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Properties

Label 16.0.52155530347...1509.1
Degree $16$
Signature $[0, 8]$
Discriminant $149^{9}\cdot 331^{8}$
Root discriminant $303.62$
Ramified primes $149, 331$
Class number $1200$ (GRH)
Class group $[2, 2, 300]$ (GRH)
Galois group 16T1675

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![153463978421, 5295191495, 20404848082, 1301275602, 1059195894, 74545252, 41589315, -1059869, 79776, -107121, -25164, 3357, 1429, 145, -32, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 32*x^14 + 145*x^13 + 1429*x^12 + 3357*x^11 - 25164*x^10 - 107121*x^9 + 79776*x^8 - 1059869*x^7 + 41589315*x^6 + 74545252*x^5 + 1059195894*x^4 + 1301275602*x^3 + 20404848082*x^2 + 5295191495*x + 153463978421)
 
gp: K = bnfinit(x^16 - 4*x^15 - 32*x^14 + 145*x^13 + 1429*x^12 + 3357*x^11 - 25164*x^10 - 107121*x^9 + 79776*x^8 - 1059869*x^7 + 41589315*x^6 + 74545252*x^5 + 1059195894*x^4 + 1301275602*x^3 + 20404848082*x^2 + 5295191495*x + 153463978421, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 32 x^{14} + 145 x^{13} + 1429 x^{12} + 3357 x^{11} - 25164 x^{10} - 107121 x^{9} + 79776 x^{8} - 1059869 x^{7} + 41589315 x^{6} + 74545252 x^{5} + 1059195894 x^{4} + 1301275602 x^{3} + 20404848082 x^{2} + 5295191495 x + 153463978421 \)

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:  \(5215553034727934460124227765005778951509=149^{9}\cdot 331^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $303.62$
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:  $149, 331$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{46} a^{13} - \frac{11}{46} a^{12} - \frac{11}{23} a^{11} + \frac{9}{23} a^{10} + \frac{7}{23} a^{9} - \frac{1}{2} a^{8} - \frac{13}{46} a^{7} - \frac{3}{46} a^{6} - \frac{8}{23} a^{5} + \frac{17}{46} a^{4} - \frac{8}{23} a^{3} + \frac{5}{23} a^{2} - \frac{11}{46} a + \frac{11}{46}$, $\frac{1}{46} a^{14} - \frac{5}{46} a^{12} + \frac{3}{23} a^{11} - \frac{9}{23} a^{10} - \frac{7}{46} a^{9} + \frac{5}{23} a^{8} - \frac{4}{23} a^{7} - \frac{3}{46} a^{6} - \frac{21}{46} a^{5} - \frac{13}{46} a^{4} + \frac{9}{23} a^{3} + \frac{7}{46} a^{2} - \frac{9}{23} a - \frac{17}{46}$, $\frac{1}{1028725612755685508784449526630273174900915666043941173811831029501926670} a^{15} + \frac{9663783906830042464304575938513654073693634547917248513915352765038149}{1028725612755685508784449526630273174900915666043941173811831029501926670} a^{14} + \frac{14521455759145693188399499092900003488099729826516261973379640190080}{7913273944274503913726534820232870576160889738799547490860238688476359} a^{13} - \frac{14922309051319518087237358652847171165027214868403596586332196358363642}{102872561275568550878444952663027317490091566604394117381183102950192667} a^{12} - \frac{132283328896534624392923703930607312251371457189320584716511067094986008}{514362806377842754392224763315136587450457833021970586905915514750963335} a^{11} + \frac{33249790091526042926430317531675599058392548572583776902397231186719003}{79132739442745039137265348202328705761608897387995474908602386884763590} a^{10} - \frac{287037440976444507462978438167751712301377001708488676834716728769038717}{1028725612755685508784449526630273174900915666043941173811831029501926670} a^{9} - \frac{194901187452273478245811654352609541578668341821946756952060526674253}{538881934392711109892325577071908420587174261940252055427884248036630} a^{8} - \frac{7677183477326603214387726860410273502826627536887929764423571257007921}{102872561275568550878444952663027317490091566604394117381183102950192667} a^{7} - \frac{416850679660326555707326988707479690588159180165535822517779875612411269}{1028725612755685508784449526630273174900915666043941173811831029501926670} a^{6} - \frac{195410888710959122414507770443451468954970249336193369640874092290388511}{514362806377842754392224763315136587450457833021970586905915514750963335} a^{5} + \frac{77584583466165420752947490271859904626408666471213703051311533720974678}{514362806377842754392224763315136587450457833021970586905915514750963335} a^{4} + \frac{399736611870765848790875695032328987154859904867142485465369731781094827}{1028725612755685508784449526630273174900915666043941173811831029501926670} a^{3} + \frac{61304160080837479673163447606638518907432273539600961572794781484313563}{1028725612755685508784449526630273174900915666043941173811831029501926670} a^{2} - \frac{225616294200007675143773980079632008691692769479436278489292200130510927}{514362806377842754392224763315136587450457833021970586905915514750963335} a - \frac{15516020606597567244737709567387228768763228808666958719834497319529902}{39566369721372519568632674101164352880804448693997737454301193442381795}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{300}$, which has order $1200$ (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:  \( 32330546487.4 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1675:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 6144
The 54 conjugacy class representatives for t16n1675 are not computed
Character table for t16n1675 is not computed

Intermediate fields

\(\Q(\sqrt{-49319}) \), 4.2.331.1, 8.0.5916393465826065121.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
149Data not computed
331Data not computed