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Properties

Label 16.0.23062278680...7533.1
Degree $16$
Signature $[0, 8]$
Discriminant $37^{7}\cdot 149^{8}$
Root discriminant $59.25$
Ramified primes $37, 149$
Class number $47$ (GRH)
Class group $[47]$ (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![34425, -61830, 17064, 59010, -72479, 27438, 12273, -22656, 15545, -6153, 1152, 156, -145, 18, 12, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 12*x^14 + 18*x^13 - 145*x^12 + 156*x^11 + 1152*x^10 - 6153*x^9 + 15545*x^8 - 22656*x^7 + 12273*x^6 + 27438*x^5 - 72479*x^4 + 59010*x^3 + 17064*x^2 - 61830*x + 34425)
 
gp: K = bnfinit(x^16 - 6*x^15 + 12*x^14 + 18*x^13 - 145*x^12 + 156*x^11 + 1152*x^10 - 6153*x^9 + 15545*x^8 - 22656*x^7 + 12273*x^6 + 27438*x^5 - 72479*x^4 + 59010*x^3 + 17064*x^2 - 61830*x + 34425, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 12 x^{14} + 18 x^{13} - 145 x^{12} + 156 x^{11} + 1152 x^{10} - 6153 x^{9} + 15545 x^{8} - 22656 x^{7} + 12273 x^{6} + 27438 x^{5} - 72479 x^{4} + 59010 x^{3} + 17064 x^{2} - 61830 x + 34425 \)

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:  \(23062278680241626549323607533=37^{7}\cdot 149^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.25$
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:  $37, 149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{4}{9} a^{4} + \frac{1}{3} a^{3} - \frac{4}{9} a^{2}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{11} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{4}{9} a^{5} + \frac{1}{3} a^{4} - \frac{4}{9} a^{3} + \frac{1}{3} a$, $\frac{1}{5535} a^{14} - \frac{59}{1107} a^{13} - \frac{68}{5535} a^{12} + \frac{127}{1107} a^{11} - \frac{50}{369} a^{10} - \frac{88}{1845} a^{9} + \frac{1}{1845} a^{8} + \frac{10}{41} a^{7} + \frac{76}{1107} a^{6} - \frac{86}{5535} a^{5} + \frac{2492}{5535} a^{4} - \frac{49}{1107} a^{3} + \frac{632}{1845} a^{2} - \frac{121}{615} a - \frac{14}{41}$, $\frac{1}{398121588664591434213555} a^{15} - \frac{1163779014322230214}{398121588664591434213555} a^{14} + \frac{4229119325208633155302}{398121588664591434213555} a^{13} + \frac{19476224555810545268822}{398121588664591434213555} a^{12} - \frac{650341054675402247234}{26541439244306095614237} a^{11} - \frac{6763904194502016703721}{44235732073843492690395} a^{10} + \frac{648127725996011323168}{132707196221530478071185} a^{9} - \frac{17999062307022312362503}{44235732073843492690395} a^{8} - \frac{21960924943789479735401}{79624317732918286842711} a^{7} + \frac{143642398564068659033644}{398121588664591434213555} a^{6} - \frac{27143757455992034446729}{398121588664591434213555} a^{5} - \frac{114639818824613080845278}{398121588664591434213555} a^{4} + \frac{22047535331019650116367}{132707196221530478071185} a^{3} + \frac{59145731666348338867604}{132707196221530478071185} a^{2} - \frac{4152049776014697375182}{14745244024614497563465} a - \frac{584937842256333098068}{2949048804922899512693}$

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

Class group and class number

$C_{47}$, which has order $47$ (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:  \( 2748487.80142 \) (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

\(\Q(\sqrt{149}) \), 4.4.821437.1, 8.8.24966073563853.1

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 $16$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ $16$

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
$37$37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$149$149.2.1.1$x^{2} - 149$$2$$1$$1$$C_2$$[\ ]_{2}$
149.2.1.1$x^{2} - 149$$2$$1$$1$$C_2$$[\ ]_{2}$
149.2.1.1$x^{2} - 149$$2$$1$$1$$C_2$$[\ ]_{2}$
149.2.1.1$x^{2} - 149$$2$$1$$1$$C_2$$[\ ]_{2}$
149.2.1.1$x^{2} - 149$$2$$1$$1$$C_2$$[\ ]_{2}$
149.2.1.1$x^{2} - 149$$2$$1$$1$$C_2$$[\ ]_{2}$
149.2.1.1$x^{2} - 149$$2$$1$$1$$C_2$$[\ ]_{2}$
149.2.1.1$x^{2} - 149$$2$$1$$1$$C_2$$[\ ]_{2}$