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Properties

Label 16.0.69770860676...7376.6
Degree $16$
Signature $[0, 8]$
Discriminant $2^{64}\cdot 3^{8}\cdot 7^{8}$
Root discriminant $73.32$
Ramified primes $2, 3, 7$
Class number $18688$ (GRH)
Class group $[2, 8, 1168]$ (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![14371681, 0, 5000000, 0, 5250000, 0, 2100000, 0, 412500, 0, 44000, 0, 2600, 0, 80, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 80*x^14 + 2600*x^12 + 44000*x^10 + 412500*x^8 + 2100000*x^6 + 5250000*x^4 + 5000000*x^2 + 14371681)
 
gp: K = bnfinit(x^16 + 80*x^14 + 2600*x^12 + 44000*x^10 + 412500*x^8 + 2100000*x^6 + 5250000*x^4 + 5000000*x^2 + 14371681, 1)
 

Normalized defining polynomial

\( x^{16} + 80 x^{14} + 2600 x^{12} + 44000 x^{10} + 412500 x^{8} + 2100000 x^{6} + 5250000 x^{4} + 5000000 x^{2} + 14371681 \)

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:  \(697708606768276588334338277376=2^{64}\cdot 3^{8}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.32$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(672=2^{5}\cdot 3\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{672}(1,·)$, $\chi_{672}(587,·)$, $\chi_{672}(589,·)$, $\chi_{672}(335,·)$, $\chi_{672}(337,·)$, $\chi_{672}(83,·)$, $\chi_{672}(85,·)$, $\chi_{672}(671,·)$, $\chi_{672}(419,·)$, $\chi_{672}(421,·)$, $\chi_{672}(167,·)$, $\chi_{672}(169,·)$, $\chi_{672}(503,·)$, $\chi_{672}(505,·)$, $\chi_{672}(251,·)$, $\chi_{672}(253,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{781} a^{8} + \frac{40}{781} a^{6} - \frac{281}{781} a^{4} - \frac{343}{781} a^{2} - \frac{312}{781}$, $\frac{1}{2960771} a^{9} + \frac{363986}{2960771} a^{7} + \frac{52678}{174163} a^{5} + \frac{69947}{2960771} a^{3} - \frac{1309268}{2960771} a$, $\frac{1}{2960771} a^{10} + \frac{50}{2960771} a^{8} + \frac{67173}{174163} a^{6} - \frac{1291022}{2960771} a^{4} - \frac{831602}{2960771} a^{2} + \frac{274}{781}$, $\frac{1}{2960771} a^{11} + \frac{64297}{269161} a^{7} + \frac{1305014}{2960771} a^{5} - \frac{1368181}{2960771} a^{3} + \frac{1365172}{2960771} a$, $\frac{1}{2960771} a^{12} - \frac{150}{269161} a^{8} - \frac{404727}{2960771} a^{6} - \frac{534161}{2960771} a^{4} - \frac{1220290}{2960771} a^{2} - \frac{21}{71}$, $\frac{1}{2960771} a^{13} - \frac{864340}{2960771} a^{7} - \frac{340990}{2960771} a^{5} - \frac{1277809}{2960771} a^{3} + \frac{17719}{269161} a$, $\frac{1}{2960771} a^{14} + \frac{8}{2960771} a^{8} - \frac{1296322}{2960771} a^{6} - \frac{125125}{269161} a^{4} - \frac{199355}{2960771} a^{2} - \frac{65}{781}$, $\frac{1}{2960771} a^{15} - \frac{1247439}{2960771} a^{7} + \frac{341730}{2960771} a^{5} - \frac{44643}{174163} a^{3} + \frac{1345416}{2960771} a$

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

Class group and class number

$C_{2}\times C_{8}\times C_{1168}$, which has order $18688$ (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

\(\Q(\sqrt{-21}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-42}) \), \(\Q(\sqrt{2}, \sqrt{-21})\), \(\Q(\zeta_{16})^+\), 4.0.903168.5, 8.0.3262849744896.5, 8.0.417644767346688.52, \(\Q(\zeta_{32})^+\)

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

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
2Data not computed
3Data not computed
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$