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Properties

Label 16.0.98855585443...5424.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{62}\cdot 11^{8}$
Root discriminant $48.66$
Ramified primes $2, 11$
Class number $565$ (GRH)
Class group $[565]$ (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![214303, -164920, 233836, -157696, 132922, -77960, 50016, -25432, 13211, -5896, 2640, -984, 346, -112, 36, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 36*x^14 - 112*x^13 + 346*x^12 - 984*x^11 + 2640*x^10 - 5896*x^9 + 13211*x^8 - 25432*x^7 + 50016*x^6 - 77960*x^5 + 132922*x^4 - 157696*x^3 + 233836*x^2 - 164920*x + 214303)
 
gp: K = bnfinit(x^16 - 8*x^15 + 36*x^14 - 112*x^13 + 346*x^12 - 984*x^11 + 2640*x^10 - 5896*x^9 + 13211*x^8 - 25432*x^7 + 50016*x^6 - 77960*x^5 + 132922*x^4 - 157696*x^3 + 233836*x^2 - 164920*x + 214303, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 36 x^{14} - 112 x^{13} + 346 x^{12} - 984 x^{11} + 2640 x^{10} - 5896 x^{9} + 13211 x^{8} - 25432 x^{7} + 50016 x^{6} - 77960 x^{5} + 132922 x^{4} - 157696 x^{3} + 233836 x^{2} - 164920 x + 214303 \)

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:  \(988555854433440250854375424=2^{62}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.66$
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:  $2, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(352=2^{5}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{352}(1,·)$, $\chi_{352}(133,·)$, $\chi_{352}(65,·)$, $\chi_{352}(265,·)$, $\chi_{352}(109,·)$, $\chi_{352}(21,·)$, $\chi_{352}(89,·)$, $\chi_{352}(153,·)$, $\chi_{352}(221,·)$, $\chi_{352}(197,·)$, $\chi_{352}(177,·)$, $\chi_{352}(45,·)$, $\chi_{352}(285,·)$, $\chi_{352}(241,·)$, $\chi_{352}(309,·)$, $\chi_{352}(329,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{17} a^{13} + \frac{2}{17} a^{12} - \frac{6}{17} a^{11} - \frac{6}{17} a^{10} - \frac{3}{17} a^{9} - \frac{2}{17} a^{8} - \frac{5}{17} a^{7} + \frac{6}{17} a^{6} - \frac{4}{17} a^{5} + \frac{8}{17} a^{4} - \frac{4}{17} a^{3} + \frac{6}{17} a^{2} - \frac{8}{17} a - \frac{1}{17}$, $\frac{1}{88490485260817} a^{14} - \frac{7}{88490485260817} a^{13} + \frac{22141174490392}{88490485260817} a^{12} + \frac{44133923579373}{88490485260817} a^{11} + \frac{2146916862594}{5205322662401} a^{10} - \frac{26609159479735}{88490485260817} a^{9} + \frac{8359768326183}{88490485260817} a^{8} - \frac{455550228156}{5205322662401} a^{7} + \frac{11852175622059}{88490485260817} a^{6} + \frac{4901430820452}{88490485260817} a^{5} - \frac{3398143464909}{88490485260817} a^{4} + \frac{5327184690890}{88490485260817} a^{3} - \frac{8029990484848}{88490485260817} a^{2} + \frac{1058888375520}{88490485260817} a - \frac{660762422144}{2854531782607}$, $\frac{1}{560764205097797329} a^{15} + \frac{3161}{560764205097797329} a^{14} - \frac{326615443912651}{560764205097797329} a^{13} - \frac{208104790790803434}{560764205097797329} a^{12} - \frac{88926368849068589}{560764205097797329} a^{11} - \frac{51415032230753356}{560764205097797329} a^{10} + \frac{71615161598098054}{560764205097797329} a^{9} + \frac{192208269951654531}{560764205097797329} a^{8} - \frac{89234832250916709}{560764205097797329} a^{7} - \frac{183376915958782279}{560764205097797329} a^{6} + \frac{202572399246444561}{560764205097797329} a^{5} - \frac{211149297846592119}{560764205097797329} a^{4} + \frac{276921206001146231}{560764205097797329} a^{3} + \frac{143875579272294383}{560764205097797329} a^{2} + \frac{163460210479340458}{560764205097797329} a + \frac{3432910264140689}{18089167906380559}$

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

Class group and class number

$C_{565}$, which has order $565$ (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:  \( 15753.94986242651 \) (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

\(\Q(\sqrt{-11}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-22}) \), \(\Q(\sqrt{2}, \sqrt{-11})\), \(\Q(\zeta_{16})^+\), 4.0.247808.2, 8.0.61408804864.2, 8.0.31441308090368.8, \(\Q(\zeta_{32})^+\)

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
11Data not computed