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Properties

Label 16.0.72399383432...5584.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{8}\cdot 17^{14}$
Root discriminant $41.33$
Ramified primes $2, 3, 17$
Class number $16$ (GRH)
Class group $[2, 8]$ (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![10609, 0, -9739, 0, 720, 0, 955, 0, 155, 0, -173, 0, 108, 0, -19, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 19*x^14 + 108*x^12 - 173*x^10 + 155*x^8 + 955*x^6 + 720*x^4 - 9739*x^2 + 10609)
 
gp: K = bnfinit(x^16 - 19*x^14 + 108*x^12 - 173*x^10 + 155*x^8 + 955*x^6 + 720*x^4 - 9739*x^2 + 10609, 1)
 

Normalized defining polynomial

\( x^{16} - 19 x^{14} + 108 x^{12} - 173 x^{10} + 155 x^{8} + 955 x^{6} + 720 x^{4} - 9739 x^{2} + 10609 \)

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:  \(72399383432805056195395584=2^{16}\cdot 3^{8}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.33$
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:  $2, 3, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(204=2^{2}\cdot 3\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{204}(1,·)$, $\chi_{204}(67,·)$, $\chi_{204}(13,·)$, $\chi_{204}(77,·)$, $\chi_{204}(115,·)$, $\chi_{204}(83,·)$, $\chi_{204}(155,·)$, $\chi_{204}(157,·)$, $\chi_{204}(161,·)$, $\chi_{204}(103,·)$, $\chi_{204}(169,·)$, $\chi_{204}(179,·)$, $\chi_{204}(53,·)$, $\chi_{204}(55,·)$, $\chi_{204}(185,·)$, $\chi_{204}(59,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{214858386788867} a^{14} - \frac{77630309559595}{214858386788867} a^{12} - \frac{27553167254957}{214858386788867} a^{10} + \frac{29017577363057}{214858386788867} a^{8} + \frac{26665218546345}{214858386788867} a^{6} - \frac{59570555356266}{214858386788867} a^{4} - \frac{20761332285453}{214858386788867} a^{2} + \frac{47990413619971}{214858386788867}$, $\frac{1}{22130413839253301} a^{15} - \frac{937063856715063}{22130413839253301} a^{13} - \frac{10340755733120573}{22130413839253301} a^{11} + \frac{4755902086718131}{22130413839253301} a^{9} - \frac{8137953479430601}{22130413839253301} a^{7} - \frac{3712163130767005}{22130413839253301} a^{5} + \frac{1912964148814350}{22130413839253301} a^{3} + \frac{10361192979485587}{22130413839253301} a$

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

Class group and class number

$C_{2}\times C_{8}$, which has order $16$ (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:  \( \frac{6819109}{27699268717} a^{15} - \frac{117936534}{27699268717} a^{13} + \frac{528550444}{27699268717} a^{11} - \frac{147762556}{27699268717} a^{9} + \frac{17757918}{27699268717} a^{7} + \frac{7698220038}{27699268717} a^{5} + \frac{21171709746}{27699268717} a^{3} - \frac{46414726094}{27699268717} a \) (order $4$)
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:  \( 175623.897279 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-17}) \), \(\Q(i, \sqrt{17})\), 4.4.4913.1, 4.0.78608.1, 8.0.6179217664.1, 8.8.8508782723328.1, 8.0.33237432513.1

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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$3$3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
$17$17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$