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Properties

Label 16.0.30555453980...7521.1
Degree $16$
Signature $[0, 8]$
Discriminant $13^{10}\cdot 53^{6}$
Root discriminant $22.02$
Ramified primes $13, 53$
Class number $1$
Class group Trivial
Galois group $C_2^4.D_4$ (as 16T339)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, -33, 118, -261, 487, -775, 925, -900, 834, -736, 591, -397, 216, -93, 30, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 30*x^14 - 93*x^13 + 216*x^12 - 397*x^11 + 591*x^10 - 736*x^9 + 834*x^8 - 900*x^7 + 925*x^6 - 775*x^5 + 487*x^4 - 261*x^3 + 118*x^2 - 33*x + 9)
 
gp: K = bnfinit(x^16 - 7*x^15 + 30*x^14 - 93*x^13 + 216*x^12 - 397*x^11 + 591*x^10 - 736*x^9 + 834*x^8 - 900*x^7 + 925*x^6 - 775*x^5 + 487*x^4 - 261*x^3 + 118*x^2 - 33*x + 9, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 30 x^{14} - 93 x^{13} + 216 x^{12} - 397 x^{11} + 591 x^{10} - 736 x^{9} + 834 x^{8} - 900 x^{7} + 925 x^{6} - 775 x^{5} + 487 x^{4} - 261 x^{3} + 118 x^{2} - 33 x + 9 \)

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:  \(3055545398040538937521=13^{10}\cdot 53^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.02$
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:  $13, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{11} + \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{1}{9} a^{8} + \frac{1}{9} a^{7} + \frac{2}{9} a^{6} - \frac{1}{3} a^{5} - \frac{4}{9} a^{4} + \frac{4}{9} a^{2} - \frac{4}{9} a + \frac{1}{3}$, $\frac{1}{1071} a^{14} + \frac{8}{1071} a^{13} - \frac{26}{1071} a^{12} + \frac{44}{357} a^{11} - \frac{11}{1071} a^{10} + \frac{2}{357} a^{9} - \frac{1}{1071} a^{8} + \frac{454}{1071} a^{7} + \frac{26}{63} a^{6} - \frac{394}{1071} a^{5} + \frac{523}{1071} a^{4} - \frac{341}{1071} a^{3} - \frac{524}{1071} a^{2} + \frac{130}{1071} a - \frac{25}{357}$, $\frac{1}{25081749} a^{15} - \frac{3803}{25081749} a^{14} + \frac{174047}{25081749} a^{13} - \frac{4645}{3583107} a^{12} + \frac{969205}{25081749} a^{11} + \frac{3652268}{25081749} a^{10} + \frac{1444403}{25081749} a^{9} + \frac{2407589}{25081749} a^{8} + \frac{2532886}{8360583} a^{7} - \frac{7216571}{25081749} a^{6} - \frac{827252}{2280159} a^{5} - \frac{3524068}{8360583} a^{4} - \frac{640072}{2786861} a^{3} - \frac{2108192}{8360583} a^{2} + \frac{1086362}{25081749} a + \frac{897097}{8360583}$

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

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 18908.1207162 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.D_4$ (as 16T339):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 23 conjugacy class representatives for $C_2^4.D_4$
Character table for $C_2^4.D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 4.4.8957.1, 8.4.1042962037.1, 8.0.4252075997.1, 8.4.55276987961.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
53Data not computed