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Properties

Label 16.0.27254242451...000.20
Degree $16$
Signature $[0, 8]$
Discriminant $2^{64}\cdot 3^{8}\cdot 5^{8}\cdot 7^{8}$
Root discriminant $163.95$
Ramified primes $2, 3, 5, 7$
Class number $12789760$ (GRH)
Class group $[2, 8, 8, 99920]$ (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![1461431956609, 0, 514035851264, 0, 103795700736, 0, 7984284672, 0, 301604160, 0, 6186752, 0, 70304, 0, 416, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 416*x^14 + 70304*x^12 + 6186752*x^10 + 301604160*x^8 + 7984284672*x^6 + 103795700736*x^4 + 514035851264*x^2 + 1461431956609)
 
gp: K = bnfinit(x^16 + 416*x^14 + 70304*x^12 + 6186752*x^10 + 301604160*x^8 + 7984284672*x^6 + 103795700736*x^4 + 514035851264*x^2 + 1461431956609, 1)
 

Normalized defining polynomial

\( x^{16} + 416 x^{14} + 70304 x^{12} + 6186752 x^{10} + 301604160 x^{8} + 7984284672 x^{6} + 103795700736 x^{4} + 514035851264 x^{2} + 1461431956609 \)

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:  \(272542424518858042318100889600000000=2^{64}\cdot 3^{8}\cdot 5^{8}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $163.95$
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:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3360=2^{5}\cdot 3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{3360}(1,·)$, $\chi_{3360}(839,·)$, $\chi_{3360}(841,·)$, $\chi_{3360}(1679,·)$, $\chi_{3360}(1681,·)$, $\chi_{3360}(2519,·)$, $\chi_{3360}(2521,·)$, $\chi_{3360}(3359,·)$, $\chi_{3360}(419,·)$, $\chi_{3360}(421,·)$, $\chi_{3360}(1259,·)$, $\chi_{3360}(1261,·)$, $\chi_{3360}(2099,·)$, $\chi_{3360}(2101,·)$, $\chi_{3360}(2939,·)$, $\chi_{3360}(2941,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{77221} a^{8} + \frac{208}{77221} a^{6} + \frac{13520}{77221} a^{4} - \frac{27668}{77221} a^{2} - \frac{12700}{77221}$, $\frac{1}{93352235237} a^{9} + \frac{37811880626}{93352235237} a^{7} - \frac{26303157942}{93352235237} a^{5} + \frac{32518893747}{93352235237} a^{3} - \frac{44018299330}{93352235237} a$, $\frac{1}{93352235237} a^{10} + \frac{260}{93352235237} a^{8} + \frac{43765721075}{93352235237} a^{6} + \frac{12736503239}{93352235237} a^{4} + \frac{29939601336}{93352235237} a^{2} + \frac{5776}{77221}$, $\frac{1}{93352235237} a^{11} + \frac{14661458200}{93352235237} a^{7} + \frac{36844395858}{93352235237} a^{5} - \frac{23271601554}{93352235237} a^{3} - \frac{30584519279}{93352235237} a$, $\frac{1}{93352235237} a^{12} - \frac{44616}{93352235237} a^{8} - \frac{25476662286}{93352235237} a^{6} + \frac{33357969514}{93352235237} a^{4} + \frac{8413289044}{93352235237} a^{2} - \frac{30295}{77221}$, $\frac{1}{93352235237} a^{13} + \frac{21146379503}{93352235237} a^{7} + \frac{22612393569}{93352235237} a^{5} - \frac{9063348258}{93352235237} a^{3} - \frac{12741525889}{93352235237} a$, $\frac{1}{93352235237} a^{14} + \frac{353179}{93352235237} a^{8} + \frac{11793974316}{93352235237} a^{6} + \frac{34557282193}{93352235237} a^{4} + \frac{17056576264}{93352235237} a^{2} - \frac{16417}{77221}$, $\frac{1}{93352235237} a^{15} - \frac{33086277177}{93352235237} a^{7} - \frac{109969670}{3011362427} a^{5} - \frac{34521357813}{93352235237} a^{3} - \frac{2050350537}{93352235237} a$

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

Class group and class number

$C_{2}\times C_{8}\times C_{8}\times C_{99920}$, which has order $12789760$ (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:  \( 15753.94986242651 \) (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

\(\Q(\sqrt{-210}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-105}) \), \(\Q(\sqrt{2}, \sqrt{-105})\), \(\Q(\zeta_{16})^+\), 4.0.22579200.7, 8.0.2039281090560000.290, 8.0.261027979591680000.1, \(\Q(\zeta_{32})^+\)

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 R R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
5Data not computed
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$