↑

Properties

Label 16.8.12913449667...0169.1
Degree $16$
Signature $[8, 4]$
Discriminant $17^{12}\cdot 53^{6}$
Root discriminant $37.11$
Ramified primes $17, 53$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_2\times C_4).D_4$ (as 16T121)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5776, 0, -24300, 0, 23349, 0, -5011, 0, 2777, 0, -449, 0, 67, 0, -18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 18*x^14 + 67*x^12 - 449*x^10 + 2777*x^8 - 5011*x^6 + 23349*x^4 - 24300*x^2 + 5776)
 
gp: K = bnfinit(x^16 - 18*x^14 + 67*x^12 - 449*x^10 + 2777*x^8 - 5011*x^6 + 23349*x^4 - 24300*x^2 + 5776, 1)
 

Normalized defining polynomial

\( x^{16} - 18 x^{14} + 67 x^{12} - 449 x^{10} + 2777 x^{8} - 5011 x^{6} + 23349 x^{4} - 24300 x^{2} + 5776 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(12913449667746331350360169=17^{12}\cdot 53^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.11$
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:  $17, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{56} a^{12} - \frac{1}{4} a^{11} - \frac{1}{56} a^{10} - \frac{1}{4} a^{9} + \frac{1}{28} a^{8} - \frac{3}{56} a^{6} - \frac{1}{4} a^{5} - \frac{1}{28} a^{4} + \frac{27}{56} a^{2} - \frac{1}{4} a + \frac{5}{14}$, $\frac{1}{56} a^{13} - \frac{1}{56} a^{11} + \frac{1}{28} a^{9} - \frac{3}{56} a^{7} + \frac{13}{28} a^{5} - \frac{1}{2} a^{4} + \frac{27}{56} a^{3} + \frac{5}{14} a$, $\frac{1}{221666540479888} a^{14} - \frac{1}{112} a^{13} + \frac{1178866722689}{221666540479888} a^{12} - \frac{27}{112} a^{11} + \frac{11893079801287}{55416635119972} a^{10} - \frac{1}{56} a^{9} + \frac{31026869511885}{221666540479888} a^{8} - \frac{25}{112} a^{7} - \frac{5011671032303}{55416635119972} a^{6} + \frac{1}{56} a^{5} - \frac{59957747278001}{221666540479888} a^{4} + \frac{29}{112} a^{3} + \frac{7211581754633}{15833324319992} a^{2} + \frac{9}{28} a + \frac{5591334184146}{13854158779993}$, $\frac{1}{4211664269117872} a^{15} - \frac{1584411507163}{526458033639734} a^{13} + \frac{726426099424805}{4211664269117872} a^{11} - \frac{162931353408017}{4211664269117872} a^{9} - \frac{1031400275068701}{4211664269117872} a^{7} - \frac{1860998388677091}{4211664269117872} a^{5} - \frac{54847630390699}{601666324159696} a^{3} + \frac{61007969304118}{263229016819867} a - \frac{1}{2}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 9055120.25088 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_4).D_4$ (as 16T121):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $(C_2\times C_4).D_4$
Character table for $(C_2\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.15317.1, 4.4.4913.1, 4.4.260389.1, 8.4.3593528860013.1, 8.4.12434355917.1, 8.8.67802431321.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
17Data not computed
$53$53.4.0.1$x^{4} - x + 18$$1$$4$$0$$C_4$$[\ ]^{4}$
53.4.0.1$x^{4} - x + 18$$1$$4$$0$$C_4$$[\ ]^{4}$
53.8.6.2$x^{8} + 477 x^{4} + 70225$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$