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Properties

Label 16.0.36114567590...1024.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{62}\cdot 23^{8}$
Root discriminant $70.36$
Ramified primes $2, 23$
Class number $30840$ (GRH)
Class group $[30840]$ (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![13289086, -8376112, 11014984, -5816080, 4331632, -1934192, 1046352, -396544, 169097, -54040, 18684, -4920, 1366, -280, 60, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 60*x^14 - 280*x^13 + 1366*x^12 - 4920*x^11 + 18684*x^10 - 54040*x^9 + 169097*x^8 - 396544*x^7 + 1046352*x^6 - 1934192*x^5 + 4331632*x^4 - 5816080*x^3 + 11014984*x^2 - 8376112*x + 13289086)
 
gp: K = bnfinit(x^16 - 8*x^15 + 60*x^14 - 280*x^13 + 1366*x^12 - 4920*x^11 + 18684*x^10 - 54040*x^9 + 169097*x^8 - 396544*x^7 + 1046352*x^6 - 1934192*x^5 + 4331632*x^4 - 5816080*x^3 + 11014984*x^2 - 8376112*x + 13289086, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 60 x^{14} - 280 x^{13} + 1366 x^{12} - 4920 x^{11} + 18684 x^{10} - 54040 x^{9} + 169097 x^{8} - 396544 x^{7} + 1046352 x^{6} - 1934192 x^{5} + 4331632 x^{4} - 5816080 x^{3} + 11014984 x^{2} - 8376112 x + 13289086 \)

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:  \(361145675909660668917421441024=2^{62}\cdot 23^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $70.36$
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:  $2, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(736=2^{5}\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{736}(1,·)$, $\chi_{736}(645,·)$, $\chi_{736}(321,·)$, $\chi_{736}(137,·)$, $\chi_{736}(461,·)$, $\chi_{736}(597,·)$, $\chi_{736}(185,·)$, $\chi_{736}(93,·)$, $\chi_{736}(229,·)$, $\chi_{736}(689,·)$, $\chi_{736}(553,·)$, $\chi_{736}(45,·)$, $\chi_{736}(413,·)$, $\chi_{736}(369,·)$, $\chi_{736}(505,·)$, $\chi_{736}(277,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{17} a^{11} + \frac{3}{17} a^{10} - \frac{3}{17} a^{9} + \frac{8}{17} a^{8} - \frac{5}{17} a^{7} + \frac{4}{17} a^{6} - \frac{2}{17} a^{5} - \frac{7}{17} a^{4} - \frac{6}{17} a^{2} - \frac{2}{17} a - \frac{4}{17}$, $\frac{1}{17} a^{12} + \frac{5}{17} a^{10} + \frac{5}{17} a^{8} + \frac{2}{17} a^{7} + \frac{3}{17} a^{6} - \frac{1}{17} a^{5} + \frac{4}{17} a^{4} - \frac{6}{17} a^{3} - \frac{1}{17} a^{2} + \frac{2}{17} a - \frac{5}{17}$, $\frac{1}{17} a^{13} + \frac{2}{17} a^{10} + \frac{3}{17} a^{9} - \frac{4}{17} a^{8} - \frac{6}{17} a^{7} - \frac{4}{17} a^{6} - \frac{3}{17} a^{5} - \frac{5}{17} a^{4} - \frac{1}{17} a^{3} - \frac{2}{17} a^{2} + \frac{5}{17} a + \frac{3}{17}$, $\frac{1}{7814377697137553} a^{14} - \frac{7}{7814377697137553} a^{13} + \frac{65747311996591}{7814377697137553} a^{12} + \frac{65185404322754}{7814377697137553} a^{11} + \frac{891444755883213}{7814377697137553} a^{10} - \frac{184674588300918}{459669276302209} a^{9} + \frac{3304333364386512}{7814377697137553} a^{8} - \frac{2397202253043803}{7814377697137553} a^{7} + \frac{68942615300433}{459669276302209} a^{6} + \frac{3842069189556940}{7814377697137553} a^{5} + \frac{3584549717675991}{7814377697137553} a^{4} - \frac{3373069998310917}{7814377697137553} a^{3} - \frac{849373051983961}{7814377697137553} a^{2} + \frac{511113310942603}{7814377697137553} a - \frac{668676592060674}{7814377697137553}$, $\frac{1}{396478081219668026561} a^{15} + \frac{25361}{396478081219668026561} a^{14} + \frac{10644167509365770619}{396478081219668026561} a^{13} + \frac{2447542088742389706}{396478081219668026561} a^{12} + \frac{9298355177244940146}{396478081219668026561} a^{11} - \frac{197093415540712186280}{396478081219668026561} a^{10} - \frac{77030096775921270421}{396478081219668026561} a^{9} - \frac{115548125278671605912}{396478081219668026561} a^{8} + \frac{35109971832332073723}{396478081219668026561} a^{7} + \frac{29403877355702895890}{396478081219668026561} a^{6} + \frac{89080691127159117870}{396478081219668026561} a^{5} - \frac{187573098280870951313}{396478081219668026561} a^{4} + \frac{164907657582528170937}{396478081219668026561} a^{3} - \frac{59032413951863323995}{396478081219668026561} a^{2} - \frac{92804187012462096461}{396478081219668026561} a + \frac{165470094922149227679}{396478081219668026561}$

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

Class group and class number

$C_{30840}$, which has order $30840$ (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

\(\Q(\sqrt{-46}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{2}, \sqrt{-23})\), \(\Q(\zeta_{16})^+\), 4.0.1083392.5, 8.0.1173738225664.2, \(\Q(\zeta_{32})^+\), 8.0.600953971539968.31

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.8.31.6$x^{8} + 16 x^{7} + 28 x^{4} + 16 x^{3} + 2$$8$$1$$31$$C_8$$[3, 4, 5]$
2.8.31.6$x^{8} + 16 x^{7} + 28 x^{4} + 16 x^{3} + 2$$8$$1$$31$$C_8$$[3, 4, 5]$
$23$23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$