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Properties

Label 16.4.95735899582...8944.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{54}\cdot 3^{12}$
Root discriminant $23.65$
Ramified primes $2, 3$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4^2:C_2$ (as 16T27)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8, 64, -256, 576, -624, -256, 2032, -3296, 2740, -1008, -216, 352, -156, 24, 16, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 16*x^14 + 24*x^13 - 156*x^12 + 352*x^11 - 216*x^10 - 1008*x^9 + 2740*x^8 - 3296*x^7 + 2032*x^6 - 256*x^5 - 624*x^4 + 576*x^3 - 256*x^2 + 64*x - 8)
 
gp: K = bnfinit(x^16 - 8*x^15 + 16*x^14 + 24*x^13 - 156*x^12 + 352*x^11 - 216*x^10 - 1008*x^9 + 2740*x^8 - 3296*x^7 + 2032*x^6 - 256*x^5 - 624*x^4 + 576*x^3 - 256*x^2 + 64*x - 8, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 16 x^{14} + 24 x^{13} - 156 x^{12} + 352 x^{11} - 216 x^{10} - 1008 x^{9} + 2740 x^{8} - 3296 x^{7} + 2032 x^{6} - 256 x^{5} - 624 x^{4} + 576 x^{3} - 256 x^{2} + 64 x - 8 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(9573589958277615058944=2^{54}\cdot 3^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.65$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{4} a^{12}$, $\frac{1}{4} a^{13}$, $\frac{1}{292} a^{14} + \frac{3}{146} a^{13} + \frac{15}{292} a^{12} + \frac{4}{73} a^{11} + \frac{17}{146} a^{10} + \frac{13}{73} a^{9} - \frac{21}{146} a^{8} - \frac{15}{146} a^{7} + \frac{25}{146} a^{6} + \frac{25}{73} a^{5} - \frac{22}{73} a^{4} - \frac{15}{73} a^{3} - \frac{29}{73} a^{2} - \frac{9}{73} a + \frac{12}{73}$, $\frac{1}{3582778147684} a^{15} + \frac{1660196805}{1791389073842} a^{14} - \frac{207300211651}{1791389073842} a^{13} + \frac{368892930829}{3582778147684} a^{12} + \frac{347267997029}{3582778147684} a^{11} - \frac{381780540405}{1791389073842} a^{10} - \frac{201761597583}{1791389073842} a^{9} - \frac{11532318478}{895694536921} a^{8} - \frac{189753563277}{895694536921} a^{7} - \frac{220979850230}{895694536921} a^{6} - \frac{342837082692}{895694536921} a^{5} - \frac{102966826076}{895694536921} a^{4} - \frac{281981123012}{895694536921} a^{3} + \frac{16257716197}{895694536921} a^{2} + \frac{191863491741}{895694536921} a + \frac{285503610695}{895694536921}$

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:  $9$
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:  \( 84674.9713264 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2:C_2$ (as 16T27):

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 $C_4^2:C_2$
Character table for $C_4^2:C_2$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}, \sqrt{3})\), 8.4.12230590464.2, 8.4.1358954496.1, 8.4.3057647616.1

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

Sibling fields

Galois closure: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
3Data not computed