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Properties

Label 16.0.19360527043...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{38}\cdot 5^{8}\cdot 89^{4}\cdot 130201^{4}$
Root discriminant $676.76$
Ramified primes $2, 5, 89, 130201$
Class number $259312$ (GRH)
Class group $[2, 129656]$ (GRH)
Galois group 16T1605

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![432808725924, 0, 50079521052, 0, 5837598720, 0, -27766746, 0, 1792727, 0, 13224, 0, 2637, 0, -20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 20*x^14 + 2637*x^12 + 13224*x^10 + 1792727*x^8 - 27766746*x^6 + 5837598720*x^4 + 50079521052*x^2 + 432808725924)
 
gp: K = bnfinit(x^16 - 20*x^14 + 2637*x^12 + 13224*x^10 + 1792727*x^8 - 27766746*x^6 + 5837598720*x^4 + 50079521052*x^2 + 432808725924, 1)
 

Normalized defining polynomial

\( x^{16} - 20 x^{14} + 2637 x^{12} + 13224 x^{10} + 1792727 x^{8} - 27766746 x^{6} + 5837598720 x^{4} + 50079521052 x^{2} + 432808725924 \)

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:  \(1936052704382448567172058182790309478400000000=2^{38}\cdot 5^{8}\cdot 89^{4}\cdot 130201^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $676.76$
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:  $2, 5, 89, 130201$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a$, $\frac{1}{9} a^{10} - \frac{2}{9} a^{8} + \frac{1}{3} a^{4} - \frac{1}{9} a^{2}$, $\frac{1}{27} a^{11} - \frac{2}{27} a^{9} + \frac{1}{3} a^{7} - \frac{2}{9} a^{5} + \frac{8}{27} a^{3} + \frac{1}{3} a$, $\frac{1}{162} a^{12} - \frac{1}{81} a^{10} + \frac{1}{18} a^{8} - \frac{10}{27} a^{6} - \frac{73}{162} a^{4} + \frac{2}{9} a^{2}$, $\frac{1}{15066} a^{13} - \frac{64}{7533} a^{11} + \frac{5}{54} a^{9} - \frac{307}{2511} a^{7} + \frac{2951}{15066} a^{5} - \frac{5}{31} a^{3} - \frac{6}{31} a$, $\frac{1}{21000980731084342579618643175333516} a^{14} - \frac{24308317890615637993007114408647}{10500490365542171289809321587666758} a^{12} - \frac{56725651266548533987649615777}{1475928085676037850841144365404} a^{10} - \frac{1203373432874796640980415981079689}{3500163455180723763269773862555586} a^{8} + \frac{6862810873719726306438919658220653}{21000980731084342579618643175333516} a^{6} - \frac{195545335515541475067834491356321}{583360575863453960544962310425931} a^{4} + \frac{58129727707510882822873644655799}{129635683525211991232213846761318} a^{2} + \frac{31821541120143970533404933906}{232322013486042995039809761221}$, $\frac{1}{8253385427316146633790126767906071788} a^{15} + \frac{9146052051374553292725491207177}{4126692713658073316895063383953035894} a^{13} - \frac{793311755312446852835886124328893}{53943695603373507410392985411150796} a^{11} - \frac{91818716216997978512297894867240971}{1375564237886024438965021127984345298} a^{9} + \frac{3348778732636344387246876124499555177}{8253385427316146633790126767906071788} a^{7} - \frac{65297516920614967674108445209988604}{229260706314337406494170187997390883} a^{5} + \frac{4801680599065536755545709349266177}{50946823625408312554260041777197974} a^{3} + \frac{1316858352159671986992018040506830}{2830379090300461808570002320955443} a$

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

Class group and class number

$C_{2}\times C_{129656}$, which has order $259312$ (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:  \( 52031774306.7 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1605:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 124 conjugacy class representatives for t16n1605 are not computed
Character table for t16n1605 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2225.1, 8.0.85938669744845440000.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.8.18.47$x^{8} + 4 x^{7} + 2 x^{6} + 8 x^{5} + 2 x^{4} + 28$$4$$2$$18$$(C_4^2 : C_2):C_2$$[2, 2, 3, 7/2, 7/2]^{2}$
2.8.20.37$x^{8} + 14 x^{4} + 8 x^{3} + 20$$4$$2$$20$$(C_4^2 : C_2):C_2$$[2, 2, 3, 7/2, 7/2]^{2}$
5Data not computed
$89$89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
130201Data not computed