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Properties

Label 16.8.81753664774...3873.1
Degree $16$
Signature $[8, 4]$
Discriminant $13^{4}\cdot 17^{15}$
Root discriminant $27.04$
Ramified primes $13, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^3.C_8$ (as 16T104)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-67, 130, 325, -1593, 2265, -500, -2253, 2757, -953, -606, 729, -209, -72, 58, -8, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 8*x^14 + 58*x^13 - 72*x^12 - 209*x^11 + 729*x^10 - 606*x^9 - 953*x^8 + 2757*x^7 - 2253*x^6 - 500*x^5 + 2265*x^4 - 1593*x^3 + 325*x^2 + 130*x - 67)
 
gp: K = bnfinit(x^16 - 3*x^15 - 8*x^14 + 58*x^13 - 72*x^12 - 209*x^11 + 729*x^10 - 606*x^9 - 953*x^8 + 2757*x^7 - 2253*x^6 - 500*x^5 + 2265*x^4 - 1593*x^3 + 325*x^2 + 130*x - 67, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 8 x^{14} + 58 x^{13} - 72 x^{12} - 209 x^{11} + 729 x^{10} - 606 x^{9} - 953 x^{8} + 2757 x^{7} - 2253 x^{6} - 500 x^{5} + 2265 x^{4} - 1593 x^{3} + 325 x^{2} + 130 x - 67 \)

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:  \(81753664774171848863873=13^{4}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.04$
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:  $13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{111237872907898952087} a^{15} - \frac{12102471454861338578}{111237872907898952087} a^{14} - \frac{41788610935766666816}{111237872907898952087} a^{13} - \frac{52566676447784732220}{111237872907898952087} a^{12} + \frac{50756556225962512142}{111237872907898952087} a^{11} + \frac{4636645644778321762}{111237872907898952087} a^{10} + \frac{13225128678226508992}{111237872907898952087} a^{9} + \frac{36652279601567955688}{111237872907898952087} a^{8} + \frac{6645281353638674480}{111237872907898952087} a^{7} + \frac{31831170982342892553}{111237872907898952087} a^{6} - \frac{51469628721665715807}{111237872907898952087} a^{5} + \frac{47345246026844959515}{111237872907898952087} a^{4} + \frac{10974925954322824933}{111237872907898952087} a^{3} + \frac{31823890639249616677}{111237872907898952087} a^{2} - \frac{54797145029996351448}{111237872907898952087} a - \frac{54345171875194322284}{111237872907898952087}$

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:  \( 188249.182135 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_8$ (as 16T104):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $C_2^3.C_8$
Character table for $C_2^3.C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ R R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
17Data not computed