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Properties

Label 16.0.64265535869...4624.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{62}\cdot 139^{8}$
Root discriminant $172.98$
Ramified primes $2, 139$
Class number $12406149$ (GRH)
Class group $[3, 3, 1378461]$ (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![3472637458463, -675681491256, 734957493100, -121613242624, 68561518138, -9593148040, 3683233888, -429720664, 124608155, -11803912, 2716496, -198872, 37210, -1904, 292, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 292*x^14 - 1904*x^13 + 37210*x^12 - 198872*x^11 + 2716496*x^10 - 11803912*x^9 + 124608155*x^8 - 429720664*x^7 + 3683233888*x^6 - 9593148040*x^5 + 68561518138*x^4 - 121613242624*x^3 + 734957493100*x^2 - 675681491256*x + 3472637458463)
 
gp: K = bnfinit(x^16 - 8*x^15 + 292*x^14 - 1904*x^13 + 37210*x^12 - 198872*x^11 + 2716496*x^10 - 11803912*x^9 + 124608155*x^8 - 429720664*x^7 + 3683233888*x^6 - 9593148040*x^5 + 68561518138*x^4 - 121613242624*x^3 + 734957493100*x^2 - 675681491256*x + 3472637458463, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 292 x^{14} - 1904 x^{13} + 37210 x^{12} - 198872 x^{11} + 2716496 x^{10} - 11803912 x^{9} + 124608155 x^{8} - 429720664 x^{7} + 3683233888 x^{6} - 9593148040 x^{5} + 68561518138 x^{4} - 121613242624 x^{3} + 734957493100 x^{2} - 675681491256 x + 3472637458463 \)

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:  \(642655358696704749650787270733594624=2^{62}\cdot 139^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $172.98$
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:  $2, 139$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4448=2^{5}\cdot 139\)
Dirichlet character group:    $\lbrace$$\chi_{4448}(1,·)$, $\chi_{4448}(1669,·)$, $\chi_{4448}(833,·)$, $\chi_{4448}(3337,·)$, $\chi_{4448}(1389,·)$, $\chi_{4448}(277,·)$, $\chi_{4448}(1113,·)$, $\chi_{4448}(1945,·)$, $\chi_{4448}(2781,·)$, $\chi_{4448}(2501,·)$, $\chi_{4448}(2225,·)$, $\chi_{4448}(557,·)$, $\chi_{4448}(3613,·)$, $\chi_{4448}(3057,·)$, $\chi_{4448}(3893,·)$, $\chi_{4448}(4169,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{72782914696103338467646481} a^{14} - \frac{7}{72782914696103338467646481} a^{13} - \frac{34753946407992268639502751}{72782914696103338467646481} a^{12} - \frac{9825065640356403565922846}{72782914696103338467646481} a^{11} - \frac{20206292814429350565287193}{72782914696103338467646481} a^{10} + \frac{9137279035155439344945684}{72782914696103338467646481} a^{9} - \frac{17123913659407484939500822}{72782914696103338467646481} a^{8} - \frac{21620826921119797069175355}{72782914696103338467646481} a^{7} - \frac{34395415893690319991010957}{72782914696103338467646481} a^{6} + \frac{34179777112150180318483568}{72782914696103338467646481} a^{5} + \frac{8248733667694938125848675}{72782914696103338467646481} a^{4} + \frac{35217778616408420659323616}{72782914696103338467646481} a^{3} + \frac{12883485691380644851820262}{72782914696103338467646481} a^{2} - \frac{2030853381288078646921668}{4281347923300196380449793} a - \frac{6519619308310361992513499}{72782914696103338467646481}$, $\frac{1}{33885280813881475152120442581828577} a^{15} + \frac{232783201}{33885280813881475152120442581828577} a^{14} - \frac{12668705379051349147571995315668175}{33885280813881475152120442581828577} a^{13} + \frac{6485888183181328817879693574193527}{33885280813881475152120442581828577} a^{12} + \frac{7158778480212120627885598877317957}{33885280813881475152120442581828577} a^{11} - \frac{5209325593588802659304668415693014}{33885280813881475152120442581828577} a^{10} - \frac{16827516165051999386087278193466532}{33885280813881475152120442581828577} a^{9} - \frac{4775617220055191354452397994880195}{33885280813881475152120442581828577} a^{8} + \frac{10368024233253554421424187286158080}{33885280813881475152120442581828577} a^{7} - \frac{9699907353494592142792610022557271}{33885280813881475152120442581828577} a^{6} + \frac{15292912710852832595329563077400555}{33885280813881475152120442581828577} a^{5} + \frac{277031865671852254820258557925824}{33885280813881475152120442581828577} a^{4} - \frac{7803002243541358431522174205088989}{33885280813881475152120442581828577} a^{3} + \frac{13984212504225126546628600118963806}{33885280813881475152120442581828577} a^{2} - \frac{4027859556950520954461246742783139}{33885280813881475152120442581828577} a - \frac{14737755540559274948361731984998648}{33885280813881475152120442581828577}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{1378461}$, which has order $12406149$ (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

\(\Q(\sqrt{-139}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-278}) \), \(\Q(\sqrt{2}, \sqrt{-139})\), \(\Q(\zeta_{16})^+\), 4.0.39569408.2, 8.0.1565738049470464.11, \(\Q(\zeta_{32})^+\), 8.0.801657881328877568.6

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
139Data not computed