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Properties

Label 16.0.17606641095...625.13
Degree $16$
Signature $[0, 8]$
Discriminant $5^{6}\cdot 101^{12}$
Root discriminant $58.26$
Ramified primes $5, 101$
Class number $50$ (GRH)
Class group $[5, 10]$ (GRH)
Galois group $D_4:C_4$ (as 16T26)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1171175, 1532951, 1487871, 329316, 145099, -11405, 106205, 10737, 15052, -3367, 1205, -442, 47, -73, 3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 3*x^14 - 73*x^13 + 47*x^12 - 442*x^11 + 1205*x^10 - 3367*x^9 + 15052*x^8 + 10737*x^7 + 106205*x^6 - 11405*x^5 + 145099*x^4 + 329316*x^3 + 1487871*x^2 + 1532951*x + 1171175)
 
gp: K = bnfinit(x^16 + 3*x^14 - 73*x^13 + 47*x^12 - 442*x^11 + 1205*x^10 - 3367*x^9 + 15052*x^8 + 10737*x^7 + 106205*x^6 - 11405*x^5 + 145099*x^4 + 329316*x^3 + 1487871*x^2 + 1532951*x + 1171175, 1)
 

Normalized defining polynomial

\( x^{16} + 3 x^{14} - 73 x^{13} + 47 x^{12} - 442 x^{11} + 1205 x^{10} - 3367 x^{9} + 15052 x^{8} + 10737 x^{7} + 106205 x^{6} - 11405 x^{5} + 145099 x^{4} + 329316 x^{3} + 1487871 x^{2} + 1532951 x + 1171175 \)

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:  \(17606641095812026885331265625=5^{6}\cdot 101^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.26$
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:  $5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5} a^{6} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{5} a^{7} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{5} a^{8} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{25} a^{9} - \frac{1}{25} a^{6} - \frac{12}{25} a^{5} + \frac{7}{25} a^{4} + \frac{3}{25} a^{3} - \frac{7}{25} a^{2} + \frac{9}{25} a$, $\frac{1}{25} a^{10} - \frac{1}{25} a^{7} - \frac{2}{25} a^{6} + \frac{7}{25} a^{5} + \frac{8}{25} a^{4} - \frac{12}{25} a^{3} + \frac{4}{25} a^{2} - \frac{1}{5} a$, $\frac{1}{25} a^{11} - \frac{1}{25} a^{8} - \frac{2}{25} a^{7} + \frac{2}{25} a^{6} + \frac{8}{25} a^{5} - \frac{2}{25} a^{4} - \frac{6}{25} a^{3} + \frac{2}{5} a^{2} - \frac{2}{5} a$, $\frac{1}{3250} a^{12} + \frac{43}{3250} a^{11} - \frac{11}{650} a^{10} + \frac{6}{325} a^{9} + \frac{11}{325} a^{8} + \frac{11}{125} a^{7} + \frac{59}{1625} a^{6} + \frac{41}{650} a^{5} - \frac{47}{650} a^{4} - \frac{79}{650} a^{3} - \frac{776}{1625} a^{2} - \frac{768}{1625} a - \frac{27}{130}$, $\frac{1}{3250} a^{13} + \frac{23}{1625} a^{11} - \frac{9}{650} a^{10} + \frac{53}{1625} a^{8} + \frac{4}{325} a^{7} + \frac{201}{3250} a^{6} + \frac{57}{325} a^{5} + \frac{74}{325} a^{4} + \frac{483}{3250} a^{3} + \frac{98}{325} a^{2} - \frac{147}{3250} a - \frac{9}{130}$, $\frac{1}{21823750} a^{14} - \frac{823}{10911875} a^{13} - \frac{1149}{21823750} a^{12} - \frac{187008}{10911875} a^{11} - \frac{85167}{4364750} a^{10} - \frac{185297}{10911875} a^{9} - \frac{179983}{10911875} a^{8} - \frac{71489}{21823750} a^{7} + \frac{1086717}{10911875} a^{6} - \frac{36633}{872950} a^{5} - \frac{1109591}{10911875} a^{4} + \frac{5088067}{21823750} a^{3} - \frac{593771}{1283750} a^{2} - \frac{330311}{1678750} a - \frac{4349}{11050}$, $\frac{1}{12970911491050700651311250} a^{15} + \frac{3153327978888101}{762994793591217685371250} a^{14} + \frac{887593387358584137414}{6485455745525350325655625} a^{13} - \frac{11692645245049420548}{498881211194257717358125} a^{12} + \frac{71460901124162436751486}{6485455745525350325655625} a^{11} - \frac{247652927506622531626999}{12970911491050700651311250} a^{10} + \frac{70489800390148592867506}{6485455745525350325655625} a^{9} - \frac{99956499642426825623747}{12970911491050700651311250} a^{8} + \frac{1251601470167519255283607}{12970911491050700651311250} a^{7} - \frac{586538094629128539889834}{6485455745525350325655625} a^{6} - \frac{43044592109139400207267}{281976336761971753289375} a^{5} + \frac{2318951547991304971737363}{6485455745525350325655625} a^{4} - \frac{1574322735052770939630193}{6485455745525350325655625} a^{3} + \frac{84317823361011686002931}{498881211194257717358125} a^{2} - \frac{377536483478065621254289}{12970911491050700651311250} a - \frac{1355462684537925944927}{6567550122050987671550}$

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

Class group and class number

$C_{5}\times C_{10}$, which has order $50$ (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:  \( 4045208.26311 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4:C_4$ (as 16T26):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $D_4:C_4$
Character table for $D_4:C_4$

Intermediate fields

\(\Q(\sqrt{101}) \), 4.4.51005.1, 4.0.1030301.1, 4.0.5151505.1, 8.0.13007550125.1, 8.8.132690018825125.1, 8.0.26538003765025.1

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

Sibling fields

Galois closure: data not computed
Degree 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
$101$101.8.6.1$x^{8} - 707 x^{4} + 826281$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
101.8.6.1$x^{8} - 707 x^{4} + 826281$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$