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Properties

Label 16.0.56119884623...000.44
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{8}\cdot 5^{12}\cdot 13^{8}$
Root discriminant $83.53$
Ramified primes $2, 3, 5, 13$
Class number $133120$ (GRH)
Class group $[2, 4, 8, 2080]$ (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![382482481, -189953172, 234422198, -95384228, 64443548, -21926452, 10483974, -2996084, 1107333, -262972, 77626, -14860, 3508, -504, 92, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 92*x^14 - 504*x^13 + 3508*x^12 - 14860*x^11 + 77626*x^10 - 262972*x^9 + 1107333*x^8 - 2996084*x^7 + 10483974*x^6 - 21926452*x^5 + 64443548*x^4 - 95384228*x^3 + 234422198*x^2 - 189953172*x + 382482481)
 
gp: K = bnfinit(x^16 - 8*x^15 + 92*x^14 - 504*x^13 + 3508*x^12 - 14860*x^11 + 77626*x^10 - 262972*x^9 + 1107333*x^8 - 2996084*x^7 + 10483974*x^6 - 21926452*x^5 + 64443548*x^4 - 95384228*x^3 + 234422198*x^2 - 189953172*x + 382482481, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 92 x^{14} - 504 x^{13} + 3508 x^{12} - 14860 x^{11} + 77626 x^{10} - 262972 x^{9} + 1107333 x^{8} - 2996084 x^{7} + 10483974 x^{6} - 21926452 x^{5} + 64443548 x^{4} - 95384228 x^{3} + 234422198 x^{2} - 189953172 x + 382482481 \)

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:  \(5611988462318125056000000000000=2^{32}\cdot 3^{8}\cdot 5^{12}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $83.53$
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:  $2, 3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1560=2^{3}\cdot 3\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{1560}(1,·)$, $\chi_{1560}(389,·)$, $\chi_{1560}(1481,·)$, $\chi_{1560}(1483,·)$, $\chi_{1560}(781,·)$, $\chi_{1560}(1169,·)$, $\chi_{1560}(467,·)$, $\chi_{1560}(469,·)$, $\chi_{1560}(1327,·)$, $\chi_{1560}(1247,·)$, $\chi_{1560}(1249,·)$, $\chi_{1560}(547,·)$, $\chi_{1560}(623,·)$, $\chi_{1560}(1403,·)$, $\chi_{1560}(701,·)$, $\chi_{1560}(703,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a$, $\frac{1}{4757893582676181418} a^{14} - \frac{7}{4757893582676181418} a^{13} + \frac{563730373568230345}{4757893582676181418} a^{12} - \frac{644053562305065}{3053846972192671} a^{11} - \frac{313859293980189991}{2378946791338090709} a^{10} + \frac{419254203660648979}{2378946791338090709} a^{9} + \frac{20716387892670401}{116046184943321498} a^{8} - \frac{217376839202605762}{2378946791338090709} a^{7} + \frac{312150889342897816}{2378946791338090709} a^{6} + \frac{324130544633695745}{2378946791338090709} a^{5} - \frac{1775058287954727537}{4757893582676181418} a^{4} + \frac{555207561262299977}{2378946791338090709} a^{3} + \frac{1346636417608762303}{4757893582676181418} a^{2} + \frac{12562510692322995}{250415451719799022} a + \frac{11315768339626209}{58023092471660749}$, $\frac{1}{22382278065262182349993738} a^{15} + \frac{2352113}{22382278065262182349993738} a^{14} + \frac{2365333550189845260800500}{11191139032631091174996869} a^{13} - \frac{722763401373883908369562}{11191139032631091174996869} a^{12} - \frac{32213863881271207805916}{141659987754823938924011} a^{11} + \frac{2055874852311532275934133}{22382278065262182349993738} a^{10} - \frac{243523132141486575573082}{11191139032631091174996869} a^{9} - \frac{3201689921039064920553237}{22382278065262182349993738} a^{8} - \frac{718890132726879531557091}{11191139032631091174996869} a^{7} + \frac{1906357010133412641316177}{22382278065262182349993738} a^{6} + \frac{2757015038945108268099909}{11191139032631091174996869} a^{5} + \frac{8088509656295422104332525}{22382278065262182349993738} a^{4} + \frac{3236556665715110301292437}{22382278065262182349993738} a^{3} + \frac{1274710861702555609443723}{11191139032631091174996869} a^{2} + \frac{5530292966468268193177116}{11191139032631091174996869} a + \frac{127540001681852723901249}{545909221103955667073018}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{8}\times C_{2080}$, which has order $133120$ (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:  \( 7114.135357253273 \) (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

\(\Q(\sqrt{-390}) \), \(\Q(\sqrt{-195}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-78}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{-195})\), \(\Q(\sqrt{5}, \sqrt{-78})\), \(\Q(\sqrt{10}, \sqrt{-39})\), \(\Q(\sqrt{5}, \sqrt{-39})\), \(\Q(\sqrt{10}, \sqrt{-78})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-39})\), 4.4.8000.1, 4.0.3042000.1, 4.0.12168000.2, \(\Q(\zeta_{20})^+\), 8.0.5922408960000.17, 8.0.2368963584000000.120, 8.0.2368963584000000.134, 8.0.148060224000000.143, 8.0.9253764000000.5, \(\Q(\zeta_{40})^+\), 8.0.2368963584000000.146

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$13$13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$