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Properties

Label 16.0.94779071185...3856.6
Degree $16$
Signature $[0, 8]$
Discriminant $2^{64}\cdot 3^{8}\cdot 23^{8}$
Root discriminant $132.91$
Ramified primes $2, 3, 23$
Class number $1648912$ (GRH)
Class group $[2, 2, 2, 206114]$ (GRH)
Galois group $C_8\times C_2$ (as 16T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![78310985281, 0, 217908828608, 0, 49740058704, 0, 4325222496, 0, 184695060, 0, 4282784, 0, 55016, 0, 368, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 368*x^14 + 55016*x^12 + 4282784*x^10 + 184695060*x^8 + 4325222496*x^6 + 49740058704*x^4 + 217908828608*x^2 + 78310985281)
 
gp: K = bnfinit(x^16 + 368*x^14 + 55016*x^12 + 4282784*x^10 + 184695060*x^8 + 4325222496*x^6 + 49740058704*x^4 + 217908828608*x^2 + 78310985281, 1)
 

Normalized defining polynomial

\( x^{16} + 368 x^{14} + 55016 x^{12} + 4282784 x^{10} + 184695060 x^{8} + 4325222496 x^{6} + 49740058704 x^{4} + 217908828608 x^{2} + 78310985281 \)

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:  \(9477907118573134595068808298233856=2^{64}\cdot 3^{8}\cdot 23^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $132.91$
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:  $2, 3, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2208=2^{5}\cdot 3\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{2208}(1,·)$, $\chi_{2208}(1931,·)$, $\chi_{2208}(781,·)$, $\chi_{2208}(1105,·)$, $\chi_{2208}(275,·)$, $\chi_{2208}(599,·)$, $\chi_{2208}(1885,·)$, $\chi_{2208}(1379,·)$, $\chi_{2208}(229,·)$, $\chi_{2208}(1703,·)$, $\chi_{2208}(553,·)$, $\chi_{2208}(47,·)$, $\chi_{2208}(1333,·)$, $\chi_{2208}(1657,·)$, $\chi_{2208}(827,·)$, $\chi_{2208}(1151,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{23} a^{2}$, $\frac{1}{23} a^{3}$, $\frac{1}{529} a^{4}$, $\frac{1}{529} a^{5}$, $\frac{1}{12167} a^{6}$, $\frac{1}{12167} a^{7}$, $\frac{1}{279841} a^{8}$, $\frac{1}{279841} a^{9}$, $\frac{1}{6436343} a^{10}$, $\frac{1}{6436343} a^{11}$, $\frac{1}{148035889} a^{12}$, $\frac{1}{148035889} a^{13}$, $\frac{1}{3404825447} a^{14}$, $\frac{1}{3404825447} a^{15}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{206114}$, which has order $1648912$ (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:  \( 11964.310642723332 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_8$ (as 16T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\zeta_{16})^+\), 4.4.18432.1, \(\Q(\zeta_{48})^+\), 8.0.600953971539968.31, 8.0.48677271694737408.34

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
3Data not computed
$23$23.4.2.2$x^{4} - 23 x^{2} + 3703$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
23.4.2.2$x^{4} - 23 x^{2} + 3703$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
23.4.2.2$x^{4} - 23 x^{2} + 3703$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
23.4.2.2$x^{4} - 23 x^{2} + 3703$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$