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Properties

Label 16.0.25405371337890625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 101^{4}$
Root discriminant $10.60$
Ramified primes $5, 101$
Class number $1$
Class group Trivial
Galois group $C_2\wr C_4$ (as 16T158)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 11, -23, 47, -69, 97, -108, 121, -108, 97, -69, 47, -23, 11, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 11*x^14 - 23*x^13 + 47*x^12 - 69*x^11 + 97*x^10 - 108*x^9 + 121*x^8 - 108*x^7 + 97*x^6 - 69*x^5 + 47*x^4 - 23*x^3 + 11*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 + 11*x^14 - 23*x^13 + 47*x^12 - 69*x^11 + 97*x^10 - 108*x^9 + 121*x^8 - 108*x^7 + 97*x^6 - 69*x^5 + 47*x^4 - 23*x^3 + 11*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 11 x^{14} - 23 x^{13} + 47 x^{12} - 69 x^{11} + 97 x^{10} - 108 x^{9} + 121 x^{8} - 108 x^{7} + 97 x^{6} - 69 x^{5} + 47 x^{4} - 23 x^{3} + 11 x^{2} - 4 x + 1 \)

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:  \(25405371337890625=5^{12}\cdot 101^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $10.60$
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:  $5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{9} + \frac{1}{5} a^{6} + \frac{1}{5} a^{3} + \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{10} + \frac{1}{5} a^{7} + \frac{1}{5} a^{4} + \frac{1}{5} a$, $\frac{1}{5} a^{14} + \frac{1}{5} a^{11} + \frac{1}{5} a^{8} + \frac{1}{5} a^{5} + \frac{1}{5} a^{2}$, $\frac{1}{145} a^{15} - \frac{1}{29} a^{14} - \frac{13}{145} a^{13} - \frac{2}{29} a^{12} - \frac{6}{29} a^{11} - \frac{68}{145} a^{10} + \frac{4}{29} a^{9} - \frac{14}{29} a^{8} + \frac{17}{145} a^{7} + \frac{4}{29} a^{6} - \frac{2}{29} a^{5} + \frac{57}{145} a^{4} - \frac{2}{29} a^{3} + \frac{9}{29} a^{2} - \frac{63}{145} a + \frac{59}{145}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{1029}{145} a^{15} + \frac{2709}{145} a^{14} - \frac{1257}{29} a^{13} + \frac{11276}{145} a^{12} - \frac{23911}{145} a^{11} + \frac{4320}{29} a^{10} - \frac{35399}{145} a^{9} + \frac{26674}{145} a^{8} - \frac{6677}{29} a^{7} + \frac{18831}{145} a^{6} - \frac{19696}{145} a^{5} + \frac{1621}{29} a^{4} - \frac{3514}{145} a^{3} + \frac{1719}{145} a^{2} - \frac{102}{29} a - \frac{26}{29} \) (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:  \( 118.594048145 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T158):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.4.2525.1, 4.0.12625.1, 8.0.1578125.1 x2, 8.4.31878125.1 x2, 8.0.159390625.1

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

Sibling fields

Degree 8 siblings: data not computed
Degree 16 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$101$$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$