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Properties

Label 16.0.72057594037...0000.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{64}\cdot 5^{8}$
Root discriminant $35.78$
Ramified primes $2, 5$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_8\times C_2$ (as 16T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2209, 0, -64, 0, 336, 0, -672, 0, 660, 0, -352, 0, 104, 0, -16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 16*x^14 + 104*x^12 - 352*x^10 + 660*x^8 - 672*x^6 + 336*x^4 - 64*x^2 + 2209)
 
gp: K = bnfinit(x^16 - 16*x^14 + 104*x^12 - 352*x^10 + 660*x^8 - 672*x^6 + 336*x^4 - 64*x^2 + 2209, 1)
 

Normalized defining polynomial

\( x^{16} - 16 x^{14} + 104 x^{12} - 352 x^{10} + 660 x^{8} - 672 x^{6} + 336 x^{4} - 64 x^{2} + 2209 \)

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:  \(7205759403792793600000000=2^{64}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.78$
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:  $2, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(160=2^{5}\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{160}(1,·)$, $\chi_{160}(131,·)$, $\chi_{160}(69,·)$, $\chi_{160}(11,·)$, $\chi_{160}(79,·)$, $\chi_{160}(81,·)$, $\chi_{160}(149,·)$, $\chi_{160}(91,·)$, $\chi_{160}(29,·)$, $\chi_{160}(159,·)$, $\chi_{160}(39,·)$, $\chi_{160}(41,·)$, $\chi_{160}(109,·)$, $\chi_{160}(51,·)$, $\chi_{160}(119,·)$, $\chi_{160}(121,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{21} a^{8} - \frac{8}{21} a^{6} - \frac{1}{21} a^{4} + \frac{5}{21} a^{2} + \frac{2}{21}$, $\frac{1}{987} a^{9} - \frac{386}{987} a^{7} - \frac{295}{987} a^{5} - \frac{373}{987} a^{3} - \frac{313}{987} a$, $\frac{1}{987} a^{10} - \frac{10}{987} a^{8} - \frac{114}{329} a^{6} + \frac{34}{141} a^{4} - \frac{407}{987} a^{2} - \frac{5}{21}$, $\frac{1}{987} a^{11} - \frac{254}{987} a^{7} + \frac{83}{329} a^{5} - \frac{9}{47} a^{3} - \frac{404}{987} a$, $\frac{1}{987} a^{12} - \frac{19}{987} a^{8} + \frac{49}{141} a^{6} - \frac{424}{987} a^{4} - \frac{72}{329} a^{2} + \frac{10}{21}$, $\frac{1}{987} a^{13} - \frac{82}{987} a^{7} - \frac{107}{987} a^{5} - \frac{394}{987} a^{3} + \frac{445}{987} a$, $\frac{1}{987} a^{14} + \frac{4}{329} a^{8} + \frac{128}{987} a^{6} - \frac{488}{987} a^{4} - \frac{24}{329} a^{2} + \frac{4}{21}$, $\frac{1}{987} a^{15} - \frac{25}{141} a^{7} + \frac{13}{141} a^{5} + \frac{152}{329} a^{3} - \frac{4}{987} a$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 320942.011738 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_8$ (as 16T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{-10}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\zeta_{16})^+\), 4.0.51200.2, 8.0.10485760000.2, 8.0.2147483648.1, 8.8.1342177280000.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
5Data not computed