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Properties

Label 16.0.34807074150...6608.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 41^{4}\cdot 73^{3}\cdot 1697^{2}$
Root discriminant $81.07$
Ramified primes $2, 41, 73, 1697$
Class number $18584$ (GRH)
Class group $[2, 2, 4646]$ (GRH)
Galois group 16T1781

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3398063, -6736240, 11409096, -11087660, 10434076, -6196184, 3801328, -1276532, 605820, -119824, 57440, -6396, 3132, -216, 88, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 88*x^14 - 216*x^13 + 3132*x^12 - 6396*x^11 + 57440*x^10 - 119824*x^9 + 605820*x^8 - 1276532*x^7 + 3801328*x^6 - 6196184*x^5 + 10434076*x^4 - 11087660*x^3 + 11409096*x^2 - 6736240*x + 3398063)
 
gp: K = bnfinit(x^16 - 4*x^15 + 88*x^14 - 216*x^13 + 3132*x^12 - 6396*x^11 + 57440*x^10 - 119824*x^9 + 605820*x^8 - 1276532*x^7 + 3801328*x^6 - 6196184*x^5 + 10434076*x^4 - 11087660*x^3 + 11409096*x^2 - 6736240*x + 3398063, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 88 x^{14} - 216 x^{13} + 3132 x^{12} - 6396 x^{11} + 57440 x^{10} - 119824 x^{9} + 605820 x^{8} - 1276532 x^{7} + 3801328 x^{6} - 6196184 x^{5} + 10434076 x^{4} - 11087660 x^{3} + 11409096 x^{2} - 6736240 x + 3398063 \)

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:  \(3480707415024772072868946116608=2^{40}\cdot 41^{4}\cdot 73^{3}\cdot 1697^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.07$
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:  $2, 41, 73, 1697$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{6}$, $\frac{1}{935552763998631959513780615467952296899023786203970} a^{15} - \frac{12936615776898691944073340758915025161613036862640}{93555276399863195951378061546795229689902378620397} a^{14} - \frac{111360996198957843555909752886489455420522993411667}{935552763998631959513780615467952296899023786203970} a^{13} + \frac{44039972025503971493970351172760620963057114998998}{467776381999315979756890307733976148449511893101985} a^{12} - \frac{85241784892556155913454091538463375621752974535979}{935552763998631959513780615467952296899023786203970} a^{11} + \frac{14356945723350889350588562397802493413945241251439}{467776381999315979756890307733976148449511893101985} a^{10} + \frac{42982338716212234763638030103157054435592529314841}{467776381999315979756890307733976148449511893101985} a^{9} + \frac{216629727425912324567481408766339173687418595706869}{935552763998631959513780615467952296899023786203970} a^{8} - \frac{50113623425147784600566030681617000560078292883539}{935552763998631959513780615467952296899023786203970} a^{7} + \frac{48295218456437935971067676279598452912188917476491}{467776381999315979756890307733976148449511893101985} a^{6} - \frac{360464944752220390385124877508191128950183501866699}{935552763998631959513780615467952296899023786203970} a^{5} + \frac{2077208886801711890372166053015622113775295562708}{13365039485694742278768294506685032812843196945771} a^{4} - \frac{229803715465788034203322668300037194448530189004479}{935552763998631959513780615467952296899023786203970} a^{3} - \frac{31389561718634955742762701052193794909319511928529}{66825197428473711393841472533425164064215984728855} a^{2} + \frac{87500135297740795710113182711495905787371476571101}{467776381999315979756890307733976148449511893101985} a + \frac{462313358013619167181588291666137015563215113847333}{935552763998631959513780615467952296899023786203970}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4646}$, which has order $18584$ (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:  \( 26238.1534853 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1781:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16384
The 148 conjugacy class representatives for t16n1781 are not computed
Character table for t16n1781 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.2624.1, 8.8.8042119168.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$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}$
41.8.4.1$x^{8} + 57154 x^{4} - 68921 x^{2} + 816644929$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$73$73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.4.3.3$x^{4} + 365$$4$$1$$3$$C_4$$[\ ]_{4}$
73.4.0.1$x^{4} - x + 13$$1$$4$$0$$C_4$$[\ ]^{4}$
1697Data not computed