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Properties

Label 16.0.132120176259825664.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 17^{10}$
Root discriminant $11.75$
Ramified primes $2, 17$
Class number $1$
Class group Trivial
Galois group $C_4\wr C_2$ (as 16T42)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 7, -12, 9, 6, -24, 16, 19, -54, 50, -24, 8, -2, 4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 4*x^14 - 2*x^13 + 8*x^12 - 24*x^11 + 50*x^10 - 54*x^9 + 19*x^8 + 16*x^7 - 24*x^6 + 6*x^5 + 9*x^4 - 12*x^3 + 7*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^16 - 2*x^15 + 4*x^14 - 2*x^13 + 8*x^12 - 24*x^11 + 50*x^10 - 54*x^9 + 19*x^8 + 16*x^7 - 24*x^6 + 6*x^5 + 9*x^4 - 12*x^3 + 7*x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 4 x^{14} - 2 x^{13} + 8 x^{12} - 24 x^{11} + 50 x^{10} - 54 x^{9} + 19 x^{8} + 16 x^{7} - 24 x^{6} + 6 x^{5} + 9 x^{4} - 12 x^{3} + 7 x^{2} - 2 x + 1 \)

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:  \(132120176259825664=2^{16}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11.75$
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:  $2, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{149} a^{14} - \frac{14}{149} a^{13} - \frac{48}{149} a^{12} - \frac{71}{149} a^{11} - \frac{53}{149} a^{10} - \frac{9}{149} a^{9} + \frac{47}{149} a^{8} + \frac{21}{149} a^{7} + \frac{6}{149} a^{6} - \frac{57}{149} a^{5} - \frac{64}{149} a^{4} + \frac{53}{149} a^{3} + \frac{43}{149} a^{2} + \frac{30}{149} a + \frac{21}{149}$, $\frac{1}{1239829} a^{15} + \frac{756}{1239829} a^{14} + \frac{15545}{1239829} a^{13} - \frac{249058}{1239829} a^{12} + \frac{391979}{1239829} a^{11} - \frac{530433}{1239829} a^{10} - \frac{571593}{1239829} a^{9} - \frac{508980}{1239829} a^{8} - \frac{386422}{1239829} a^{7} + \frac{382427}{1239829} a^{6} + \frac{134101}{1239829} a^{5} - \frac{475069}{1239829} a^{4} + \frac{413353}{1239829} a^{3} - \frac{147150}{1239829} a^{2} - \frac{371133}{1239829} a - \frac{485066}{1239829}$

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:  \( \frac{18899}{23393} a^{15} - \frac{30913}{23393} a^{14} + \frac{67628}{23393} a^{13} - \frac{13823}{23393} a^{12} + \frac{1036}{157} a^{11} - \frac{387271}{23393} a^{10} + \frac{837522}{23393} a^{9} - \frac{738958}{23393} a^{8} + \frac{168789}{23393} a^{7} + \frac{372028}{23393} a^{6} - \frac{350951}{23393} a^{5} + \frac{34993}{23393} a^{4} + \frac{132112}{23393} a^{3} - \frac{1248}{157} a^{2} + \frac{76716}{23393} a - \frac{32962}{23393} \) (order $4$)
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:  \( 127.734112859 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\wr C_2$ (as 16T42):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_4\wr C_2$
Character table for $C_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{17}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-17}) \), 4.2.1156.1 x2, 4.0.272.1 x2, \(\Q(i, \sqrt{17})\), 8.0.1257728.1, 8.0.363483392.1, 8.0.21381376.2

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 sibling: 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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$17$17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$