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Properties

Label 16.0.19244198650...721.18
Degree $16$
Signature $[0, 8]$
Discriminant $31^{8}\cdot 41^{12}$
Root discriminant $90.21$
Ramified primes $31, 41$
Class number $64$ (GRH)
Class group $[2, 2, 16]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T172)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![26924347, -12569079, 39990424, -14162372, 14735693, -4148370, 1470441, -160713, 39799, 5649, 7028, 3369, 80, -71, -27, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 27*x^14 - 71*x^13 + 80*x^12 + 3369*x^11 + 7028*x^10 + 5649*x^9 + 39799*x^8 - 160713*x^7 + 1470441*x^6 - 4148370*x^5 + 14735693*x^4 - 14162372*x^3 + 39990424*x^2 - 12569079*x + 26924347)
 
gp: K = bnfinit(x^16 - 2*x^15 - 27*x^14 - 71*x^13 + 80*x^12 + 3369*x^11 + 7028*x^10 + 5649*x^9 + 39799*x^8 - 160713*x^7 + 1470441*x^6 - 4148370*x^5 + 14735693*x^4 - 14162372*x^3 + 39990424*x^2 - 12569079*x + 26924347, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 27 x^{14} - 71 x^{13} + 80 x^{12} + 3369 x^{11} + 7028 x^{10} + 5649 x^{9} + 39799 x^{8} - 160713 x^{7} + 1470441 x^{6} - 4148370 x^{5} + 14735693 x^{4} - 14162372 x^{3} + 39990424 x^{2} - 12569079 x + 26924347 \)

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:  \(19244198650569257148820544058721=31^{8}\cdot 41^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $90.21$
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:  $31, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{10} a^{10} + \frac{1}{10} a^{6} + \frac{1}{5} a^{5} + \frac{1}{10} a^{4} + \frac{1}{10} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5}$, $\frac{1}{10} a^{11} + \frac{1}{10} a^{7} + \frac{1}{5} a^{6} + \frac{1}{10} a^{5} + \frac{1}{10} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a$, $\frac{1}{20} a^{12} - \frac{1}{20} a^{10} + \frac{1}{20} a^{8} + \frac{1}{10} a^{7} - \frac{1}{4} a^{6} + \frac{1}{5} a^{5} - \frac{1}{2} a^{4} + \frac{1}{5} a^{3} - \frac{1}{10} a^{2} - \frac{9}{20}$, $\frac{1}{20} a^{13} - \frac{1}{20} a^{11} + \frac{1}{20} a^{9} + \frac{1}{10} a^{8} - \frac{1}{4} a^{7} + \frac{1}{5} a^{6} - \frac{1}{2} a^{5} + \frac{1}{5} a^{4} - \frac{1}{10} a^{3} - \frac{9}{20} a$, $\frac{1}{100} a^{14} + \frac{1}{100} a^{13} + \frac{3}{100} a^{11} + \frac{1}{25} a^{10} - \frac{7}{100} a^{9} - \frac{11}{50} a^{8} + \frac{3}{20} a^{7} - \frac{19}{100} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{3}{10} a^{3} + \frac{3}{100} a^{2} - \frac{3}{100} a - \frac{3}{100}$, $\frac{1}{22041046042676974819724317555873481310613892913947500} a^{15} - \frac{17167722058421390698221810870285194570367363425362}{5510261510669243704931079388968370327653473228486875} a^{14} + \frac{439724460442770485251753015944179137455482469517281}{22041046042676974819724317555873481310613892913947500} a^{13} + \frac{377729659592158861120436744846191857226308852680353}{22041046042676974819724317555873481310613892913947500} a^{12} + \frac{842770018857529293878762084471191041044914988527267}{22041046042676974819724317555873481310613892913947500} a^{11} - \frac{860817076340860607606609538503680806704459145674463}{22041046042676974819724317555873481310613892913947500} a^{10} - \frac{2040823537845537518524725457416498546261265766191849}{22041046042676974819724317555873481310613892913947500} a^{9} + \frac{3996855768359285100011654661963508965391199053653803}{22041046042676974819724317555873481310613892913947500} a^{8} + \frac{281492379384874549675408049003028571698177496215893}{11020523021338487409862158777936740655306946456973750} a^{7} + \frac{4716102026072352652066055084181555004067459898418231}{22041046042676974819724317555873481310613892913947500} a^{6} + \frac{126349349875160481371157778211401736848086984574927}{1102052302133848740986215877793674065530694645697375} a^{5} + \frac{196632783848174713448832373337646478629945198691477}{1102052302133848740986215877793674065530694645697375} a^{4} + \frac{48226858393049285657287307118818642664039290592649}{111883482450136928018905165258241021881288796517500} a^{3} + \frac{865245943019461920114399615561958641289982760721319}{2204104604267697481972431755587348131061389291394750} a^{2} + \frac{2586083276801199167951045212023984296488421474503546}{5510261510669243704931079388968370327653473228486875} a - \frac{8164701460023313076058330070713457434381364693469393}{22041046042676974819724317555873481310613892913947500}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{16}$, which has order $64$ (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:  \( 17590199.4078 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T172):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{41}) \), 4.2.52111.1, 4.0.66233081.2, 4.2.2136551.1, 8.4.111337809161.1, 8.0.106995634603721.2, 8.0.4386821018752561.4

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$31$31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$41$41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$