↑

Properties

Label 16.0.90116382360...0000.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{62}\cdot 5^{8}\cdot 29^{8}$
Root discriminant $176.68$
Ramified primes $2, 5, 29$
Class number $16886916$ (GRH)
Class group $[6, 2814486]$ (GRH)
Galois group $C_8\times C_2$ (as 16T5)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4002399015119, -1174884720024, 1380990979420, -426048616864, 248075557402, -43360708648, 16644202576, -1928520760, 546280763, -43714024, 9696320, -528440, 95050, -3248, 484, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 484*x^14 - 3248*x^13 + 95050*x^12 - 528440*x^11 + 9696320*x^10 - 43714024*x^9 + 546280763*x^8 - 1928520760*x^7 + 16644202576*x^6 - 43360708648*x^5 + 248075557402*x^4 - 426048616864*x^3 + 1380990979420*x^2 - 1174884720024*x + 4002399015119)
 
gp: K = bnfinit(x^16 - 8*x^15 + 484*x^14 - 3248*x^13 + 95050*x^12 - 528440*x^11 + 9696320*x^10 - 43714024*x^9 + 546280763*x^8 - 1928520760*x^7 + 16644202576*x^6 - 43360708648*x^5 + 248075557402*x^4 - 426048616864*x^3 + 1380990979420*x^2 - 1174884720024*x + 4002399015119, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 484 x^{14} - 3248 x^{13} + 95050 x^{12} - 528440 x^{11} + 9696320 x^{10} - 43714024 x^{9} + 546280763 x^{8} - 1928520760 x^{7} + 16644202576 x^{6} - 43360708648 x^{5} + 248075557402 x^{4} - 426048616864 x^{3} + 1380990979420 x^{2} - 1174884720024 x + 4002399015119 \)

