Properties

Label 16.0.68719476736...000.13
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 5^{8}$
Root discriminant $15.04$
Ramified primes $2, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2\wr C_2$ (as 16T39)

Related objects

Downloads

Learn more about

Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 + 38*x^12 - 116*x^10 + 203*x^8 - 188*x^6 + 106*x^4 - 60*x^2 + 25)
 
gp: K = bnfinit(x^16 - 8*x^14 + 38*x^12 - 116*x^10 + 203*x^8 - 188*x^6 + 106*x^4 - 60*x^2 + 25, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, 0, -60, 0, 106, 0, -188, 0, 203, 0, -116, 0, 38, 0, -8, 0, 1]);
 

Normalized defining polynomial

\( x^{16} - 8 x^{14} + 38 x^{12} - 116 x^{10} + 203 x^{8} - 188 x^{6} + 106 x^{4} - 60 x^{2} + 25 \)

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(6871947673600000000=2^{44}\cdot 5^{8}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $15.04$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 5$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4}$, $\frac{1}{340} a^{12} - \frac{3}{170} a^{10} - \frac{1}{4} a^{9} + \frac{11}{340} a^{8} + \frac{83}{170} a^{6} - \frac{1}{4} a^{5} - \frac{11}{68} a^{4} + \frac{3}{10} a^{2} + \frac{1}{4} a + \frac{3}{34}$, $\frac{1}{340} a^{13} - \frac{3}{170} a^{11} - \frac{37}{170} a^{9} - \frac{1}{4} a^{8} + \frac{83}{170} a^{7} - \frac{7}{17} a^{5} - \frac{1}{4} a^{4} + \frac{3}{10} a^{3} + \frac{23}{68} a + \frac{1}{4}$, $\frac{1}{340} a^{14} - \frac{5}{68} a^{10} - \frac{23}{340} a^{8} - \frac{79}{340} a^{6} - \frac{143}{340} a^{4} - \frac{19}{170} a^{2} + \frac{19}{68}$, $\frac{1}{1700} a^{15} + \frac{1}{850} a^{13} - \frac{207}{1700} a^{11} - \frac{171}{1700} a^{9} - \frac{597}{1700} a^{7} - \frac{423}{1700} a^{5} + \frac{169}{425} a^{3} - \frac{71}{340} a$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( \frac{3}{17} a^{14} - \frac{21}{17} a^{12} + \frac{11}{2} a^{10} - \frac{515}{34} a^{8} + \frac{43}{2} a^{6} - \frac{469}{34} a^{4} + \frac{265}{34} a^{2} - \frac{163}{34} \) (order $4$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 2187.081297 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Galois group

$C_2^2\wr C_2$ (as 16T39):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_2^2\wr C_2$
Character table for $C_2^2\wr C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-5}) \), 4.2.1600.1 x2, 4.0.1280.1 x2, 4.0.12800.1, 4.0.512.1, 4.0.2560.2, 4.0.2560.1, \(\Q(i, \sqrt{5})\), 8.0.163840000.4, 8.0.2621440000.7, 8.0.104857600.2, 8.0.104857600.1, 8.0.2621440000.6, 8.0.40960000.2, 8.0.163840000.3

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

Sibling fields

Galois closure: Deg 32
Degree 8 siblings: 8.0.26214400.2, 8.0.104857600.2, 8.0.26214400.1, 8.0.104857600.1, 8.0.655360000.1, 8.0.655360000.2, 8.0.2621440000.7, 8.0.2621440000.6
Degree 16 siblings: 16.0.429496729600000000.4, 16.4.6871947673600000000.8, 16.0.43980465111040000.1, 16.0.27487790694400000000.3, 16.0.27487790694400000000.2, 16.0.27487790694400000000.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$