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Properties

Label 16.0.69690127210...4593.1
Degree $16$
Signature $[0, 8]$
Discriminant $17^{7}\cdot 19^{8}$
Root discriminant $15.06$
Ramified primes $17, 19$
Class number $1$
Class group Trivial
Galois group $C_4.D_4:C_4$ (as 16T289)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 16, 14, -62, 39, 122, -275, 250, -72, -69, 86, -34, -4, 11, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 11*x^14 - 4*x^13 - 34*x^12 + 86*x^11 - 69*x^10 - 72*x^9 + 250*x^8 - 275*x^7 + 122*x^6 + 39*x^5 - 62*x^4 + 14*x^3 + 16*x^2 - 8*x + 1)
 
gp: K = bnfinit(x^16 - 5*x^15 + 11*x^14 - 4*x^13 - 34*x^12 + 86*x^11 - 69*x^10 - 72*x^9 + 250*x^8 - 275*x^7 + 122*x^6 + 39*x^5 - 62*x^4 + 14*x^3 + 16*x^2 - 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 11 x^{14} - 4 x^{13} - 34 x^{12} + 86 x^{11} - 69 x^{10} - 72 x^{9} + 250 x^{8} - 275 x^{7} + 122 x^{6} + 39 x^{5} - 62 x^{4} + 14 x^{3} + 16 x^{2} - 8 x + 1 \)

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:  \(6969012721055784593=17^{7}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.06$
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:  $17, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{11} a^{13} - \frac{5}{11} a^{12} + \frac{4}{11} a^{11} + \frac{5}{11} a^{10} + \frac{2}{11} a^{9} + \frac{2}{11} a^{8} - \frac{4}{11} a^{7} + \frac{5}{11} a^{6} - \frac{5}{11} a^{5} + \frac{3}{11} a^{4} + \frac{5}{11} a^{3} + \frac{5}{11} a^{2} + \frac{1}{11} a + \frac{3}{11}$, $\frac{1}{55} a^{14} - \frac{21}{55} a^{12} - \frac{19}{55} a^{11} - \frac{17}{55} a^{10} - \frac{21}{55} a^{9} - \frac{1}{11} a^{8} - \frac{3}{11} a^{7} + \frac{4}{11} a^{6} - \frac{13}{55} a^{4} + \frac{19}{55} a^{3} + \frac{4}{55} a^{2} - \frac{14}{55} a - \frac{7}{55}$, $\frac{1}{62315} a^{15} + \frac{519}{62315} a^{14} - \frac{1086}{62315} a^{13} + \frac{3097}{62315} a^{12} + \frac{9402}{62315} a^{11} + \frac{2716}{62315} a^{10} + \frac{14796}{62315} a^{9} - \frac{2510}{12463} a^{8} - \frac{3176}{12463} a^{7} + \frac{5763}{12463} a^{6} - \frac{256}{605} a^{5} + \frac{22802}{62315} a^{4} + \frac{3086}{12463} a^{3} + \frac{25172}{62315} a^{2} + \frac{287}{5665} a + \frac{4612}{62315}$

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:  \( 746.1190743 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T289):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 44 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-19}) \), 4.0.6137.1, 8.0.640267073.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$17$17.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
17.8.7.8$x^{8} + 4131$$8$$1$$7$$C_8$$[\ ]_{8}$
19Data not computed