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:  \(901163823601834744225359462400000000=2^{62}\cdot 5^{8}\cdot 29^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $176.68$
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:  $2, 5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4640=2^{5}\cdot 5\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{4640}(1,·)$, $\chi_{4640}(579,·)$, $\chi_{4640}(1161,·)$, $\chi_{4640}(1739,·)$, $\chi_{4640}(2321,·)$, $\chi_{4640}(4291,·)$, $\chi_{4640}(3481,·)$, $\chi_{4640}(2899,·)$, $\chi_{4640}(4059,·)$, $\chi_{4640}(929,·)$, $\chi_{4640}(2089,·)$, $\chi_{4640}(811,·)$, $\chi_{4640}(3249,·)$, $\chi_{4640}(1971,·)$, $\chi_{4640}(4409,·)$, $\chi_{4640}(3131,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{29} a^{4} - \frac{2}{29} a^{3} - \frac{1}{29} a^{2} + \frac{2}{29} a + \frac{1}{29}$, $\frac{1}{29} a^{5} - \frac{5}{29} a^{3} + \frac{5}{29} a + \frac{2}{29}$, $\frac{1}{29} a^{6} - \frac{10}{29} a^{3} + \frac{12}{29} a + \frac{5}{29}$, $\frac{1}{29} a^{7} + \frac{9}{29} a^{3} + \frac{2}{29} a^{2} - \frac{4}{29} a + \frac{10}{29}$, $\frac{1}{841} a^{8} - \frac{4}{841} a^{7} + \frac{2}{841} a^{6} + \frac{8}{841} a^{5} - \frac{5}{841} a^{4} - \frac{8}{841} a^{3} + \frac{2}{841} a^{2} + \frac{4}{841} a + \frac{1}{841}$, $\frac{1}{841} a^{9} - \frac{14}{841} a^{7} - \frac{13}{841} a^{6} - \frac{2}{841} a^{5} + \frac{1}{841} a^{4} + \frac{347}{841} a^{3} - \frac{17}{841} a^{2} - \frac{418}{841} a - \frac{170}{841}$, $\frac{1}{841} a^{10} - \frac{11}{841} a^{7} - \frac{3}{841} a^{6} - \frac{3}{841} a^{5} - \frac{13}{841} a^{4} + \frac{161}{841} a^{3} + \frac{16}{841} a^{2} - \frac{172}{841} a - \frac{73}{841}$, $\frac{1}{841} a^{11} + \frac{11}{841} a^{7} - \frac{10}{841} a^{6} - \frac{12}{841} a^{5} - \frac{10}{841} a^{4} - \frac{275}{841} a^{3} + \frac{82}{841} a^{2} + \frac{14}{29} a + \frac{156}{841}$, $\frac{1}{24389} a^{12} - \frac{6}{24389} a^{11} + \frac{9}{24389} a^{10} + \frac{10}{24389} a^{9} - \frac{1}{24389} a^{8} - \frac{122}{24389} a^{7} + \frac{99}{24389} a^{6} + \frac{238}{24389} a^{5} - \frac{175}{24389} a^{4} - \frac{242}{24389} a^{3} + \frac{67}{24389} a^{2} + \frac{122}{24389} a + \frac{30}{24389}$, $\frac{1}{24389} a^{13} + \frac{2}{24389} a^{11} + \frac{6}{24389} a^{10} + \frac{1}{24389} a^{9} - \frac{12}{24389} a^{8} - \frac{169}{24389} a^{7} + \frac{20}{24389} a^{6} - \frac{400}{24389} a^{5} + \frac{216}{24389} a^{4} + \frac{3139}{24389} a^{3} - \frac{172}{24389} a^{2} - \frac{3240}{24389} a - \frac{1270}{24389}$, $\frac{1}{193161844675161488223180277} a^{14} - \frac{7}{193161844675161488223180277} a^{13} - \frac{1930756168179963280739}{193161844675161488223180277} a^{12} + \frac{11584537009079779684525}{193161844675161488223180277} a^{11} - \frac{65625666578410833167915}{193161844675161488223180277} a^{10} - \frac{7744403415110744712868}{193161844675161488223180277} a^{9} - \frac{86323411766589737187314}{193161844675161488223180277} a^{8} + \frac{2356639151115036434442901}{193161844675161488223180277} a^{7} - \frac{1770232713019119387790808}{193161844675161488223180277} a^{6} - \frac{1443791032476290164704112}{193161844675161488223180277} a^{5} - \frac{92736328562416775734628}{193161844675161488223180277} a^{4} - \frac{11575997734304732884348709}{193161844675161488223180277} a^{3} + \frac{2124587431600751265203413}{6660753264660740973213113} a^{2} - \frac{42505805040651578370996988}{193161844675161488223180277} a + \frac{54918418087706487769877}{6231027247585854458812267}$, $\frac{1}{95391852211406957493671656217697029} a^{15} + \frac{246922081}{95391852211406957493671656217697029} a^{14} + \frac{101762416203882766089162757563}{95391852211406957493671656217697029} a^{13} - \frac{1138202050756491756724178582474}{95391852211406957493671656217697029} a^{12} + \frac{49004500682929361487696647452534}{95391852211406957493671656217697029} a^{11} + \frac{20846204353231549726553488605349}{95391852211406957493671656217697029} a^{10} - \frac{19918916726240912423841499359643}{95391852211406957493671656217697029} a^{9} + \frac{26131095628081713077004234206470}{95391852211406957493671656217697029} a^{8} - \frac{262937788077193766791148111470308}{95391852211406957493671656217697029} a^{7} - \frac{50572233301456930310227887181896}{3077156522948611532053924394119259} a^{6} - \frac{240141337388472135405636084296015}{95391852211406957493671656217697029} a^{5} + \frac{669196081876463560523155851577291}{95391852211406957493671656217697029} a^{4} + \frac{20476663880849602509199519572730191}{95391852211406957493671656217697029} a^{3} + \frac{44116192880674797374434965621823852}{95391852211406957493671656217697029} a^{2} + \frac{18026843126490348273501922183798170}{95391852211406957493671656217697029} a - \frac{48972156500368357217645227045095}{106108845618917639036342220486871}$

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

Class group and class number

$C_{6}\times C_{2814486}$, which has order $16886916$ (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:  \( 12198.951274811623 \) (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

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\zeta_{16})^+\), 4.4.51200.1, 8.8.2621440000.1, 8.0.1518874382041088.26, 8.0.949296488775680000.1

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

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
5Data not computed
$29$29.8.4.2$x^{8} - 24389 x^{2} + 13438339$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
29.8.4.2$x^{8} - 24389 x^{2} + 13438339$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$