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Properties

Label 16.0.76573533698...000.62
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{8}\cdot 5^{12}\cdot 19^{8}$
Root discriminant $201.96$
Ramified primes $2, 3, 5, 19$
Class number $18186240$ (GRH)
Class group $[2, 2, 4, 8, 8, 17760]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![18803667681, 8505140040, 5238445584, 1554591360, 556550752, 101274600, 24453672, 1732800, 213330, -96600, 17968, 6720, 1632, -120, -24, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 24*x^14 - 120*x^13 + 1632*x^12 + 6720*x^11 + 17968*x^10 - 96600*x^9 + 213330*x^8 + 1732800*x^7 + 24453672*x^6 + 101274600*x^5 + 556550752*x^4 + 1554591360*x^3 + 5238445584*x^2 + 8505140040*x + 18803667681)
 
gp: K = bnfinit(x^16 - 24*x^14 - 120*x^13 + 1632*x^12 + 6720*x^11 + 17968*x^10 - 96600*x^9 + 213330*x^8 + 1732800*x^7 + 24453672*x^6 + 101274600*x^5 + 556550752*x^4 + 1554591360*x^3 + 5238445584*x^2 + 8505140040*x + 18803667681, 1)
 

Normalized defining polynomial

\( x^{16} - 24 x^{14} - 120 x^{13} + 1632 x^{12} + 6720 x^{11} + 17968 x^{10} - 96600 x^{9} + 213330 x^{8} + 1732800 x^{7} + 24453672 x^{6} + 101274600 x^{5} + 556550752 x^{4} + 1554591360 x^{3} + 5238445584 x^{2} + 8505140040 x + 18803667681 \)

