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Properties

Label 16.0.20315598351...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 13^{8}\cdot 101^{2}$
Root discriminant $21.47$
Ramified primes $5, 13, 101$
Class number $1$
Class group Trivial
Galois group $C_2^4.C_2^3$ (as 16T208)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, -405, 792, -687, 232, -458, 1449, -1816, 1030, -209, 2, -16, 7, -8, 12, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 12*x^14 - 8*x^13 + 7*x^12 - 16*x^11 + 2*x^10 - 209*x^9 + 1030*x^8 - 1816*x^7 + 1449*x^6 - 458*x^5 + 232*x^4 - 687*x^3 + 792*x^2 - 405*x + 81)
 
gp: K = bnfinit(x^16 - 6*x^15 + 12*x^14 - 8*x^13 + 7*x^12 - 16*x^11 + 2*x^10 - 209*x^9 + 1030*x^8 - 1816*x^7 + 1449*x^6 - 458*x^5 + 232*x^4 - 687*x^3 + 792*x^2 - 405*x + 81, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 12 x^{14} - 8 x^{13} + 7 x^{12} - 16 x^{11} + 2 x^{10} - 209 x^{9} + 1030 x^{8} - 1816 x^{7} + 1449 x^{6} - 458 x^{5} + 232 x^{4} - 687 x^{3} + 792 x^{2} - 405 x + 81 \)

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:  \(2031559835185791015625=5^{12}\cdot 13^{8}\cdot 101^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{6} a^{13} + \frac{1}{6} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{6} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{6} a^{4} - \frac{1}{3} a^{2} + \frac{1}{6} a$, $\frac{1}{18} a^{14} + \frac{1}{6} a^{12} + \frac{1}{18} a^{11} + \frac{2}{9} a^{10} - \frac{1}{18} a^{9} + \frac{5}{18} a^{8} - \frac{4}{9} a^{7} + \frac{1}{18} a^{6} - \frac{1}{18} a^{5} - \frac{1}{3} a^{4} + \frac{1}{18} a^{3} - \frac{5}{18} a^{2} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{1358136008018694} a^{15} - \frac{417994704632}{25150666815161} a^{14} - \frac{11304058826197}{226356001336449} a^{13} - \frac{133469372145253}{679068004009347} a^{12} - \frac{201248430430966}{679068004009347} a^{11} + \frac{154468326870823}{679068004009347} a^{10} + \frac{205589335629337}{679068004009347} a^{9} - \frac{38339858147791}{679068004009347} a^{8} + \frac{75916276432718}{679068004009347} a^{7} + \frac{32603195210152}{679068004009347} a^{6} - \frac{14342588826868}{226356001336449} a^{5} - \frac{174548640066619}{679068004009347} a^{4} + \frac{72584189917193}{679068004009347} a^{3} + \frac{12669553122461}{226356001336449} a^{2} - \frac{5865872317984}{75452000445483} a + \frac{6081932414061}{50301333630322}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{422734426708763}{1358136008018694} a^{15} - \frac{354274381307000}{226356001336449} a^{14} + \frac{503029563369511}{226356001336449} a^{13} - \frac{504634900039963}{1358136008018694} a^{12} + \frac{1274046699607054}{679068004009347} a^{11} - \frac{2149527094891270}{679068004009347} a^{10} - \frac{3266771579462129}{1358136008018694} a^{9} - \frac{45821514269369216}{679068004009347} a^{8} + \frac{173135003867276494}{679068004009347} a^{7} - \frac{433590289972859705}{1358136008018694} a^{6} + \frac{11128375348627619}{75452000445483} a^{5} - \frac{5142468494110616}{679068004009347} a^{4} + \frac{93150480918289721}{1358136008018694} a^{3} - \frac{33337419127831705}{226356001336449} a^{2} + \frac{7937655724604092}{75452000445483} a - \frac{696780724936276}{25150666815161} \) (order $10$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 39722.3107785 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T208):

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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{65}) \), \(\Q(\zeta_{5})\), 4.0.21125.1, \(\Q(\sqrt{5}, \sqrt{13})\), 8.8.45072828125.1, 8.0.446265625.1, 8.0.1802913125.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ 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 ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$5.8.6.1$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$$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
101.2.0.1$x^{2} - x + 3$$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}$