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Properties

Label 16.0.12464512845...2704.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{62}\cdot 151^{8}$
Root discriminant $180.29$
Ramified primes $2, 151$
Class number $63904008$ (GRH)
Class group $[63904008]$ (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![6489839655038, -1177122876720, 1273174269256, -196706722576, 110004994672, -14381845360, 5469084496, -596127232, 171097481, -15127576, 3446780, -235064, 43606, -2072, 316, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 316*x^14 - 2072*x^13 + 43606*x^12 - 235064*x^11 + 3446780*x^10 - 15127576*x^9 + 171097481*x^8 - 596127232*x^7 + 5469084496*x^6 - 14381845360*x^5 + 110004994672*x^4 - 196706722576*x^3 + 1273174269256*x^2 - 1177122876720*x + 6489839655038)
 
gp: K = bnfinit(x^16 - 8*x^15 + 316*x^14 - 2072*x^13 + 43606*x^12 - 235064*x^11 + 3446780*x^10 - 15127576*x^9 + 171097481*x^8 - 596127232*x^7 + 5469084496*x^6 - 14381845360*x^5 + 110004994672*x^4 - 196706722576*x^3 + 1273174269256*x^2 - 1177122876720*x + 6489839655038, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 316 x^{14} - 2072 x^{13} + 43606 x^{12} - 235064 x^{11} + 3446780 x^{10} - 15127576 x^{9} + 171097481 x^{8} - 596127232 x^{7} + 5469084496 x^{6} - 14381845360 x^{5} + 110004994672 x^{4} - 196706722576 x^{3} + 1273174269256 x^{2} - 1177122876720 x + 6489839655038 \)

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:  \(1246451284576930712479346307528392704=2^{62}\cdot 151^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $180.29$
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:  $2, 151$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4832=2^{5}\cdot 151\)
Dirichlet character group:    $\lbrace$$\chi_{4832}(1,·)$, $\chi_{4832}(4229,·)$, $\chi_{4832}(2113,·)$, $\chi_{4832}(905,·)$, $\chi_{4832}(3021,·)$, $\chi_{4832}(1813,·)$, $\chi_{4832}(3321,·)$, $\chi_{4832}(2717,·)$, $\chi_{4832}(1509,·)$, $\chi_{4832}(4529,·)$, $\chi_{4832}(3625,·)$, $\chi_{4832}(301,·)$, $\chi_{4832}(605,·)$, $\chi_{4832}(2417,·)$, $\chi_{4832}(1209,·)$, $\chi_{4832}(3925,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{191546083225848628222312097} a^{14} - \frac{7}{191546083225848628222312097} a^{13} - \frac{39174665825651874831146840}{191546083225848628222312097} a^{12} + \frac{43501911728062620764569034}{191546083225848628222312097} a^{11} + \frac{29030742940214748099937262}{191546083225848628222312097} a^{10} - \frac{38946335540107017581237}{6178905910511246071687487} a^{9} + \frac{26748744397166660696610141}{191546083225848628222312097} a^{8} - \frac{4322096621215165550695016}{191546083225848628222312097} a^{7} - \frac{82228810704497981289388550}{191546083225848628222312097} a^{6} - \frac{11077380059830247759195219}{191546083225848628222312097} a^{5} - \frac{15568148048913919845471464}{191546083225848628222312097} a^{4} + \frac{30836559812826771145874511}{191546083225848628222312097} a^{3} - \frac{87325853393690180374953063}{191546083225848628222312097} a^{2} - \frac{80759751048576741733434540}{191546083225848628222312097} a - \frac{66433560287432812779329083}{191546083225848628222312097}$, $\frac{1}{122112828729428682266613607981182433} a^{15} + \frac{318755737}{122112828729428682266613607981182433} a^{14} + \frac{29155393443625078428340633743400081}{122112828729428682266613607981182433} a^{13} - \frac{18827964060214356943916744636735518}{122112828729428682266613607981182433} a^{12} - \frac{41838096617161856313542812944924471}{122112828729428682266613607981182433} a^{11} + \frac{49567589081662806973376941200515698}{122112828729428682266613607981182433} a^{10} - \frac{2024747640084843677889162337656633}{122112828729428682266613607981182433} a^{9} - \frac{24733121150950279285168653119314698}{122112828729428682266613607981182433} a^{8} - \frac{40040849087210752175081804969021248}{122112828729428682266613607981182433} a^{7} + \frac{44923521042020297143994460234702241}{122112828729428682266613607981182433} a^{6} + \frac{32820978663157841165033985432069551}{122112828729428682266613607981182433} a^{5} - \frac{46912406983726804661039259170752620}{122112828729428682266613607981182433} a^{4} + \frac{47511671958255736914726501841891057}{122112828729428682266613607981182433} a^{3} - \frac{14606935429541902377105159526080392}{122112828729428682266613607981182433} a^{2} - \frac{12057821590480220668816314018997559}{122112828729428682266613607981182433} a - \frac{22644684783881652492660462086092907}{122112828729428682266613607981182433}$

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

Class group and class number

$C_{63904008}$, which has order $63904008$ (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

\(\Q(\sqrt{-151}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-302}) \), \(\Q(\sqrt{2}, \sqrt{-151})\), \(\Q(\zeta_{16})^+\), 4.0.46696448.5, 8.0.2180558255816704.18, 8.0.1116445826978152448.20, \(\Q(\zeta_{32})^+\)

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$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]$
$151$151.8.4.1$x^{8} + 273612 x^{4} - 3442951 x^{2} + 18715881636$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
151.8.4.1$x^{8} + 273612 x^{4} - 3442951 x^{2} + 18715881636$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$