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Properties

Label 16.0.68963216043...4889.2
Degree $16$
Signature $[0, 8]$
Discriminant $13^{10}\cdot 29^{8}$
Root discriminant $26.76$
Ramified primes $13, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2.D_4$ (as 16T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![368, -1328, 2452, -3820, 5155, -5434, 5144, -4331, 3366, -2228, 1265, -586, 234, -75, 20, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 20*x^14 - 75*x^13 + 234*x^12 - 586*x^11 + 1265*x^10 - 2228*x^9 + 3366*x^8 - 4331*x^7 + 5144*x^6 - 5434*x^5 + 5155*x^4 - 3820*x^3 + 2452*x^2 - 1328*x + 368)
 
gp: K = bnfinit(x^16 - 4*x^15 + 20*x^14 - 75*x^13 + 234*x^12 - 586*x^11 + 1265*x^10 - 2228*x^9 + 3366*x^8 - 4331*x^7 + 5144*x^6 - 5434*x^5 + 5155*x^4 - 3820*x^3 + 2452*x^2 - 1328*x + 368, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 20 x^{14} - 75 x^{13} + 234 x^{12} - 586 x^{11} + 1265 x^{10} - 2228 x^{9} + 3366 x^{8} - 4331 x^{7} + 5144 x^{6} - 5434 x^{5} + 5155 x^{4} - 3820 x^{3} + 2452 x^{2} - 1328 x + 368 \)

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:  \(68963216043675506454889=13^{10}\cdot 29^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.76$
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:  $13, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{9} a^{10} + \frac{1}{9} a^{9} - \frac{1}{9} a^{6} + \frac{1}{3} a^{4} - \frac{2}{9} a^{3} - \frac{2}{9} a^{2} - \frac{1}{3} a + \frac{4}{9}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{9} - \frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{2}{9} a^{4} - \frac{1}{3} a^{3} - \frac{4}{9} a^{2} - \frac{2}{9} a - \frac{1}{9}$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{9} - \frac{1}{9} a^{8} + \frac{1}{9} a^{7} - \frac{1}{9} a^{6} + \frac{1}{9} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{9} a^{2} - \frac{4}{9} a + \frac{1}{9}$, $\frac{1}{18} a^{13} - \frac{1}{18} a^{10} - \frac{1}{9} a^{8} - \frac{1}{18} a^{7} - \frac{1}{6} a^{4} + \frac{7}{18} a + \frac{2}{9}$, $\frac{1}{117936} a^{14} - \frac{233}{58968} a^{13} + \frac{841}{29484} a^{12} - \frac{5051}{117936} a^{11} - \frac{787}{29484} a^{10} + \frac{8657}{58968} a^{9} + \frac{1559}{39312} a^{8} + \frac{103}{6552} a^{7} + \frac{1529}{19656} a^{6} + \frac{14137}{117936} a^{5} - \frac{6791}{58968} a^{4} - \frac{17113}{58968} a^{3} - \frac{4879}{16848} a^{2} - \frac{23971}{58968} a + \frac{7361}{29484}$, $\frac{1}{596129919840} a^{15} - \frac{469733}{149032479960} a^{14} + \frac{109166479}{5732018460} a^{13} + \frac{10281709679}{198709973280} a^{12} - \frac{5239283611}{298064959920} a^{11} + \frac{1055587409}{59612991984} a^{10} - \frac{16060166963}{119225983968} a^{9} + \frac{4778323891}{49677493320} a^{8} - \frac{2590412381}{19870997328} a^{7} - \frac{5601237407}{45856147680} a^{6} + \frac{2428533941}{18629059995} a^{5} - \frac{18953951935}{59612991984} a^{4} - \frac{53663769841}{119225983968} a^{3} - \frac{6400137941}{29806495992} a^{2} - \frac{836265719}{5322588570} a + \frac{8055349067}{74516239980}$

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:  $7$
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:  \( 162678.83983 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.D_4$ (as 16T33):

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 $C_2^2.D_4$
Character table for $C_2^2.D_4$

Intermediate fields

\(\Q(\sqrt{377}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{29}) \), 4.0.10933.1 x2, \(\Q(\sqrt{13}, \sqrt{29})\), 4.0.4901.1 x2, 8.0.20200652641.2 x2, 8.4.262608484333.1 x2, 8.0.20200652641.1

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 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$