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Properties

Label 16.0.19670421429...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 5^{8}\cdot 17^{15}$
Root discriminant $214.22$
Ramified primes $2, 5, 17$
Class number $13491472$ (GRH)
Class group $[2, 2, 3372868]$ (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![17956250000, 0, 38547500000, 0, 16679125000, 0, 2011950000, 0, 107950000, 0, 3009000, 0, 45050, 0, 340, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 340*x^14 + 45050*x^12 + 3009000*x^10 + 107950000*x^8 + 2011950000*x^6 + 16679125000*x^4 + 38547500000*x^2 + 17956250000)
 
gp: K = bnfinit(x^16 + 340*x^14 + 45050*x^12 + 3009000*x^10 + 107950000*x^8 + 2011950000*x^6 + 16679125000*x^4 + 38547500000*x^2 + 17956250000, 1)
 

Normalized defining polynomial

\( x^{16} + 340 x^{14} + 45050 x^{12} + 3009000 x^{10} + 107950000 x^{8} + 2011950000 x^{6} + 16679125000 x^{4} + 38547500000 x^{2} + 17956250000 \)

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:  \(19670421429681891606270889164800000000=2^{44}\cdot 5^{8}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $214.22$
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:  \(1360=2^{4}\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{1360}(1,·)$, $\chi_{1360}(1349,·)$, $\chi_{1360}(841,·)$, $\chi_{1360}(269,·)$, $\chi_{1360}(1229,·)$, $\chi_{1360}(81,·)$, $\chi_{1360}(789,·)$, $\chi_{1360}(281,·)$, $\chi_{1360}(989,·)$, $\chi_{1360}(1121,·)$, $\chi_{1360}(1041,·)$, $\chi_{1360}(1001,·)$, $\chi_{1360}(29,·)$, $\chi_{1360}(1269,·)$, $\chi_{1360}(121,·)$, $\chi_{1360}(469,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{5} a^{2}$, $\frac{1}{5} a^{3}$, $\frac{1}{50} a^{4}$, $\frac{1}{50} a^{5}$, $\frac{1}{250} a^{6}$, $\frac{1}{250} a^{7}$, $\frac{1}{2500} a^{8}$, $\frac{1}{2500} a^{9}$, $\frac{1}{12500} a^{10}$, $\frac{1}{162500} a^{11} + \frac{3}{16250} a^{9} + \frac{1}{650} a^{7} + \frac{2}{325} a^{5} - \frac{1}{65} a^{3} - \frac{6}{13} a$, $\frac{1}{6500000} a^{12} + \frac{1}{40625} a^{10} - \frac{1}{16250} a^{8} + \frac{1}{6500} a^{6} + \frac{3}{650} a^{4} - \frac{4}{65} a^{2} + \frac{1}{4}$, $\frac{1}{6500000} a^{13} - \frac{1}{500} a^{7} + \frac{5}{52} a$, $\frac{1}{86346428642500000} a^{14} + \frac{565829253}{17269285728500000} a^{12} - \frac{3929348577}{107933035803125} a^{10} - \frac{6832590249}{86346428642500} a^{8} - \frac{12026022487}{17269285728500} a^{6} + \frac{2635174837}{345385714570} a^{4} - \frac{20620648543}{690771429140} a^{2} - \frac{1021958411}{10627252756}$, $\frac{1}{86346428642500000} a^{15} + \frac{565829253}{17269285728500000} a^{13} + \frac{111742413}{215866071606250} a^{11} - \frac{3700757454}{21586607160625} a^{9} + \frac{369139321}{690771429140} a^{7} + \frac{7862247807}{1726928572850} a^{5} + \frac{53770120749}{690771429140} a^{3} + \frac{18596298925}{138154285828} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{3372868}$, which has order $13491472$ (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:  \( 308865.41107064753 \) (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, 8.8.1680747204608.1

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 $16$ $16$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{16}$ 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$2.8.22.32$x^{8} + 8 x^{7} + 2 x^{4} + 16 x^{3} + 16 x + 20$$4$$2$$22$$C_8$$[3, 4]^{2}$
2.8.22.32$x^{8} + 8 x^{7} + 2 x^{4} + 16 x^{3} + 16 x + 20$$4$$2$$22$$C_8$$[3, 4]^{2}$
5Data not computed
17Data not computed