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Properties

Label 16.0.27848315074...3209.2
Degree $16$
Signature $[0, 8]$
Discriminant $13^{14}\cdot 29^{4}$
Root discriminant $21.89$
Ramified primes $13, 29$
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![9, -36, 313, -756, 1150, -1500, 1357, -709, 94, 231, -223, 103, -11, -17, 12, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 12*x^14 - 17*x^13 - 11*x^12 + 103*x^11 - 223*x^10 + 231*x^9 + 94*x^8 - 709*x^7 + 1357*x^6 - 1500*x^5 + 1150*x^4 - 756*x^3 + 313*x^2 - 36*x + 9)
 
gp: K = bnfinit(x^16 - 5*x^15 + 12*x^14 - 17*x^13 - 11*x^12 + 103*x^11 - 223*x^10 + 231*x^9 + 94*x^8 - 709*x^7 + 1357*x^6 - 1500*x^5 + 1150*x^4 - 756*x^3 + 313*x^2 - 36*x + 9, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 12 x^{14} - 17 x^{13} - 11 x^{12} + 103 x^{11} - 223 x^{10} + 231 x^{9} + 94 x^{8} - 709 x^{7} + 1357 x^{6} - 1500 x^{5} + 1150 x^{4} - 756 x^{3} + 313 x^{2} - 36 x + 9 \)

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:  \(2784831507453778823209=13^{14}\cdot 29^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.89$
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:  $13, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{6} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{9} - \frac{1}{6} a^{6} - \frac{1}{2} a^{5} + \frac{1}{6} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} + \frac{1}{6} a^{5} + \frac{1}{3} a^{4} - \frac{1}{6} a^{2} + \frac{1}{6} a$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{72} a^{12} + \frac{1}{72} a^{10} + \frac{1}{72} a^{9} - \frac{1}{12} a^{8} + \frac{1}{24} a^{7} + \frac{1}{36} a^{6} - \frac{1}{24} a^{5} + \frac{1}{12} a^{4} - \frac{11}{72} a^{3} - \frac{29}{72} a^{2} - \frac{5}{24} a - \frac{3}{8}$, $\frac{1}{72} a^{13} + \frac{1}{72} a^{11} + \frac{1}{72} a^{10} - \frac{1}{12} a^{9} + \frac{1}{24} a^{8} + \frac{1}{36} a^{7} - \frac{1}{24} a^{6} + \frac{1}{12} a^{5} - \frac{11}{72} a^{4} - \frac{29}{72} a^{3} - \frac{5}{24} a^{2} - \frac{3}{8} a$, $\frac{1}{216} a^{14} - \frac{1}{216} a^{13} + \frac{2}{27} a^{10} - \frac{1}{54} a^{9} - \frac{7}{216} a^{8} + \frac{7}{54} a^{7} - \frac{17}{216} a^{6} - \frac{37}{108} a^{5} + \frac{1}{3} a^{4} + \frac{73}{216} a^{3} + \frac{65}{216} a^{2} + \frac{1}{4} a + \frac{11}{24}$, $\frac{1}{351153576} a^{15} + \frac{79765}{87788394} a^{14} - \frac{285977}{87788394} a^{13} + \frac{289243}{117051192} a^{12} - \frac{27518435}{351153576} a^{11} + \frac{3135635}{175576788} a^{10} - \frac{415565}{14631399} a^{9} - \frac{9786341}{175576788} a^{8} - \frac{501164}{4877133} a^{7} + \frac{620566}{14631399} a^{6} + \frac{87635849}{351153576} a^{5} + \frac{48713}{43894197} a^{4} - \frac{32500327}{175576788} a^{3} - \frac{168536693}{351153576} a^{2} + \frac{1376397}{13005688} a - \frac{2125279}{4877133}$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 23909.8875954 \)
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

\(\Q(\sqrt{13}) \), 4.0.2197.1, 4.4.63713.1, 4.0.4901.1, 8.0.1819706993.1 x2, 8.4.52771502797.1 x2, 8.0.4059346369.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$13$13.8.7.2$x^{8} - 52$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
13.8.7.2$x^{8} - 52$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$