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:  \(7657353369870241665908736000000000000=2^{48}\cdot 3^{8}\cdot 5^{12}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $201.96$
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:  $2, 3, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4560=2^{4}\cdot 3\cdot 5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{4560}(1,·)$, $\chi_{4560}(4559,·)$, $\chi_{4560}(4369,·)$, $\chi_{4560}(1747,·)$, $\chi_{4560}(1559,·)$, $\chi_{4560}(1369,·)$, $\chi_{4560}(2203,·)$, $\chi_{4560}(797,·)$, $\chi_{4560}(1253,·)$, $\chi_{4560}(3307,·)$, $\chi_{4560}(3763,·)$, $\chi_{4560}(2357,·)$, $\chi_{4560}(3191,·)$, $\chi_{4560}(3001,·)$, $\chi_{4560}(2813,·)$, $\chi_{4560}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{360} a^{8} - \frac{1}{9} a^{7} - \frac{4}{45} a^{6} - \frac{4}{9} a^{5} + \frac{1}{90} a^{4} - \frac{4}{9} a^{3} - \frac{2}{15} a^{2} + \frac{1}{3} a + \frac{9}{40}$, $\frac{1}{360} a^{9} + \frac{2}{15} a^{7} - \frac{1}{10} a^{5} - \frac{11}{45} a^{3} - \frac{53}{120} a$, $\frac{1}{360} a^{10} - \frac{1}{6} a^{6} - \frac{1}{9} a^{4} - \frac{3}{8} a^{2} + \frac{1}{5}$, $\frac{1}{360} a^{11} - \frac{1}{6} a^{7} - \frac{1}{9} a^{5} - \frac{3}{8} a^{3} + \frac{1}{5} a$, $\frac{1}{115560} a^{12} - \frac{7}{6420} a^{11} - \frac{37}{115560} a^{10} - \frac{13}{9630} a^{9} + \frac{1}{963} a^{8} + \frac{68}{4815} a^{7} + \frac{709}{5778} a^{6} + \frac{2369}{4815} a^{5} - \frac{9139}{23112} a^{4} - \frac{6437}{19260} a^{3} - \frac{6431}{38520} a^{2} + \frac{607}{3210} a - \frac{1529}{3210}$, $\frac{1}{346680} a^{13} - \frac{1}{346680} a^{12} - \frac{29}{173340} a^{11} - \frac{287}{346680} a^{10} - \frac{49}{38520} a^{9} - \frac{1}{5778} a^{8} - \frac{4346}{43335} a^{7} + \frac{689}{17334} a^{6} + \frac{139261}{346680} a^{5} + \frac{25531}{69336} a^{4} - \frac{127}{1284} a^{3} - \frac{38389}{115560} a^{2} + \frac{7307}{38520} a + \frac{898}{4815}$, $\frac{1}{704752598160} a^{14} - \frac{81791}{176188149540} a^{13} - \frac{72727}{18070579440} a^{12} - \frac{8941117}{44047037385} a^{11} - \frac{474471001}{704752598160} a^{10} - \frac{142522759}{117458766360} a^{9} + \frac{63673273}{140950519632} a^{8} - \frac{282602209}{3388233645} a^{7} - \frac{5957475853}{46983506544} a^{6} + \frac{39588587}{330559380} a^{5} + \frac{105749951347}{704752598160} a^{4} + \frac{1465448753}{14682345795} a^{3} - \frac{105537190607}{234917532720} a^{2} + \frac{2371886593}{13050974040} a + \frac{331946177}{1909898640}$, $\frac{1}{6691545553429149758565249401302320} a^{15} - \frac{474583863695543566589}{3345772776714574879282624700651160} a^{14} - \frac{6397444864225096746381145463}{6691545553429149758565249401302320} a^{13} + \frac{1539133792953185835761913119}{669154555342914975856524940130232} a^{12} - \frac{3903967609353271589660806060553}{6691545553429149758565249401302320} a^{11} - \frac{6250371884128395785691412061}{6253780891055280148191821870376} a^{10} - \frac{431873656177511328453514288037}{1338309110685829951713049880260464} a^{9} + \frac{109738135318478379599805561349}{1672886388357287439641312350325580} a^{8} + \frac{6885479053092291110749148675545}{1338309110685829951713049880260464} a^{7} - \frac{546007680871751335311182594769481}{3345772776714574879282624700651160} a^{6} + \frac{2563306221407788942458982159372277}{6691545553429149758565249401302320} a^{5} - \frac{867936196763451696735717448366861}{3345772776714574879282624700651160} a^{4} + \frac{244134845945563974627053425799771}{743505061492127750951694377922480} a^{3} - \frac{9987045977123771967577047058081}{85789045556783971263657043606440} a^{2} + \frac{322081316079131458150920967443211}{743505061492127750951694377922480} a - \frac{1343094097248169775682653540329}{4533567448122730188729843767820}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{8}\times C_{8}\times C_{17760}$, which has order $18186240$ (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:  \( 1850089.3337810908 \) (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

\(\Q(\sqrt{-285}) \), \(\Q(\sqrt{-95}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-114}) \), \(\Q(\sqrt{-38}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{3}, \sqrt{-95})\), \(\Q(\sqrt{10}, \sqrt{-114})\), \(\Q(\sqrt{30}, \sqrt{-38})\), \(\Q(\sqrt{10}, \sqrt{-38})\), \(\Q(\sqrt{30}, \sqrt{-95})\), \(\Q(\sqrt{3}, \sqrt{10})\), \(\Q(\sqrt{3}, \sqrt{-38})\), 4.4.2304000.1, 4.0.92416000.4, 4.0.831744000.2, 4.4.256000.1, 8.0.432373800960000.30, 8.0.2767192326144000000.16, 8.0.2767192326144000000.6, 8.0.691798081536000000.9, 8.0.8540717056000000.25, 8.8.21233664000000.4, 8.0.2767192326144000000.22

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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.24.2$x^{8} + 14 x^{4} + 4 x^{2} + 8 x + 6$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
2.8.24.2$x^{8} + 14 x^{4} + 4 x^{2} + 8 x + 6$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$19$19.8.4.1$x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
19.8.4.1$x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$