↑

Properties

Label 16.16.4443274623...5664.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{72}\cdot 97^{2}$
Root discriminant $40.08$
Ramified primes $2, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T817

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5983, 34368, 66152, 26576, -67220, -76384, 5352, 41760, 11262, -9312, -4216, 912, 612, -32, -40, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 40*x^14 - 32*x^13 + 612*x^12 + 912*x^11 - 4216*x^10 - 9312*x^9 + 11262*x^8 + 41760*x^7 + 5352*x^6 - 76384*x^5 - 67220*x^4 + 26576*x^3 + 66152*x^2 + 34368*x + 5983)
 
gp: K = bnfinit(x^16 - 40*x^14 - 32*x^13 + 612*x^12 + 912*x^11 - 4216*x^10 - 9312*x^9 + 11262*x^8 + 41760*x^7 + 5352*x^6 - 76384*x^5 - 67220*x^4 + 26576*x^3 + 66152*x^2 + 34368*x + 5983, 1)
 

Normalized defining polynomial

\( x^{16} - 40 x^{14} - 32 x^{13} + 612 x^{12} + 912 x^{11} - 4216 x^{10} - 9312 x^{9} + 11262 x^{8} + 41760 x^{7} + 5352 x^{6} - 76384 x^{5} - 67220 x^{4} + 26576 x^{3} + 66152 x^{2} + 34368 x + 5983 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(44432746237320491815665664=2^{72}\cdot 97^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.08$
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:  $2, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3359878031141249} a^{15} + \frac{506840168758600}{3359878031141249} a^{14} + \frac{1313721977447145}{3359878031141249} a^{13} - \frac{1637146009936963}{3359878031141249} a^{12} + \frac{832342658319825}{3359878031141249} a^{11} + \frac{694577613789932}{3359878031141249} a^{10} + \frac{652271026614294}{3359878031141249} a^{9} + \frac{1198155591585719}{3359878031141249} a^{8} - \frac{362518931899687}{3359878031141249} a^{7} - \frac{508415689509563}{3359878031141249} a^{6} + \frac{755098090040754}{3359878031141249} a^{5} + \frac{572892078474821}{3359878031141249} a^{4} + \frac{964598000405921}{3359878031141249} a^{3} - \frac{915846542589988}{3359878031141249} a^{2} + \frac{1433266326900976}{3359878031141249} a - \frac{53502269631613}{108383162294879}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $15$
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:  \( 68899827.3987 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T817:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n817
Character table for t16n817 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), \(\Q(\zeta_{32})^+\)

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$97$$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$