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Properties

Label 16.0.29780125313...4944.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{62}\cdot 71^{8}$
Root discriminant $123.63$
Ramified primes $2, 71$
Class number $1223768$ (GRH)
Class group $[1223768]$ (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![24635401918, -8134929520, 9376337416, -2597020816, 1590951472, -370613360, 157205136, -30603232, 9888041, -1578616, 404700, -50904, 10486, -952, 156, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 156*x^14 - 952*x^13 + 10486*x^12 - 50904*x^11 + 404700*x^10 - 1578616*x^9 + 9888041*x^8 - 30603232*x^7 + 157205136*x^6 - 370613360*x^5 + 1590951472*x^4 - 2597020816*x^3 + 9376337416*x^2 - 8134929520*x + 24635401918)
 
gp: K = bnfinit(x^16 - 8*x^15 + 156*x^14 - 952*x^13 + 10486*x^12 - 50904*x^11 + 404700*x^10 - 1578616*x^9 + 9888041*x^8 - 30603232*x^7 + 157205136*x^6 - 370613360*x^5 + 1590951472*x^4 - 2597020816*x^3 + 9376337416*x^2 - 8134929520*x + 24635401918, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 156 x^{14} - 952 x^{13} + 10486 x^{12} - 50904 x^{11} + 404700 x^{10} - 1578616 x^{9} + 9888041 x^{8} - 30603232 x^{7} + 157205136 x^{6} - 370613360 x^{5} + 1590951472 x^{4} - 2597020816 x^{3} + 9376337416 x^{2} - 8134929520 x + 24635401918 \)

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:  \(2978012531396189373689422302674944=2^{62}\cdot 71^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $123.63$
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:  $2, 71$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2272=2^{5}\cdot 71\)
Dirichlet character group:    $\lbrace$$\chi_{2272}(1,·)$, $\chi_{2272}(709,·)$, $\chi_{2272}(1421,·)$, $\chi_{2272}(141,·)$, $\chi_{2272}(2129,·)$, $\chi_{2272}(853,·)$, $\chi_{2272}(1561,·)$, $\chi_{2272}(285,·)$, $\chi_{2272}(1989,·)$, $\chi_{2272}(993,·)$, $\chi_{2272}(425,·)$, $\chi_{2272}(1137,·)$, $\chi_{2272}(1845,·)$, $\chi_{2272}(1705,·)$, $\chi_{2272}(569,·)$, $\chi_{2272}(1277,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{31157231261845499101537} a^{14} - \frac{7}{31157231261845499101537} a^{13} - \frac{2776505232432509862171}{31157231261845499101537} a^{12} - \frac{14498199867250439928420}{31157231261845499101537} a^{11} - \frac{3979164828334177689158}{31157231261845499101537} a^{10} - \frac{8183038594735157568468}{31157231261845499101537} a^{9} - \frac{13126493745477917814021}{31157231261845499101537} a^{8} + \frac{4438470534258775657632}{31157231261845499101537} a^{7} + \frac{7796211148709276513092}{31157231261845499101537} a^{6} + \frac{14526524223588985150231}{31157231261845499101537} a^{5} - \frac{11617993663689555982760}{31157231261845499101537} a^{4} + \frac{3143769687983268716224}{31157231261845499101537} a^{3} - \frac{10226374252168828736766}{31157231261845499101537} a^{2} + \frac{196797842806046026062}{1832778309520323476561} a - \frac{1025972979333457539816}{31157231261845499101537}$, $\frac{1}{1201147687947489087752701046753} a^{15} + \frac{19275577}{1201147687947489087752701046753} a^{14} - \frac{337046436517413841667932340922}{1201147687947489087752701046753} a^{13} + \frac{335561544219511354816271142933}{1201147687947489087752701046753} a^{12} - \frac{122354050454409787541801217638}{1201147687947489087752701046753} a^{11} + \frac{484777685985413324428524092082}{1201147687947489087752701046753} a^{10} + \frac{139771338624553278318885514427}{1201147687947489087752701046753} a^{9} - \frac{584200448943960578001447942806}{1201147687947489087752701046753} a^{8} - \frac{83563686482398676564630585991}{1201147687947489087752701046753} a^{7} - \frac{439906977508722301316903317889}{1201147687947489087752701046753} a^{6} + \frac{403233982450735764458555807533}{1201147687947489087752701046753} a^{5} + \frac{279781423838206145561749263099}{1201147687947489087752701046753} a^{4} + \frac{29144823826630526913315739624}{1201147687947489087752701046753} a^{3} + \frac{70663298631790216272743157356}{1201147687947489087752701046753} a^{2} - \frac{582007258923149632968545327875}{1201147687947489087752701046753} a - \frac{417277692204996254716637575584}{1201147687947489087752701046753}$

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

Class group and class number

$C_{1223768}$, which has order $1223768$ (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

\(\Q(\sqrt{-71}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-142}) \), \(\Q(\sqrt{2}, \sqrt{-71})\), \(\Q(\zeta_{16})^+\), 4.0.10323968.5, 8.0.106584315265024.29, \(\Q(\zeta_{32})^+\), 8.0.54571169415692288.25

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.8.31.6$x^{8} + 16 x^{7} + 28 x^{4} + 16 x^{3} + 2$$8$$1$$31$$C_8$$[3, 4, 5]$
2.8.31.6$x^{8} + 16 x^{7} + 28 x^{4} + 16 x^{3} + 2$$8$$1$$31$$C_8$$[3, 4, 5]$
$71$71.8.4.1$x^{8} + 110902 x^{4} - 357911 x^{2} + 3074813401$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
71.8.4.1$x^{8} + 110902 x^{4} - 357911 x^{2} + 3074813401$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$