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Properties

Label 16.0.40092926956...969.21
Degree $16$
Signature $[0, 8]$
Discriminant $17^{14}\cdot 47^{8}$
Root discriminant $81.79$
Ramified primes $17, 47$
Class number $104960$ (GRH)
Class group $[4, 4, 6560]$ (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![1198719131, -389008031, 601821142, -173184835, 139303886, -35078044, 19309779, -4186200, 1747266, -318139, 105440, -15442, 4135, -446, 96, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 96*x^14 - 446*x^13 + 4135*x^12 - 15442*x^11 + 105440*x^10 - 318139*x^9 + 1747266*x^8 - 4186200*x^7 + 19309779*x^6 - 35078044*x^5 + 139303886*x^4 - 173184835*x^3 + 601821142*x^2 - 389008031*x + 1198719131)
 
gp: K = bnfinit(x^16 - 6*x^15 + 96*x^14 - 446*x^13 + 4135*x^12 - 15442*x^11 + 105440*x^10 - 318139*x^9 + 1747266*x^8 - 4186200*x^7 + 19309779*x^6 - 35078044*x^5 + 139303886*x^4 - 173184835*x^3 + 601821142*x^2 - 389008031*x + 1198719131, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 96 x^{14} - 446 x^{13} + 4135 x^{12} - 15442 x^{11} + 105440 x^{10} - 318139 x^{9} + 1747266 x^{8} - 4186200 x^{7} + 19309779 x^{6} - 35078044 x^{5} + 139303886 x^{4} - 173184835 x^{3} + 601821142 x^{2} - 389008031 x + 1198719131 \)

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:  \(4009292695690170390860412175969=17^{14}\cdot 47^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.79$
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:  $17, 47$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(799=17\cdot 47\)
Dirichlet character group:    $\lbrace$$\chi_{799}(1,·)$, $\chi_{799}(706,·)$, $\chi_{799}(518,·)$, $\chi_{799}(140,·)$, $\chi_{799}(659,·)$, $\chi_{799}(281,·)$, $\chi_{799}(93,·)$, $\chi_{799}(798,·)$, $\chi_{799}(610,·)$, $\chi_{799}(424,·)$, $\chi_{799}(234,·)$, $\chi_{799}(236,·)$, $\chi_{799}(563,·)$, $\chi_{799}(565,·)$, $\chi_{799}(375,·)$, $\chi_{799}(189,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{84931405249464923861733741290369791467797099674} a^{15} - \frac{189628381398151009590775987494645762184100105}{3266592509594804763912836203475761210299888449} a^{14} + \frac{3460411585764627539008359128370952006939594175}{42465702624732461930866870645184895733898549837} a^{13} - \frac{1074529759271382383335999910119681831673829217}{6533185019189609527825672406951522420599776898} a^{12} + \frac{674096105675419406496736527496521382597375016}{42465702624732461930866870645184895733898549837} a^{11} + \frac{1835354996817904544140989936321578420058547980}{42465702624732461930866870645184895733898549837} a^{10} - \frac{20724093244451724972528486118710942864917144629}{84931405249464923861733741290369791467797099674} a^{9} - \frac{17243305451486833995334870699138555382694023203}{42465702624732461930866870645184895733898549837} a^{8} + \frac{8709796341684916776733005347135255930913605533}{42465702624732461930866870645184895733898549837} a^{7} - \frac{38564009747403388903707087579141842136671150847}{84931405249464923861733741290369791467797099674} a^{6} - \frac{9220478019170303924272579951882236755074896568}{42465702624732461930866870645184895733898549837} a^{5} + \frac{13071743542394954056291614730612218347502422161}{42465702624732461930866870645184895733898549837} a^{4} + \frac{3031217565088071564392869472400678531808963305}{84931405249464923861733741290369791467797099674} a^{3} - \frac{13470042619231721232063456364417118195269368187}{42465702624732461930866870645184895733898549837} a^{2} - \frac{15472953182260237109591395883599053167404161351}{42465702624732461930866870645184895733898549837} a + \frac{413839297214797775802532116436735312096728943}{42465702624732461930866870645184895733898549837}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{6560}$, which has order $104960$ (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

\(\Q(\sqrt{-799}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-47}) \), \(\Q(\sqrt{17}, \sqrt{-47})\), 4.4.4913.1, 4.0.10852817.2, 8.0.117783636835489.4, \(\Q(\zeta_{17})^+\), 8.0.2002321826203313.5

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

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}$
$47$47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$