↑

Properties

Label 16.0.24967943722...0000.5
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 5^{12}\cdot 47^{8}$
Root discriminant $91.69$
Ramified primes $2, 5, 47$
Class number $176800$ (GRH)
Class group $[10, 17680]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1353436561, -598345980, 722028526, -263040692, 172046412, -52453964, 24095662, -6156340, 2174117, -459508, 129218, -21852, 4916, -616, 108, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 108*x^14 - 616*x^13 + 4916*x^12 - 21852*x^11 + 129218*x^10 - 459508*x^9 + 2174117*x^8 - 6156340*x^7 + 24095662*x^6 - 52453964*x^5 + 172046412*x^4 - 263040692*x^3 + 722028526*x^2 - 598345980*x + 1353436561)
 
gp: K = bnfinit(x^16 - 8*x^15 + 108*x^14 - 616*x^13 + 4916*x^12 - 21852*x^11 + 129218*x^10 - 459508*x^9 + 2174117*x^8 - 6156340*x^7 + 24095662*x^6 - 52453964*x^5 + 172046412*x^4 - 263040692*x^3 + 722028526*x^2 - 598345980*x + 1353436561, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 108 x^{14} - 616 x^{13} + 4916 x^{12} - 21852 x^{11} + 129218 x^{10} - 459508 x^{9} + 2174117 x^{8} - 6156340 x^{7} + 24095662 x^{6} - 52453964 x^{5} + 172046412 x^{4} - 263040692 x^{3} + 722028526 x^{2} - 598345980 x + 1353436561 \)

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:  \(24967943722642702336000000000000=2^{32}\cdot 5^{12}\cdot 47^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $91.69$
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:  $2, 5, 47$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1880=2^{3}\cdot 5\cdot 47\)
Dirichlet character group:    $\lbrace$$\chi_{1880}(1409,·)$, $\chi_{1880}(1221,·)$, $\chi_{1880}(1,·)$, $\chi_{1880}(847,·)$, $\chi_{1880}(469,·)$, $\chi_{1880}(281,·)$, $\chi_{1880}(283,·)$, $\chi_{1880}(1503,·)$, $\chi_{1880}(187,·)$, $\chi_{1880}(1127,·)$, $\chi_{1880}(1129,·)$, $\chi_{1880}(1223,·)$, $\chi_{1880}(941,·)$, $\chi_{1880}(563,·)$, $\chi_{1880}(1787,·)$, $\chi_{1880}(189,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{10} a^{12} - \frac{1}{10} a^{11} + \frac{1}{5} a^{10} - \frac{1}{10} a^{8} + \frac{3}{10} a^{7} - \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{10} a^{4} + \frac{1}{10} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{10} a^{13} + \frac{1}{10} a^{11} + \frac{1}{5} a^{10} - \frac{1}{10} a^{9} + \frac{1}{5} a^{8} - \frac{1}{10} a^{7} - \frac{1}{2} a^{5} + \frac{1}{5} a^{4} - \frac{3}{10} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{7475447125321027810} a^{14} - \frac{7}{7475447125321027810} a^{13} + \frac{155477706952695696}{3737723562660513905} a^{12} - \frac{1865732483432348261}{7475447125321027810} a^{11} - \frac{797943506175114214}{3737723562660513905} a^{10} - \frac{541041056037964818}{3737723562660513905} a^{9} + \frac{1738758570284046999}{7475447125321027810} a^{8} + \frac{288148489905328533}{1495089425064205562} a^{7} + \frac{1020700861348891308}{3737723562660513905} a^{6} - \frac{584424497328246662}{3737723562660513905} a^{5} - \frac{2649681852022029071}{7475447125321027810} a^{4} + \frac{2076854972413412081}{7475447125321027810} a^{3} + \frac{2253312550217917507}{7475447125321027810} a^{2} + \frac{2237930338152349371}{7475447125321027810} a + \frac{47599514623538149}{7475447125321027810}$, $\frac{1}{66605523719133452289458050} a^{15} + \frac{890989}{13321104743826690457891610} a^{14} + \frac{1701385912613541180165393}{66605523719133452289458050} a^{13} + \frac{158555862224179057456154}{33302761859566726144729025} a^{12} - \frac{222786508692352593812621}{6660552371913345228945805} a^{11} + \frac{2240887432433560781643119}{33302761859566726144729025} a^{10} - \frac{13471755251659958416472603}{66605523719133452289458050} a^{9} + \frac{197861602324955780381269}{33302761859566726144729025} a^{8} - \frac{341093854026046492048452}{33302761859566726144729025} a^{7} - \frac{15684556751456368667540886}{33302761859566726144729025} a^{6} - \frac{3193739362837945959451929}{66605523719133452289458050} a^{5} - \frac{598109368196890056698888}{33302761859566726144729025} a^{4} + \frac{1242073280050507018548649}{66605523719133452289458050} a^{3} - \frac{163901488433771180191839}{701110775990878445152190} a^{2} + \frac{353550908140077053455458}{33302761859566726144729025} a - \frac{5159042765634899440018226}{33302761859566726144729025}$

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

Class group and class number

$C_{10}\times C_{17680}$, which has order $176800$ (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:  \( 7114.135357253273 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_4$ (as 16T2):

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 $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{-94}) \), \(\Q(\sqrt{-47}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-470}) \), \(\Q(\sqrt{-235}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{-47})\), \(\Q(\sqrt{5}, \sqrt{-94})\), \(\Q(\sqrt{10}, \sqrt{-94})\), \(\Q(\sqrt{5}, \sqrt{-47})\), \(\Q(\sqrt{10}, \sqrt{-47})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-235})\), 4.0.4418000.1, 4.4.8000.1, \(\Q(\zeta_{20})^+\), 4.0.17672000.5, 8.0.12491983360000.26, 8.0.4996793344000000.29, 8.0.4996793344000000.12, 8.0.19518724000000.17, 8.0.312299584000000.18, 8.0.4996793344000000.20, \(\Q(\zeta_{40})^+\)

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$47$47.8.4.1$x^{8} + 172302 x^{4} - 103823 x^{2} + 7421994801$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
47.8.4.1$x^{8} + 172302 x^{4} - 103823 x^{2} + 7421994801$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$