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Properties

Label 16.0.20723941878...0625.3
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 13^{8}\cdot 101^{4}$
Root discriminant $38.22$
Ramified primes $5, 13, 101$
Class number $16$ (GRH)
Class group $[2, 8]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T217)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![20341, 24330, 19072, 10922, 1467, -4242, -3236, 1166, 2370, 194, -468, -204, 72, 38, -8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 8*x^14 + 38*x^13 + 72*x^12 - 204*x^11 - 468*x^10 + 194*x^9 + 2370*x^8 + 1166*x^7 - 3236*x^6 - 4242*x^5 + 1467*x^4 + 10922*x^3 + 19072*x^2 + 24330*x + 20341)
 
gp: K = bnfinit(x^16 - 4*x^15 - 8*x^14 + 38*x^13 + 72*x^12 - 204*x^11 - 468*x^10 + 194*x^9 + 2370*x^8 + 1166*x^7 - 3236*x^6 - 4242*x^5 + 1467*x^4 + 10922*x^3 + 19072*x^2 + 24330*x + 20341, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 8 x^{14} + 38 x^{13} + 72 x^{12} - 204 x^{11} - 468 x^{10} + 194 x^{9} + 2370 x^{8} + 1166 x^{7} - 3236 x^{6} - 4242 x^{5} + 1467 x^{4} + 10922 x^{3} + 19072 x^{2} + 24330 x + 20341 \)

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:  \(20723941878730254150390625=5^{12}\cdot 13^{8}\cdot 101^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.22$
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:  $5, 13, 101$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{12} a^{11} + \frac{1}{12} a^{10} - \frac{1}{6} a^{7} + \frac{1}{12} a^{5} - \frac{1}{12} a^{4} - \frac{5}{12} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{24} a^{12} + \frac{1}{12} a^{10} - \frac{1}{8} a^{9} + \frac{1}{24} a^{8} - \frac{1}{6} a^{7} - \frac{5}{24} a^{6} - \frac{1}{12} a^{5} + \frac{5}{24} a^{4} - \frac{1}{24} a^{3} - \frac{3}{8} a^{2} + \frac{5}{24} a - \frac{5}{24}$, $\frac{1}{24} a^{13} + \frac{1}{24} a^{10} + \frac{1}{24} a^{9} + \frac{1}{12} a^{8} - \frac{1}{24} a^{7} - \frac{1}{12} a^{6} + \frac{1}{8} a^{5} - \frac{5}{24} a^{4} - \frac{11}{24} a^{3} - \frac{1}{24} a^{2} - \frac{5}{24} a + \frac{1}{12}$, $\frac{1}{32328} a^{14} + \frac{35}{4041} a^{13} + \frac{185}{16164} a^{12} - \frac{211}{32328} a^{11} - \frac{3385}{32328} a^{10} + \frac{49}{898} a^{9} + \frac{2387}{32328} a^{8} - \frac{233}{1796} a^{7} + \frac{2777}{32328} a^{6} - \frac{1451}{10776} a^{5} + \frac{1891}{10776} a^{4} - \frac{4817}{10776} a^{3} - \frac{3013}{10776} a^{2} + \frac{46}{4041} a - \frac{1754}{4041}$, $\frac{1}{28301705576542970623944} a^{15} + \frac{74687526487457585}{7075426394135742655986} a^{14} + \frac{273039179652670989019}{14150852788271485311972} a^{13} - \frac{74999443909045283531}{9433901858847656874648} a^{12} + \frac{368309373295886970409}{28301705576542970623944} a^{11} - \frac{16285250749160987173}{7075426394135742655986} a^{10} - \frac{860770094354843402227}{28301705576542970623944} a^{9} - \frac{393866455542675942725}{3537713197067871327993} a^{8} - \frac{1069721257134635488387}{28301705576542970623944} a^{7} - \frac{5665753369852753706827}{28301705576542970623944} a^{6} - \frac{653657030895806022277}{3144633952949218958216} a^{5} - \frac{677219092023457036063}{9433901858847656874648} a^{4} + \frac{508246911080280955321}{3144633952949218958216} a^{3} - \frac{956578096084926898817}{14150852788271485311972} a^{2} + \frac{3242370816697995500753}{14150852788271485311972} a - \frac{5802038166926715801329}{14150852788271485311972}$

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

Class group and class number

$C_{2}\times C_{8}$, which has order $16$ (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:  $7$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 79444.6215571 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T217):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{5}, \sqrt{13})\), 8.0.1802913125.1, 8.0.4552355640625.8, 8.8.45072828125.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$13$13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$101$101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$