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Properties

Label 16.0.68372792983...489.27
Degree $16$
Signature $[0, 8]$
Discriminant $17^{14}\cdot 67^{8}$
Root discriminant $97.65$
Ramified primes $17, 67$
Class number $368640$ (GRH)
Class group $[2, 2, 2, 2, 2, 24, 480]$ (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![14754909041, -3792717441, 5669585782, -1282215295, 987871066, -194147354, 101632369, -17057530, 6738726, -939579, 294420, -32502, 8265, -656, 136, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 136*x^14 - 656*x^13 + 8265*x^12 - 32502*x^11 + 294420*x^10 - 939579*x^9 + 6738726*x^8 - 17057530*x^7 + 101632369*x^6 - 194147354*x^5 + 987871066*x^4 - 1282215295*x^3 + 5669585782*x^2 - 3792717441*x + 14754909041)
 
gp: K = bnfinit(x^16 - 6*x^15 + 136*x^14 - 656*x^13 + 8265*x^12 - 32502*x^11 + 294420*x^10 - 939579*x^9 + 6738726*x^8 - 17057530*x^7 + 101632369*x^6 - 194147354*x^5 + 987871066*x^4 - 1282215295*x^3 + 5669585782*x^2 - 3792717441*x + 14754909041, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 136 x^{14} - 656 x^{13} + 8265 x^{12} - 32502 x^{11} + 294420 x^{10} - 939579 x^{9} + 6738726 x^{8} - 17057530 x^{7} + 101632369 x^{6} - 194147354 x^{5} + 987871066 x^{4} - 1282215295 x^{3} + 5669585782 x^{2} - 3792717441 x + 14754909041 \)

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:  \(68372792983010839504523425519489=17^{14}\cdot 67^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $97.65$
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:  $17, 67$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1139=17\cdot 67\)
Dirichlet character group:    $\lbrace$$\chi_{1139}(1,·)$, $\chi_{1139}(66,·)$, $\chi_{1139}(135,·)$, $\chi_{1139}(200,·)$, $\chi_{1139}(202,·)$, $\chi_{1139}(336,·)$, $\chi_{1139}(468,·)$, $\chi_{1139}(535,·)$, $\chi_{1139}(604,·)$, $\chi_{1139}(671,·)$, $\chi_{1139}(803,·)$, $\chi_{1139}(937,·)$, $\chi_{1139}(939,·)$, $\chi_{1139}(1004,·)$, $\chi_{1139}(1073,·)$, $\chi_{1139}(1138,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{900191283628083548049826945949707099505772786413974} a^{15} - \frac{61674795541850263374907175938007309647686240543929}{900191283628083548049826945949707099505772786413974} a^{14} - \frac{57352476894761825915989334468024778991376051667651}{450095641814041774024913472974853549752886393206987} a^{13} - \frac{11196518659279720025439280378744647334860816038687}{450095641814041774024913472974853549752886393206987} a^{12} + \frac{4788376214899156492773026842707609037781953020783}{900191283628083548049826945949707099505772786413974} a^{11} - \frac{81444085535341556416393319588870521488324614695004}{450095641814041774024913472974853549752886393206987} a^{10} - \frac{6862204295149885934998090453521981244750756710003}{34622741678003213386531805613450273057914337938999} a^{9} + \frac{265213768643047008289370135186150953024716251948635}{900191283628083548049826945949707099505772786413974} a^{8} - \frac{118702388610932392705538660877140564207930463991426}{450095641814041774024913472974853549752886393206987} a^{7} + \frac{27329918523207801454279563867596900993057656905281}{450095641814041774024913472974853549752886393206987} a^{6} - \frac{370673194337812866803115731613003027704747714561549}{900191283628083548049826945949707099505772786413974} a^{5} - \frac{1222303854633918791966437122763652660202479990350}{4369860600136327903154499737619937376241615468029} a^{4} - \frac{203213302353974155119022765325925573473604000842592}{450095641814041774024913472974853549752886393206987} a^{3} + \frac{417527312192923206872333238752881577904111385764991}{900191283628083548049826945949707099505772786413974} a^{2} - \frac{17075831075820143083403183725850784951142209469204}{34622741678003213386531805613450273057914337938999} a - \frac{48578439319609004337426408788602517673408632586833}{900191283628083548049826945949707099505772786413974}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{24}\times C_{480}$, which has order $368640$ (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:  \( 3640.01221338 \) (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

\(\Q(\sqrt{-1139}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-67}) \), \(\Q(\sqrt{17}, \sqrt{-67})\), 4.4.4913.1, 4.0.22054457.2, 8.0.486399073564849.4, 8.0.8268784250602433.5, \(\Q(\zeta_{17})^+\)

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ 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
$17$17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
$67$67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$