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Properties

Label 16.0.16362991598...000.14
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 5^{12}\cdot 47^{8}$
Root discriminant $183.39$
Ramified primes $2, 5, 47$
Class number $26343200$ (GRH)
Class group $[2, 2, 2, 10, 329290]$ (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![5770367861281, 0, 1337601164112, 0, 154584535996, 0, 9928151336, 0, 361387379, 0, 7415648, 0, 83144, 0, 464, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 464*x^14 + 83144*x^12 + 7415648*x^10 + 361387379*x^8 + 9928151336*x^6 + 154584535996*x^4 + 1337601164112*x^2 + 5770367861281)
 
gp: K = bnfinit(x^16 + 464*x^14 + 83144*x^12 + 7415648*x^10 + 361387379*x^8 + 9928151336*x^6 + 154584535996*x^4 + 1337601164112*x^2 + 5770367861281, 1)
 

Normalized defining polynomial

\( x^{16} + 464 x^{14} + 83144 x^{12} + 7415648 x^{10} + 361387379 x^{8} + 9928151336 x^{6} + 154584535996 x^{4} + 1337601164112 x^{2} + 5770367861281 \)

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:  \(1636299159807112140292096000000000000=2^{48}\cdot 5^{12}\cdot 47^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $183.39$
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:  \(3760=2^{4}\cdot 5\cdot 47\)
Dirichlet character group:    $\lbrace$$\chi_{3760}(1,·)$, $\chi_{3760}(2821,·)$, $\chi_{3760}(3009,·)$, $\chi_{3760}(2443,·)$, $\chi_{3760}(2067,·)$, $\chi_{3760}(2069,·)$, $\chi_{3760}(1881,·)$, $\chi_{3760}(1503,·)$, $\chi_{3760}(1127,·)$, $\chi_{3760}(1129,·)$, $\chi_{3760}(941,·)$, $\chi_{3760}(563,·)$, $\chi_{3760}(3383,·)$, $\chi_{3760}(187,·)$, $\chi_{3760}(189,·)$, $\chi_{3760}(3007,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2402159} a^{11} - \frac{203178}{2402159} a^{9} + \frac{1073007}{2402159} a^{7} - \frac{568963}{2402159} a^{5} - \frac{79049}{2402159} a^{3} - \frac{969473}{2402159} a$, $\frac{1}{6915815761} a^{12} + \frac{1364223134}{6915815761} a^{10} + \frac{1324662616}{6915815761} a^{8} + \frac{2132548229}{6915815761} a^{6} + \frac{3365345710}{6915815761} a^{4} + \frac{2670231335}{6915815761} a^{2} - \frac{632}{2879}$, $\frac{1}{6915815761} a^{13} + \frac{347}{6915815761} a^{11} + \frac{2402194583}{6915815761} a^{9} - \frac{3071857498}{6915815761} a^{7} - \frac{927030244}{6915815761} a^{5} - \frac{68182456}{223090831} a^{3} + \frac{1949498884}{6915815761} a$, $\frac{1}{501379238229345183736989270719} a^{14} + \frac{18320090632718542138}{501379238229345183736989270719} a^{12} - \frac{103394043470063305831877738404}{501379238229345183736989270719} a^{10} + \frac{38438805969248938514658265075}{501379238229345183736989270719} a^{8} + \frac{111649989199650980154083317420}{501379238229345183736989270719} a^{6} + \frac{97823068859586554824549072905}{501379238229345183736989270719} a^{4} - \frac{228195081402386831108914581159}{501379238229345183736989270719} a^{2} + \frac{13206226843042993}{86888609232971999}$, $\frac{1}{501379238229345183736989270719} a^{15} + \frac{18320090632718542138}{501379238229345183736989270719} a^{13} - \frac{82195680988502768336393}{501379238229345183736989270719} a^{11} - \frac{51122429147367959161969670502}{501379238229345183736989270719} a^{9} - \frac{135084068008710066454671412228}{501379238229345183736989270719} a^{7} + \frac{86835199772925077439975419301}{501379238229345183736989270719} a^{5} + \frac{66901754734995900521057112440}{501379238229345183736989270719} a^{3} + \frac{68775626968101261471454}{208720254666466784145841} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{10}\times C_{329290}$, which has order $26343200$ (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

$C_4^2$ (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

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\zeta_{16})^+\), \(\Q(\sqrt{2}, \sqrt{5})\), 4.4.51200.1, 4.0.17672000.5, 4.0.4418000.1, 4.0.565504000.4, 4.0.565504000.2, 8.8.2621440000.1, 8.0.4996793344000000.20, 8.0.319794774016000000.15

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
5Data not computed
$47$47.4.2.2$x^{4} - 47 x^{2} + 28717$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
47.4.2.2$x^{4} - 47 x^{2} + 28717$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
47.4.2.2$x^{4} - 47 x^{2} + 28717$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
47.4.2.2$x^{4} - 47 x^{2} + 28717$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$