} a^{7} + \frac{2833465300223367}{31344897000673672} a^{5} + \frac{12243579416603733}{31344897000673672} a^{3} - \frac{2482953100457}{21706992382738} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{6}\times C_{2406}$, which has order $115488$ (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:  \( 3607556.75902 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.C_2^2:D_4$ (as 16T209):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_4.C_2^2:D_4$
Character table for $C_4.C_2^2:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.434656.1, 4.4.4913.1, 4.4.7389152.1, 8.8.54599567279104.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.22.5$x^{8} + 10 x^{4} + 16 x + 36$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
2.8.18.6$x^{8} + 16 x^{4} + 4$$4$$2$$18$$Q_8:C_2$$[2, 3, 7/2]^{2}$
17Data not computed
$47$47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$