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Properties

Label 16.0.21767823360...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{12}\cdot 5^{12}$
Root discriminant $21.56$
Ramified primes $2, 3, 5$
Class number $2$
Class group $[2]$
Galois group $C_4 \times D_4$ (as 16T19)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, -32, 40, -16, -52, -56, 194, -364, 355, 2, 83, 4, 47, -16, 7, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 7*x^14 - 16*x^13 + 47*x^12 + 4*x^11 + 83*x^10 + 2*x^9 + 355*x^8 - 364*x^7 + 194*x^6 - 56*x^5 - 52*x^4 - 16*x^3 + 40*x^2 - 32*x + 16)
 
gp: K = bnfinit(x^16 - 2*x^15 + 7*x^14 - 16*x^13 + 47*x^12 + 4*x^11 + 83*x^10 + 2*x^9 + 355*x^8 - 364*x^7 + 194*x^6 - 56*x^5 - 52*x^4 - 16*x^3 + 40*x^2 - 32*x + 16, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 7 x^{14} - 16 x^{13} + 47 x^{12} + 4 x^{11} + 83 x^{10} + 2 x^{9} + 355 x^{8} - 364 x^{7} + 194 x^{6} - 56 x^{5} - 52 x^{4} - 16 x^{3} + 40 x^{2} - 32 x + 16 \)

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:  \(2176782336000000000000=2^{24}\cdot 3^{12}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.56$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{20} a^{12} - \frac{1}{5} a^{11} + \frac{1}{20} a^{10} - \frac{1}{2} a^{9} + \frac{9}{20} a^{8} + \frac{1}{10} a^{7} - \frac{3}{20} a^{6} + \frac{2}{5} a^{5} + \frac{1}{20} a^{4} - \frac{3}{10} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5}$, $\frac{1}{623180} a^{13} - \frac{6933}{311590} a^{12} - \frac{16761}{623180} a^{11} - \frac{2018}{155795} a^{10} - \frac{89601}{623180} a^{9} + \frac{51231}{155795} a^{8} + \frac{203423}{623180} a^{7} - \frac{20913}{311590} a^{6} + \frac{17611}{124636} a^{5} + \frac{74693}{155795} a^{4} - \frac{61891}{311590} a^{3} + \frac{71422}{155795} a^{2} - \frac{30206}{155795} a - \frac{11308}{155795}$, $\frac{1}{1246360} a^{14} + \frac{30631}{1246360} a^{12} - \frac{21923}{124636} a^{11} - \frac{18565}{249272} a^{10} - \frac{102401}{623180} a^{9} - \frac{395101}{1246360} a^{8} - \frac{97433}{311590} a^{7} - \frac{37845}{249272} a^{6} - \frac{144017}{623180} a^{5} - \frac{250571}{623180} a^{4} - \frac{26464}{155795} a^{3} + \frac{5954}{155795} a^{2} - \frac{35372}{155795} a - \frac{64638}{155795}$, $\frac{1}{6231800} a^{15} - \frac{3}{6231800} a^{13} + \frac{20799}{3115900} a^{12} - \frac{884487}{6231800} a^{11} + \frac{79211}{623180} a^{10} - \frac{42787}{6231800} a^{9} - \frac{153693}{1557950} a^{8} - \frac{2791399}{6231800} a^{7} + \frac{997509}{3115900} a^{6} + \frac{32033}{311590} a^{5} - \frac{189937}{778975} a^{4} + \frac{326529}{1557950} a^{3} + \frac{48662}{778975} a^{2} + \frac{45309}{778975} a - \frac{288676}{778975}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{3108}{155795} a^{15} - \frac{27381}{623180} a^{14} + \frac{21756}{155795} a^{13} - \frac{49728}{155795} a^{12} + \frac{146076}{155795} a^{11} + \frac{12432}{155795} a^{10} + \frac{361919}{311590} a^{9} + \frac{6216}{155795} a^{8} + \frac{220668}{31159} a^{7} - \frac{1131312}{155795} a^{6} + \frac{602952}{155795} a^{5} + \frac{3632853}{623180} a^{4} - \frac{161616}{155795} a^{3} - \frac{49728}{155795} a^{2} + \frac{24864}{31159} a - \frac{99456}{155795} \) (order $10$)
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:  \( 23907.9456463 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times D_4$ (as 16T19):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_4 \times D_4$
Character table for $C_4 \times D_4$

Intermediate fields

\(\Q(\sqrt{30}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{6}) \), \(\Q(\zeta_{5})\), 4.0.72000.2, 4.2.8640.2, \(\Q(\sqrt{5}, \sqrt{6})\), 4.2.8640.1, 8.4.1866240000.2, 8.0.5184000000.6, 8.4.46656000000.2

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 16 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 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
3Data not computed
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$