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Properties

Label 16.0.93071566388...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{6}\cdot 5^{8}\cdot 41^{6}\cdot 97^{8}$
Root discriminant $114.96$
Ramified primes $2, 5, 41, 97$
Class number $29$ (GRH)
Class group $[29]$ (GRH)
Galois group 16T1228

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![447723604, -410243094, 263496981, -169349557, 78845290, -34075054, 12754641, -3916271, 1374044, -296277, 85725, -14666, 3251, -355, 47, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 47*x^14 - 355*x^13 + 3251*x^12 - 14666*x^11 + 85725*x^10 - 296277*x^9 + 1374044*x^8 - 3916271*x^7 + 12754641*x^6 - 34075054*x^5 + 78845290*x^4 - 169349557*x^3 + 263496981*x^2 - 410243094*x + 447723604)
 
gp: K = bnfinit(x^16 - 6*x^15 + 47*x^14 - 355*x^13 + 3251*x^12 - 14666*x^11 + 85725*x^10 - 296277*x^9 + 1374044*x^8 - 3916271*x^7 + 12754641*x^6 - 34075054*x^5 + 78845290*x^4 - 169349557*x^3 + 263496981*x^2 - 410243094*x + 447723604, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 47 x^{14} - 355 x^{13} + 3251 x^{12} - 14666 x^{11} + 85725 x^{10} - 296277 x^{9} + 1374044 x^{8} - 3916271 x^{7} + 12754641 x^{6} - 34075054 x^{5} + 78845290 x^{4} - 169349557 x^{3} + 263496981 x^{2} - 410243094 x + 447723604 \)

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:  \(930715663880146904969790025000000=2^{6}\cdot 5^{8}\cdot 41^{6}\cdot 97^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $114.96$
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:  $2, 5, 41, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{164} a^{12} - \frac{39}{164} a^{11} - \frac{1}{82} a^{10} - \frac{9}{82} a^{9} + \frac{5}{164} a^{8} - \frac{11}{82} a^{7} - \frac{67}{164} a^{6} + \frac{19}{164} a^{5} + \frac{11}{41} a^{4} + \frac{20}{41} a^{3} + \frac{67}{164} a^{2} + \frac{3}{41} a - \frac{18}{41}$, $\frac{1}{164} a^{13} + \frac{35}{164} a^{11} - \frac{7}{82} a^{10} - \frac{1}{4} a^{9} + \frac{9}{164} a^{8} - \frac{23}{164} a^{7} + \frac{15}{82} a^{6} + \frac{47}{164} a^{5} + \frac{37}{82} a^{4} + \frac{71}{164} a^{3} + \frac{1}{164} a^{2} - \frac{7}{82} a - \frac{5}{41}$, $\frac{1}{164} a^{14} + \frac{39}{164} a^{11} + \frac{29}{164} a^{10} - \frac{17}{164} a^{9} - \frac{17}{82} a^{8} - \frac{5}{41} a^{7} - \frac{17}{41} a^{6} + \frac{65}{164} a^{5} + \frac{7}{164} a^{4} - \frac{11}{164} a^{3} - \frac{63}{164} a^{2} + \frac{13}{41} a + \frac{15}{41}$, $\frac{1}{82755154230985706527264782403701992795182423011945673288} a^{15} - \frac{48769567409179122679310522598636933004081360034115307}{82755154230985706527264782403701992795182423011945673288} a^{14} - \frac{58866833803651600104640840408909504511789840673100519}{20688788557746426631816195600925498198795605752986418322} a^{13} - \frac{226301486288083207604057063039066923506994342458499867}{82755154230985706527264782403701992795182423011945673288} a^{12} - \frac{481196602307367012702768328593089120543151332643606026}{10344394278873213315908097800462749099397802876493209161} a^{11} - \frac{3709324016938195070862823950008303340553404057904164541}{41377577115492853263632391201850996397591211505972836644} a^{10} + \frac{5566547373930696972451668251837727217184101191960798985}{82755154230985706527264782403701992795182423011945673288} a^{9} + \frac{909480216666328849411554161645872051386821470825878062}{10344394278873213315908097800462749099397802876493209161} a^{8} - \frac{12041141496854654900099164093159443453855811134866490721}{41377577115492853263632391201850996397591211505972836644} a^{7} - \frac{36005527834045952470088523562732159880869940367644485981}{82755154230985706527264782403701992795182423011945673288} a^{6} - \frac{5411147701685963389715679066531414666038331574402030511}{20688788557746426631816195600925498198795605752986418322} a^{5} + \frac{4100222415505939284740823382567528476710951765055650771}{41377577115492853263632391201850996397591211505972836644} a^{4} - \frac{5266950433274743110869381187757370071881430002196128103}{41377577115492853263632391201850996397591211505972836644} a^{3} - \frac{1526830058539148130718280026332829266135045888365235545}{82755154230985706527264782403701992795182423011945673288} a^{2} + \frac{7497243006915263088079228396876445791217725907142062425}{41377577115492853263632391201850996397591211505972836644} a + \frac{8015044011014559452847617203829317396006120061119618573}{20688788557746426631816195600925498198795605752986418322}$

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

Class group and class number

$C_{29}$, which has order $29$ (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:  \( 403164653.505 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1228:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 61 conjugacy class representatives for t16n1228 are not computed
Character table for t16n1228 is not computed

Intermediate fields

\(\Q(\sqrt{97}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{485}) \), 4.0.1025.1, 4.0.9644225.1, \(\Q(\sqrt{5}, \sqrt{97})\), 8.0.93011075850625.1

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$41$41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.8.6.1$x^{8} - 9881 x^{4} + 34857216$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
97Data not computed