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Properties

Label 16.8.57681033264...3953.4
Degree $16$
Signature $[8, 4]$
Discriminant $17^{15}\cdot 67^{4}$
Root discriminant $40.74$
Ramified primes $17, 67$
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![2381, 42154, -44487, -53868, 97907, -70727, 30047, -4995, -4727, 4375, -461, -243, -106, 24, 9, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 9*x^14 + 24*x^13 - 106*x^12 - 243*x^11 - 461*x^10 + 4375*x^9 - 4727*x^8 - 4995*x^7 + 30047*x^6 - 70727*x^5 + 97907*x^4 - 53868*x^3 - 44487*x^2 + 42154*x + 2381)
 
gp: K = bnfinit(x^16 - 3*x^15 + 9*x^14 + 24*x^13 - 106*x^12 - 243*x^11 - 461*x^10 + 4375*x^9 - 4727*x^8 - 4995*x^7 + 30047*x^6 - 70727*x^5 + 97907*x^4 - 53868*x^3 - 44487*x^2 + 42154*x + 2381, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 9 x^{14} + 24 x^{13} - 106 x^{12} - 243 x^{11} - 461 x^{10} + 4375 x^{9} - 4727 x^{8} - 4995 x^{7} + 30047 x^{6} - 70727 x^{5} + 97907 x^{4} - 53868 x^{3} - 44487 x^{2} + 42154 x + 2381 \)

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:  \(57681033264163530732453953=17^{15}\cdot 67^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.74$
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:  $17, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{113825488146183927278203278534249066061793} a^{15} + \frac{13206728767389916578695512576358517129255}{113825488146183927278203278534249066061793} a^{14} - \frac{1939321617235438028807943653923756296563}{113825488146183927278203278534249066061793} a^{13} - \frac{31609648249380809295111887320484403821239}{113825488146183927278203278534249066061793} a^{12} + \frac{205451283823549874942351118354389243831}{113825488146183927278203278534249066061793} a^{11} + \frac{10578052339655936565099140554236014515222}{113825488146183927278203278534249066061793} a^{10} - \frac{22459992049675702521063520480587987274801}{113825488146183927278203278534249066061793} a^{9} - \frac{37715386148882350437224998260472207956503}{113825488146183927278203278534249066061793} a^{8} + \frac{12951212535739525760107770098110491805874}{113825488146183927278203278534249066061793} a^{7} + \frac{3869238594665783075654248350103841529043}{113825488146183927278203278534249066061793} a^{6} - \frac{33072875014890934165496862357438377398286}{113825488146183927278203278534249066061793} a^{5} + \frac{8288802875886372972772118691556546308077}{113825488146183927278203278534249066061793} a^{4} - \frac{22129396935182084365892416627717395012654}{113825488146183927278203278534249066061793} a^{3} - \frac{11344600390900901765546296525799860275858}{113825488146183927278203278534249066061793} a^{2} - \frac{47394879942784951923063946242766869731827}{113825488146183927278203278534249066061793} a - \frac{35756644653088062268639755237979750637772}{113825488146183927278203278534249066061793}$

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:  \( 5751459.89298 \) (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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ 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
17Data not computed
$67$67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$