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Properties

Label 16.8.55810694533...0625.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{8}\cdot 13^{4}\cdot 29^{8}$
Root discriminant $22.86$
Ramified primes $5, 13, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\wr C_2^2$ (as 16T150)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, -28, -13, 66, 48, 14, 43, -19, -91, -19, 26, -9, 6, -7, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 7*x^14 + 6*x^13 - 9*x^12 + 26*x^11 - 19*x^10 - 91*x^9 - 19*x^8 + 43*x^7 + 14*x^6 + 48*x^5 + 66*x^4 - 13*x^3 - 28*x^2 - x + 1)
 
gp: K = bnfinit(x^16 - 2*x^15 - 7*x^14 + 6*x^13 - 9*x^12 + 26*x^11 - 19*x^10 - 91*x^9 - 19*x^8 + 43*x^7 + 14*x^6 + 48*x^5 + 66*x^4 - 13*x^3 - 28*x^2 - x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 7 x^{14} + 6 x^{13} - 9 x^{12} + 26 x^{11} - 19 x^{10} - 91 x^{9} - 19 x^{8} + 43 x^{7} + 14 x^{6} + 48 x^{5} + 66 x^{4} - 13 x^{3} - 28 x^{2} - x + 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5581069453351219140625=5^{8}\cdot 13^{4}\cdot 29^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.86$
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:  $5, 13, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{11} - \frac{3}{8} a^{9} + \frac{1}{4} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{3}{8} a^{5} + \frac{3}{8} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a + \frac{3}{8}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{3}{8} a^{10} - \frac{1}{8} a^{9} + \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{3}{8} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{8} a + \frac{3}{8}$, $\frac{1}{208} a^{14} - \frac{1}{26} a^{13} + \frac{3}{104} a^{12} + \frac{21}{104} a^{11} + \frac{23}{208} a^{10} - \frac{25}{52} a^{9} - \frac{1}{13} a^{8} - \frac{57}{208} a^{7} + \frac{103}{208} a^{6} - \frac{5}{104} a^{5} - \frac{73}{208} a^{4} - \frac{37}{104} a^{3} - \frac{29}{208} a^{2} + \frac{99}{208} a + \frac{81}{208}$, $\frac{1}{41476852768} a^{15} + \frac{76404359}{41476852768} a^{14} + \frac{55146509}{2592303298} a^{13} - \frac{471163451}{20738426384} a^{12} - \frac{6364987619}{41476852768} a^{11} - \frac{936020081}{41476852768} a^{10} - \frac{95309104}{1296151649} a^{9} + \frac{9717674845}{41476852768} a^{8} + \frac{9402544599}{20738426384} a^{7} + \frac{11524941405}{41476852768} a^{6} - \frac{17670716409}{41476852768} a^{5} - \frac{16815141989}{41476852768} a^{4} - \frac{2944868267}{41476852768} a^{3} + \frac{224112597}{2592303298} a^{2} + \frac{1398018743}{10369213192} a + \frac{1251595531}{3190527136}$

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:  \( 67781.2302308 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_2^2$ (as 16T150):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 16 conjugacy class representatives for $C_2\wr C_2^2$
Character table for $C_2\wr C_2^2$

Intermediate fields

\(\Q(\sqrt{29}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), \(\Q(\sqrt{5}, \sqrt{29})\), 4.4.4205.1 x2, 4.4.725.1 x2, 8.8.442050625.1, 8.4.88830625.1, 8.4.74706555625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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

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$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$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}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$29.4.2.1$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}$
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}$