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Properties

Label 16.8.58189572765...5321.4
Degree $16$
Signature $[8, 4]$
Discriminant $13^{8}\cdot 61^{10}$
Root discriminant $47.08$
Ramified primes $13, 61$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4\wr C_2$ (as 16T42)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![169, 0, -4329, 0, 29604, 0, -26388, 0, 9162, 0, -1429, 0, 109, 0, -10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 10*x^14 + 109*x^12 - 1429*x^10 + 9162*x^8 - 26388*x^6 + 29604*x^4 - 4329*x^2 + 169)
 
gp: K = bnfinit(x^16 - 10*x^14 + 109*x^12 - 1429*x^10 + 9162*x^8 - 26388*x^6 + 29604*x^4 - 4329*x^2 + 169, 1)
 

Normalized defining polynomial

\( x^{16} - 10 x^{14} + 109 x^{12} - 1429 x^{10} + 9162 x^{8} - 26388 x^{6} + 29604 x^{4} - 4329 x^{2} + 169 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(581895727651002533052085321=13^{8}\cdot 61^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.08$
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:  $13, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{26} a^{10} - \frac{2}{13} a^{8} - \frac{1}{2} a^{7} + \frac{11}{26} a^{6} - \frac{1}{2} a^{5} + \frac{6}{13} a^{4} - \frac{1}{2} a^{3} + \frac{9}{26} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{26} a^{11} - \frac{2}{13} a^{9} + \frac{11}{26} a^{7} - \frac{1}{26} a^{5} - \frac{1}{2} a^{4} + \frac{9}{26} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{26} a^{12} - \frac{5}{26} a^{8} - \frac{9}{26} a^{6} - \frac{1}{2} a^{5} + \frac{5}{26} a^{4} - \frac{1}{2} a^{3} + \frac{5}{13} a^{2} - \frac{1}{2} a$, $\frac{1}{338} a^{13} + \frac{3}{169} a^{11} - \frac{3}{338} a^{9} - \frac{151}{338} a^{7} - \frac{1}{2} a^{6} - \frac{53}{338} a^{5} - \frac{1}{2} a^{4} + \frac{19}{169} a^{3} - \frac{1}{2} a^{2} + \frac{5}{13} a$, $\frac{1}{2026087934} a^{14} - \frac{11145950}{1013043967} a^{12} - \frac{30294319}{2026087934} a^{10} - \frac{392009665}{2026087934} a^{8} - \frac{1}{2} a^{7} - \frac{233624717}{2026087934} a^{6} - \frac{1}{2} a^{5} - \frac{176520273}{1013043967} a^{4} - \frac{1}{2} a^{3} - \frac{23278435}{77926459} a^{2} - \frac{577805}{5994343}$, $\frac{1}{2026087934} a^{15} + \frac{842736}{1013043967} a^{13} + \frac{17821727}{1013043967} a^{11} - \frac{152235945}{2026087934} a^{9} - \frac{329611535}{1013043967} a^{7} - \frac{266435418}{1013043967} a^{5} - \frac{395437305}{2026087934} a^{3} - \frac{1}{2} a^{2} - \frac{9028587}{155852918} a$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 27500759.1595 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\wr C_2$ (as 16T42):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_4\wr C_2$
Character table for $C_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{61}) \), \(\Q(\sqrt{793}) \), 4.4.10309.1 x2, 4.4.48373.1 x2, \(\Q(\sqrt{13}, \sqrt{61})\), 8.4.24122514952861.2, 8.4.6482804341.1, 8.8.395451064801.1

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 8 siblings: data not computed
Degree 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
61Data not computed