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Properties

Label 16.16.3482009712...9824.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{40}\cdot 3^{8}\cdot 13^{6}$
Root discriminant $25.64$
Ramified primes $2, 3, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_4\wr C_2$ (as 16T111)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 20, -8, -172, 22, 560, -40, -884, 72, 716, -88, -292, 50, 56, -12, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 12*x^14 + 56*x^13 + 50*x^12 - 292*x^11 - 88*x^10 + 716*x^9 + 72*x^8 - 884*x^7 - 40*x^6 + 560*x^5 + 22*x^4 - 172*x^3 - 8*x^2 + 20*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 - 12*x^14 + 56*x^13 + 50*x^12 - 292*x^11 - 88*x^10 + 716*x^9 + 72*x^8 - 884*x^7 - 40*x^6 + 560*x^5 + 22*x^4 - 172*x^3 - 8*x^2 + 20*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 12 x^{14} + 56 x^{13} + 50 x^{12} - 292 x^{11} - 88 x^{10} + 716 x^{9} + 72 x^{8} - 884 x^{7} - 40 x^{6} + 560 x^{5} + 22 x^{4} - 172 x^{3} - 8 x^{2} + 20 x + 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(34820097123453788749824=2^{40}\cdot 3^{8}\cdot 13^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.64$
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:  $2, 3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{29} a^{14} - \frac{8}{29} a^{13} - \frac{1}{29} a^{12} - \frac{4}{29} a^{11} - \frac{5}{29} a^{9} - \frac{10}{29} a^{8} - \frac{9}{29} a^{7} - \frac{1}{29} a^{6} + \frac{5}{29} a^{5} - \frac{10}{29} a^{4} + \frac{2}{29} a^{3} - \frac{8}{29} a^{2} - \frac{8}{29} a - \frac{11}{29}$, $\frac{1}{22649} a^{15} - \frac{292}{22649} a^{14} - \frac{450}{2059} a^{13} + \frac{8922}{22649} a^{12} + \frac{10938}{22649} a^{11} - \frac{8473}{22649} a^{10} - \frac{5956}{22649} a^{9} + \frac{869}{2059} a^{8} - \frac{5362}{22649} a^{7} - \frac{10818}{22649} a^{6} + \frac{11069}{22649} a^{5} - \frac{298}{781} a^{4} + \frac{4557}{22649} a^{3} + \frac{4178}{22649} a^{2} + \frac{7278}{22649} a - \frac{2183}{22649}$

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:  $15$
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:  \( 1510731.82986 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_4\wr C_2$ (as 16T111):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $C_2\times C_4\wr C_2$
Character table for $C_2\times C_4\wr C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), 4.4.29952.1, 4.4.7488.1, \(\Q(\sqrt{2}, \sqrt{3})\), 8.8.46650359808.1, 8.8.11662589952.1, 8.8.3588489216.1

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

Sibling fields

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 R R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$