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Properties

Label 16.0.18759175710...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{8}\cdot 17^{15}$
Root discriminant $90.07$
Ramified primes $2, 5, 17$
Class number $155912$ (GRH)
Class group $[2, 2, 38978]$ (GRH)
Galois group $C_{16}$ (as 16T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1700000000, 0, 2040000000, 0, 714000000, 0, 112200000, 0, 9350000, 0, 442000, 0, 11900, 0, 170, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 170*x^14 + 11900*x^12 + 442000*x^10 + 9350000*x^8 + 112200000*x^6 + 714000000*x^4 + 2040000000*x^2 + 1700000000)
 
gp: K = bnfinit(x^16 + 170*x^14 + 11900*x^12 + 442000*x^10 + 9350000*x^8 + 112200000*x^6 + 714000000*x^4 + 2040000000*x^2 + 1700000000, 1)
 

Normalized defining polynomial

\( x^{16} + 170 x^{14} + 11900 x^{12} + 442000 x^{10} + 9350000 x^{8} + 112200000 x^{6} + 714000000 x^{4} + 2040000000 x^{2} + 1700000000 \)

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:  \(18759175710374728781004800000000=2^{24}\cdot 5^{8}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $90.07$
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:  $2, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(680=2^{3}\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{680}(1,·)$, $\chi_{680}(321,·)$, $\chi_{680}(269,·)$, $\chi_{680}(589,·)$, $\chi_{680}(81,·)$, $\chi_{680}(469,·)$, $\chi_{680}(441,·)$, $\chi_{680}(281,·)$, $\chi_{680}(669,·)$, $\chi_{680}(161,·)$, $\chi_{680}(549,·)$, $\chi_{680}(361,·)$, $\chi_{680}(109,·)$, $\chi_{680}(29,·)$, $\chi_{680}(309,·)$, $\chi_{680}(121,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{10} a^{2}$, $\frac{1}{10} a^{3}$, $\frac{1}{100} a^{4}$, $\frac{1}{100} a^{5}$, $\frac{1}{1000} a^{6}$, $\frac{1}{1000} a^{7}$, $\frac{1}{10000} a^{8}$, $\frac{1}{10000} a^{9}$, $\frac{1}{100000} a^{10}$, $\frac{1}{100000} a^{11}$, $\frac{1}{1000000} a^{12}$, $\frac{1}{1000000} a^{13}$, $\frac{1}{10000000} a^{14}$, $\frac{1}{10000000} a^{15}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{38978}$, which has order $155912$ (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:  \( 3640.012213375973 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{16}$ (as 16T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 16
The 16 conjugacy class representatives for $C_{16}$
Character table for $C_{16}$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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 $16$ R $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.8.12.5$x^{8} + 6 x^{6} + 8 x^{5} + 80$$2$$4$$12$$C_8$$[3]^{4}$
2.8.12.5$x^{8} + 6 x^{6} + 8 x^{5} + 80$$2$$4$$12$$C_8$$[3]^{4}$
5Data not computed
17Data not computed