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Properties

Label 16.8.29327852876...5625.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{8}\cdot 29^{4}\cdot 101^{6}$
Root discriminant $29.29$
Ramified primes $5, 29, 101$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_4^2.C_2$ (as 16T376)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![464, -2320, 4128, -2754, -799, 2804, -1885, -274, 1158, -550, -144, 265, -93, -13, 20, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 20*x^14 - 13*x^13 - 93*x^12 + 265*x^11 - 144*x^10 - 550*x^9 + 1158*x^8 - 274*x^7 - 1885*x^6 + 2804*x^5 - 799*x^4 - 2754*x^3 + 4128*x^2 - 2320*x + 464)
 
gp: K = bnfinit(x^16 - 7*x^15 + 20*x^14 - 13*x^13 - 93*x^12 + 265*x^11 - 144*x^10 - 550*x^9 + 1158*x^8 - 274*x^7 - 1885*x^6 + 2804*x^5 - 799*x^4 - 2754*x^3 + 4128*x^2 - 2320*x + 464, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 20 x^{14} - 13 x^{13} - 93 x^{12} + 265 x^{11} - 144 x^{10} - 550 x^{9} + 1158 x^{8} - 274 x^{7} - 1885 x^{6} + 2804 x^{5} - 799 x^{4} - 2754 x^{3} + 4128 x^{2} - 2320 x + 464 \)

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:  \(293278528764541359765625=5^{8}\cdot 29^{4}\cdot 101^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.29$
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:  $5, 29, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{10} a^{11} + \frac{3}{10} a^{9} - \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{10} a^{6} + \frac{3}{10} a^{5} - \frac{1}{10} a^{3} + \frac{3}{10} a^{2} + \frac{1}{10} a$, $\frac{1}{10} a^{12} - \frac{1}{10} a^{10} - \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{3}{10} a^{7} - \frac{3}{10} a^{6} + \frac{2}{5} a^{5} - \frac{3}{10} a^{4} + \frac{3}{10} a^{3} - \frac{1}{2} a^{2} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{10} a^{13} - \frac{1}{2} a^{9} + \frac{1}{10} a^{8} - \frac{3}{10} a^{7} + \frac{1}{10} a^{6} - \frac{1}{5} a^{5} - \frac{1}{10} a^{4} + \frac{2}{5} a^{3} - \frac{1}{10} a^{2} + \frac{3}{10} a + \frac{1}{5}$, $\frac{1}{100} a^{14} - \frac{1}{100} a^{13} + \frac{1}{100} a^{11} + \frac{3}{100} a^{10} + \frac{49}{100} a^{9} - \frac{1}{50} a^{8} + \frac{3}{25} a^{7} + \frac{9}{50} a^{6} - \frac{11}{25} a^{5} - \frac{31}{100} a^{4} - \frac{23}{50} a^{3} + \frac{29}{100} a^{2} - \frac{23}{50} a - \frac{11}{25}$, $\frac{1}{3407202715252600} a^{15} + \frac{773432912097}{681440543050520} a^{14} + \frac{21383471568511}{851800678813150} a^{13} + \frac{98525097341491}{3407202715252600} a^{12} + \frac{94505340597919}{3407202715252600} a^{11} - \frac{74709869322443}{3407202715252600} a^{10} - \frac{152326171883891}{425900339406575} a^{9} + \frac{49178622416751}{340720271525260} a^{8} - \frac{22585644179007}{340720271525260} a^{7} + \frac{744241132534397}{1703601357626300} a^{6} - \frac{8222126533153}{681440543050520} a^{5} - \frac{123557100771873}{851800678813150} a^{4} + \frac{775389449570033}{3407202715252600} a^{3} + \frac{94936235870959}{1703601357626300} a^{2} - \frac{7041140380367}{170360135762630} a + \frac{2084437056332}{425900339406575}$

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:  \( 682887.099408 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4^2.C_2$ (as 16T376):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 20 conjugacy class representatives for $D_4^2.C_2$
Character table for $D_4^2.C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.73225.1, 4.4.2525.1, 4.4.725.1, 8.4.18674205625.2, 8.4.18674205625.1, 8.8.5361900625.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
$101$101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$