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Properties

Label 16.0.95428956661...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{12}\cdot 13^{12}$
Root discriminant $64.75$
Ramified primes $2, 5, 13$
Class number $40$ (GRH)
Class group $[2, 20]$ (GRH)
Galois group $C_4^2$ (as 16T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![394481, -655308, 537460, -568676, 530650, -365252, 222174, -128412, 63226, -23128, 11142, -2508, 1083, -152, 54, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 54*x^14 - 152*x^13 + 1083*x^12 - 2508*x^11 + 11142*x^10 - 23128*x^9 + 63226*x^8 - 128412*x^7 + 222174*x^6 - 365252*x^5 + 530650*x^4 - 568676*x^3 + 537460*x^2 - 655308*x + 394481)
 
gp: K = bnfinit(x^16 - 4*x^15 + 54*x^14 - 152*x^13 + 1083*x^12 - 2508*x^11 + 11142*x^10 - 23128*x^9 + 63226*x^8 - 128412*x^7 + 222174*x^6 - 365252*x^5 + 530650*x^4 - 568676*x^3 + 537460*x^2 - 655308*x + 394481, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 54 x^{14} - 152 x^{13} + 1083 x^{12} - 2508 x^{11} + 11142 x^{10} - 23128 x^{9} + 63226 x^{8} - 128412 x^{7} + 222174 x^{6} - 365252 x^{5} + 530650 x^{4} - 568676 x^{3} + 537460 x^{2} - 655308 x + 394481 \)

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:  \(95428956661682176000000000000=2^{24}\cdot 5^{12}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.75$
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:  $2, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(520=2^{3}\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{520}(1,·)$, $\chi_{520}(129,·)$, $\chi_{520}(77,·)$, $\chi_{520}(333,·)$, $\chi_{520}(209,·)$, $\chi_{520}(213,·)$, $\chi_{520}(281,·)$, $\chi_{520}(157,·)$, $\chi_{520}(161,·)$, $\chi_{520}(489,·)$, $\chi_{520}(493,·)$, $\chi_{520}(369,·)$, $\chi_{520}(53,·)$, $\chi_{520}(441,·)$, $\chi_{520}(317,·)$, $\chi_{520}(437,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{42} a^{12} - \frac{1}{14} a^{11} - \frac{1}{6} a^{10} - \frac{3}{14} a^{9} - \frac{3}{7} a^{7} - \frac{1}{3} a^{6} + \frac{3}{7} a^{5} - \frac{1}{14} a^{4} - \frac{3}{14} a^{3} + \frac{1}{3} a^{2} - \frac{1}{7} a + \frac{1}{21}$, $\frac{1}{1218} a^{13} + \frac{2}{609} a^{12} - \frac{19}{174} a^{11} - \frac{289}{1218} a^{10} + \frac{6}{29} a^{9} + \frac{15}{406} a^{8} + \frac{14}{87} a^{7} + \frac{191}{609} a^{6} - \frac{85}{406} a^{5} + \frac{100}{203} a^{4} - \frac{14}{87} a^{3} + \frac{407}{1218} a^{2} - \frac{230}{609} a - \frac{14}{87}$, $\frac{1}{12126345882} a^{14} + \frac{4170665}{12126345882} a^{13} - \frac{1097921}{139383286} a^{12} + \frac{94893464}{866167563} a^{11} + \frac{1219891780}{6063172941} a^{10} - \frac{48791635}{4042115294} a^{9} - \frac{2748869339}{12126345882} a^{8} - \frac{1212531515}{6063172941} a^{7} - \frac{564726241}{6063172941} a^{6} - \frac{431729331}{4042115294} a^{5} + \frac{1530454369}{6063172941} a^{4} + \frac{646203571}{1732335126} a^{3} + \frac{384600521}{866167563} a^{2} - \frac{149262265}{866167563} a + \frac{1016329049}{12126345882}$, $\frac{1}{36333186291998487654798} a^{15} - \frac{1384220176567}{36333186291998487654798} a^{14} + \frac{300442267129318669}{865075864095202087019} a^{13} - \frac{83049565509001737616}{18166593145999243827399} a^{12} - \frac{3285593489709569728247}{18166593145999243827399} a^{11} + \frac{1354739637922678102982}{6055531048666414609133} a^{10} + \frac{1099841366517231182633}{18166593145999243827399} a^{9} + \frac{6807261943458208313951}{36333186291998487654798} a^{8} + \frac{1795774088485513419653}{18166593145999243827399} a^{7} - \frac{4719707460209894807479}{12111062097332829218266} a^{6} - \frac{8158344536396235289165}{36333186291998487654798} a^{5} + \frac{96780620543006572181}{1252868492827534057062} a^{4} + \frac{580902451235244232201}{5190455184571212522114} a^{3} + \frac{537688935855705712661}{1252868492827534057062} a^{2} - \frac{9136989770300376666103}{36333186291998487654798} a - \frac{1193087440528476067611}{6055531048666414609133}$

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

Class group and class number

$C_{2}\times C_{20}$, which has order $40$ (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:  \( 1609101.0076496245 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2$ (as 16T4):

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 $C_4^2$
Character table for $C_4^2$

Intermediate fields

\(\Q(\sqrt{65}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{13}) \), 4.4.17576000.2, \(\Q(\sqrt{5}, \sqrt{13})\), 4.4.17576000.1, 4.0.8000.2, 4.0.1352000.1, 4.0.54925.1, 4.0.2197.1, 8.8.308915776000000.5, 8.0.1827904000000.2, 8.0.3016755625.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
5Data not computed
13Data not computed