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Properties

Label 16.0.36130163148...1456.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{8}\cdot 89^{14}$
Root discriminant $703.67$
Ramified primes $2, 3, 89$
Class number $1884041728$ (GRH)
Class group $[2, 2, 2, 2, 4, 29438152]$ (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![2225008900, 0, 9471488224, 0, 8488999068, 0, 2571346704, 0, 259143525, 0, 9597760, 0, 133856, 0, 712, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 712*x^14 + 133856*x^12 + 9597760*x^10 + 259143525*x^8 + 2571346704*x^6 + 8488999068*x^4 + 9471488224*x^2 + 2225008900)
 
gp: K = bnfinit(x^16 + 712*x^14 + 133856*x^12 + 9597760*x^10 + 259143525*x^8 + 2571346704*x^6 + 8488999068*x^4 + 9471488224*x^2 + 2225008900, 1)
 

Normalized defining polynomial

\( x^{16} + 712 x^{14} + 133856 x^{12} + 9597760 x^{10} + 259143525 x^{8} + 2571346704 x^{6} + 8488999068 x^{4} + 9471488224 x^{2} + 2225008900 \)

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:  \(3613016314808050004408591512141771336587411456=2^{48}\cdot 3^{8}\cdot 89^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $703.67$
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:  $2, 3, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4272=2^{4}\cdot 3\cdot 89\)
Dirichlet character group:    $\lbrace$$\chi_{4272}(1,·)$, $\chi_{4272}(1283,·)$, $\chi_{4272}(3661,·)$, $\chi_{4272}(3025,·)$, $\chi_{4272}(853,·)$, $\chi_{4272}(2135,·)$, $\chi_{4272}(1369,·)$, $\chi_{4272}(1501,·)$, $\chi_{4272}(479,·)$, $\chi_{4272}(2747,·)$, $\chi_{4272}(37,·)$, $\chi_{4272}(2099,·)$, $\chi_{4272}(3383,·)$, $\chi_{4272}(1657,·)$, $\chi_{4272}(635,·)$, $\chi_{4272}(767,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} - \frac{1}{5} a$, $\frac{1}{5} a^{6} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{7} - \frac{1}{5} a^{3}$, $\frac{1}{2225} a^{8} + \frac{1}{25} a^{6} + \frac{11}{25} a^{4} + \frac{4}{25} a^{2}$, $\frac{1}{2225} a^{9} + \frac{1}{25} a^{7} + \frac{1}{25} a^{5} + \frac{4}{25} a^{3} + \frac{2}{5} a$, $\frac{1}{2225} a^{10} + \frac{2}{25} a^{6} - \frac{11}{25} a^{2}$, $\frac{1}{2225} a^{11} + \frac{2}{25} a^{7} - \frac{11}{25} a^{3}$, $\frac{1}{445000} a^{12} + \frac{11}{111250} a^{10} + \frac{29}{222500} a^{8} - \frac{29}{625} a^{6} - \frac{231}{5000} a^{4} - \frac{241}{1250} a^{2} - \frac{33}{100}$, $\frac{1}{23585000} a^{13} + \frac{1011}{5896250} a^{11} + \frac{1629}{11792500} a^{9} + \frac{121}{33125} a^{7} + \frac{7969}{265000} a^{5} - \frac{5541}{66250} a^{3} - \frac{193}{5300} a$, $\frac{1}{83844878737240337800625000} a^{14} + \frac{21300691730957446503}{83844878737240337800625000} a^{12} - \frac{7108573868641398497073}{41922439368620168900312500} a^{10} - \frac{2800563560524732614613}{41922439368620168900312500} a^{8} + \frac{53777081600199997958281}{942077289182475705625000} a^{6} - \frac{172648593587509687863193}{942077289182475705625000} a^{4} - \frac{223978956028210333030263}{471038644591237852812500} a^{2} - \frac{104282897609201565599}{355500863842443662500}$, $\frac{1}{167689757474480675601250000} a^{15} + \frac{1762824269417631689}{83844878737240337800625000} a^{13} + \frac{13474926147836089561677}{83844878737240337800625000} a^{11} - \frac{6457281568421127406369}{41922439368620168900312500} a^{9} + \frac{111937022924823781143281}{1884154578364951411250000} a^{7} + \frac{50108047227378972644341}{942077289182475705625000} a^{5} + \frac{462884263001775067285987}{942077289182475705625000} a^{3} - \frac{1048205118604050816811}{18841545783649514112500} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{29438152}$, which has order $1884041728$ (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:  \( 785482214.9308017 \) (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

\(\Q(\sqrt{534}) \), \(\Q(\sqrt{89}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{6}, \sqrt{89})\), 4.4.45118016.1, 4.4.101515536.1, 8.8.2638183436638027776.1, 8.0.15027092855090206212096.1, 8.0.185519664877656866816.31

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 ${\href{/LocalNumberField/5.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$2.8.24.9$x^{8} + 8 x^{7} + 14 x^{4} + 4 x^{2} + 8 x + 30$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
2.8.24.9$x^{8} + 8 x^{7} + 14 x^{4} + 4 x^{2} + 8 x + 30$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
3Data not computed
89Data not computed