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Properties

Label 16.8.48774787192...5504.6
Degree $16$
Signature $[8, 4]$
Discriminant $2^{32}\cdot 3^{12}\cdot 17^{6}\cdot 97^{4}$
Root discriminant $82.80$
Ramified primes $2, 3, 17, 97$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T657)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![26863609, 37596416, 6854770, -18086136, -15979773, -236200, 1268508, 862764, 444742, -19168, -26294, -3032, -1788, -120, 26, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 26*x^14 - 120*x^13 - 1788*x^12 - 3032*x^11 - 26294*x^10 - 19168*x^9 + 444742*x^8 + 862764*x^7 + 1268508*x^6 - 236200*x^5 - 15979773*x^4 - 18086136*x^3 + 6854770*x^2 + 37596416*x + 26863609)
 
gp: K = bnfinit(x^16 - 4*x^15 + 26*x^14 - 120*x^13 - 1788*x^12 - 3032*x^11 - 26294*x^10 - 19168*x^9 + 444742*x^8 + 862764*x^7 + 1268508*x^6 - 236200*x^5 - 15979773*x^4 - 18086136*x^3 + 6854770*x^2 + 37596416*x + 26863609, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 26 x^{14} - 120 x^{13} - 1788 x^{12} - 3032 x^{11} - 26294 x^{10} - 19168 x^{9} + 444742 x^{8} + 862764 x^{7} + 1268508 x^{6} - 236200 x^{5} - 15979773 x^{4} - 18086136 x^{3} + 6854770 x^{2} + 37596416 x + 26863609 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4877478719269533554834097045504=2^{32}\cdot 3^{12}\cdot 17^{6}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $82.80$
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:  $2, 3, 17, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{7} a^{13} - \frac{3}{7} a^{12} - \frac{3}{7} a^{11} + \frac{3}{7} a^{10} - \frac{3}{7} a^{9} - \frac{2}{7} a^{8} + \frac{1}{7} a^{7} - \frac{2}{7} a^{6} - \frac{1}{7} a^{5} + \frac{1}{7} a^{4} - \frac{3}{7} a^{3} + \frac{1}{7} a^{2} + \frac{3}{7} a + \frac{3}{7}$, $\frac{1}{245} a^{14} - \frac{8}{245} a^{13} - \frac{44}{245} a^{12} - \frac{108}{245} a^{11} - \frac{53}{245} a^{10} + \frac{111}{245} a^{9} - \frac{17}{245} a^{8} - \frac{6}{35} a^{7} + \frac{9}{245} a^{6} - \frac{43}{245} a^{5} - \frac{113}{245} a^{4} - \frac{82}{245} a^{3} - \frac{44}{245} a^{2} - \frac{26}{245} a + \frac{97}{245}$, $\frac{1}{3063607886050312933518547481298025612796884228799834772095} a^{15} - \frac{1928474611928607161515751550817605071307623818645023037}{3063607886050312933518547481298025612796884228799834772095} a^{14} - \frac{7378367531342212243456278641997632055722563846500804502}{3063607886050312933518547481298025612796884228799834772095} a^{13} + \frac{7742988321666601007356006749960545832529452986340462252}{62522609919394141500378520026490318628507841404078260655} a^{12} - \frac{8024627432024672166961082986083342434752463690230837901}{3063607886050312933518547481298025612796884228799834772095} a^{11} + \frac{415735912305253815448193575605318209213167142494759265493}{3063607886050312933518547481298025612796884228799834772095} a^{10} - \frac{914969339977618481856891432243120280470113941090161542461}{3063607886050312933518547481298025612796884228799834772095} a^{9} + \frac{1524421840152081413111070866142412338679479575619805174766}{3063607886050312933518547481298025612796884228799834772095} a^{8} + \frac{572457765765610263779023873518802860498455752749367966132}{3063607886050312933518547481298025612796884228799834772095} a^{7} - \frac{1431197146172918361903926526689854314213581064337905326684}{3063607886050312933518547481298025612796884228799834772095} a^{6} + \frac{1223654620731894521325058027771936324636648656737418189134}{3063607886050312933518547481298025612796884228799834772095} a^{5} - \frac{40636702884093286704202334388157236397733307288272033630}{87531653887151798100529928037086446079910977965709564917} a^{4} + \frac{716305773504724594223496780537638382375580788919478322574}{3063607886050312933518547481298025612796884228799834772095} a^{3} - \frac{2459667721705184090142094256145802840314066426854718313}{612721577210062586703709496259605122559376845759966954419} a^{2} - \frac{1337952225647971542915671866658214013890562275881207671499}{3063607886050312933518547481298025612796884228799834772095} a + \frac{984426466898956700645088982203770809593827134772567543182}{3063607886050312933518547481298025612796884228799834772095}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $11$
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:  \( 483829365.463 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T657):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{3}) \), 4.4.4352.1, 4.4.9792.1, \(\Q(\sqrt{2}, \sqrt{3})\), 8.8.1534132224.1

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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
3Data not computed
$17$17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.8.6.2$x^{8} + 85 x^{4} + 2601$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
97Data not computed