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Properties

Label 16.0.66556997328...0649.1
Degree $16$
Signature $[0, 8]$
Discriminant $7^{12}\cdot 37^{10}$
Root discriminant $41.11$
Ramified primes $7, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
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![1681, -11193, 25749, -11592, -24042, 14679, 15615, -4788, -2123, 987, -135, -252, 192, 0, -18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 18*x^14 + 192*x^12 - 252*x^11 - 135*x^10 + 987*x^9 - 2123*x^8 - 4788*x^7 + 15615*x^6 + 14679*x^5 - 24042*x^4 - 11592*x^3 + 25749*x^2 - 11193*x + 1681)
 
gp: K = bnfinit(x^16 - 18*x^14 + 192*x^12 - 252*x^11 - 135*x^10 + 987*x^9 - 2123*x^8 - 4788*x^7 + 15615*x^6 + 14679*x^5 - 24042*x^4 - 11592*x^3 + 25749*x^2 - 11193*x + 1681, 1)
 

Normalized defining polynomial

\( x^{16} - 18 x^{14} + 192 x^{12} - 252 x^{11} - 135 x^{10} + 987 x^{9} - 2123 x^{8} - 4788 x^{7} + 15615 x^{6} + 14679 x^{5} - 24042 x^{4} - 11592 x^{3} + 25749 x^{2} - 11193 x + 1681 \)

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:  \(66556997328875790787650649=7^{12}\cdot 37^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.11$
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:  $7, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{18} a^{10} - \frac{1}{18} a^{9} - \frac{1}{18} a^{8} - \frac{1}{6} a^{6} + \frac{1}{6} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{7}{18} a^{2} - \frac{4}{9} a - \frac{5}{18}$, $\frac{1}{18} a^{11} - \frac{1}{9} a^{9} - \frac{1}{18} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{5} + \frac{1}{3} a^{4} + \frac{5}{18} a^{3} + \frac{1}{6} a^{2} + \frac{5}{18} a - \frac{5}{18}$, $\frac{1}{36} a^{12} - \frac{1}{36} a^{10} - \frac{1}{18} a^{9} - \frac{1}{9} a^{8} + \frac{1}{3} a^{6} + \frac{1}{4} a^{5} + \frac{17}{36} a^{4} - \frac{1}{12} a^{3} - \frac{1}{18} a^{2} - \frac{13}{36} a - \frac{5}{36}$, $\frac{1}{36} a^{13} - \frac{1}{36} a^{11} - \frac{1}{6} a^{9} - \frac{1}{18} a^{8} + \frac{1}{3} a^{7} + \frac{1}{12} a^{6} - \frac{13}{36} a^{5} - \frac{5}{12} a^{4} - \frac{7}{18} a^{3} + \frac{1}{4} a^{2} + \frac{5}{12} a - \frac{5}{18}$, $\frac{1}{576} a^{14} + \frac{5}{576} a^{13} + \frac{1}{144} a^{12} + \frac{5}{576} a^{11} + \frac{13}{576} a^{10} - \frac{37}{288} a^{9} + \frac{1}{18} a^{8} - \frac{13}{192} a^{7} - \frac{23}{288} a^{6} - \frac{101}{576} a^{5} + \frac{17}{36} a^{4} - \frac{85}{288} a^{3} + \frac{5}{144} a^{2} + \frac{13}{288} a + \frac{283}{576}$, $\frac{1}{7059927930235177125048768} a^{15} + \frac{4243685378887469479}{14349447012673124237904} a^{14} - \frac{61232549279835529511089}{7059927930235177125048768} a^{13} + \frac{147319316686583623145}{172193364152077490854848} a^{12} - \frac{17833424129836079557621}{882490991279397140631096} a^{11} + \frac{183308316841383114995753}{7059927930235177125048768} a^{10} - \frac{117293685578057190199913}{1176654655039196187508128} a^{9} - \frac{1023437328291901923248743}{7059927930235177125048768} a^{8} + \frac{1145110130408084990518505}{7059927930235177125048768} a^{7} + \frac{1119187391770473077567971}{2353309310078392375016256} a^{6} + \frac{1889169936196903584215125}{7059927930235177125048768} a^{5} - \frac{10996968066774057136997}{3529963965117588562524384} a^{4} - \frac{1306180478524569291931817}{3529963965117588562524384} a^{3} - \frac{69700618332415837965709}{3529963965117588562524384} a^{2} - \frac{1061892321412419523489925}{2353309310078392375016256} a - \frac{44117279464031469054979}{172193364152077490854848}$

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:  $7$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 9547939.47561 \) (assuming GRH)
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

\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{37}) \), \(\Q(\sqrt{-259}) \), 4.0.1813.1 x2, 4.2.9583.1 x2, \(\Q(\sqrt{-7}, \sqrt{37})\), 8.0.8158247197093.1, 8.0.5959274797.1, 8.0.4499860561.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$7$7.8.6.1$x^{8} + 35 x^{4} + 441$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
7.8.6.1$x^{8} + 35 x^{4} + 441$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$37$37.8.6.2$x^{8} + 333 x^{4} + 34225$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
37.8.4.1$x^{8} + 5476 x^{4} - 50653 x^{2} + 7496644$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$