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Properties

Label 16.0.43153395314...0625.2
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{8}\cdot 17^{14}$
Root discriminant $46.20$
Ramified primes $3, 5, 17$
Class number $68$ (GRH)
Class group $[2, 34]$ (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![16999, -32207, 100862, -48869, 64192, -24194, 12743, 1516, -4488, 3149, -428, -128, 153, -70, 26, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 26*x^14 - 70*x^13 + 153*x^12 - 128*x^11 - 428*x^10 + 3149*x^9 - 4488*x^8 + 1516*x^7 + 12743*x^6 - 24194*x^5 + 64192*x^4 - 48869*x^3 + 100862*x^2 - 32207*x + 16999)
 
gp: K = bnfinit(x^16 - 6*x^15 + 26*x^14 - 70*x^13 + 153*x^12 - 128*x^11 - 428*x^10 + 3149*x^9 - 4488*x^8 + 1516*x^7 + 12743*x^6 - 24194*x^5 + 64192*x^4 - 48869*x^3 + 100862*x^2 - 32207*x + 16999, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 26 x^{14} - 70 x^{13} + 153 x^{12} - 128 x^{11} - 428 x^{10} + 3149 x^{9} - 4488 x^{8} + 1516 x^{7} + 12743 x^{6} - 24194 x^{5} + 64192 x^{4} - 48869 x^{3} + 100862 x^{2} - 32207 x + 16999 \)

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:  \(431533953146964646550390625=3^{8}\cdot 5^{8}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.20$
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:  $3, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(255=3\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{255}(64,·)$, $\chi_{255}(1,·)$, $\chi_{255}(4,·)$, $\chi_{255}(134,·)$, $\chi_{255}(206,·)$, $\chi_{255}(16,·)$, $\chi_{255}(26,·)$, $\chi_{255}(154,·)$, $\chi_{255}(161,·)$, $\chi_{255}(166,·)$, $\chi_{255}(104,·)$, $\chi_{255}(169,·)$, $\chi_{255}(106,·)$, $\chi_{255}(236,·)$, $\chi_{255}(179,·)$, $\chi_{255}(59,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{20772778} a^{14} + \frac{189600}{798953} a^{13} + \frac{1544916}{10386389} a^{12} - \frac{5382365}{20772778} a^{11} - \frac{1720230}{10386389} a^{10} - \frac{1396277}{10386389} a^{9} - \frac{1333369}{20772778} a^{8} - \frac{1773142}{10386389} a^{7} + \frac{3903049}{10386389} a^{6} - \frac{5023217}{20772778} a^{5} + \frac{2945350}{10386389} a^{4} - \frac{259826}{798953} a^{3} - \frac{5064687}{20772778} a^{2} + \frac{84626}{798953} a - \frac{17}{611}$, $\frac{1}{75937017171728990671161947654254} a^{15} + \frac{230545451296296847577150}{37968508585864495335580973827127} a^{14} + \frac{12157404459809004297486822622403}{75937017171728990671161947654254} a^{13} - \frac{16339973434218832715842292057337}{75937017171728990671161947654254} a^{12} + \frac{7249986953596387790508214127317}{37968508585864495335580973827127} a^{11} + \frac{18080441372343413481617471632857}{75937017171728990671161947654254} a^{10} + \frac{3971584729264950970449580540439}{75937017171728990671161947654254} a^{9} + \frac{10667631287900245817532346874059}{37968508585864495335580973827127} a^{8} + \frac{15625615703761977297152412103985}{75937017171728990671161947654254} a^{7} + \frac{27169200459361282894532979523133}{75937017171728990671161947654254} a^{6} + \frac{927753056611266852757135130735}{2920654506604961179660074909779} a^{5} + \frac{1377962935545346607336898796101}{75937017171728990671161947654254} a^{4} + \frac{23635269898266611298760861548145}{75937017171728990671161947654254} a^{3} - \frac{15760041506719629308558777214526}{37968508585864495335580973827127} a^{2} - \frac{27329992512545156526648921096731}{75937017171728990671161947654254} a + \frac{323607223771560567728121248}{2233573068172509873262013873}$

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

Class group and class number

$C_{2}\times C_{34}$, which has order $68$ (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:  \( 55081.08216847024 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{85}) \), \(\Q(\sqrt{5}, \sqrt{17})\), 4.4.4913.1, 4.4.122825.1, 8.8.15085980625.1, 8.0.33237432513.1, 8.0.20773395320625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ 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
$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}$
5Data not computed
17Data not computed