↑

Properties

Label 16.0.10294261539...9728.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 11^{8}\cdot 17^{15}$
Root discriminant $133.59$
Ramified primes $2, 11, 17$
Class number $2945032$ (GRH)
Class group $[2, 2, 736258]$ (GRH)
Galois group $C_{16}$ (as 16T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![932889850112, 0, 508849009152, 0, 80953251456, 0, 5782375104, 0, 219029360, 0, 4706416, 0, 57596, 0, 374, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 374*x^14 + 57596*x^12 + 4706416*x^10 + 219029360*x^8 + 5782375104*x^6 + 80953251456*x^4 + 508849009152*x^2 + 932889850112)
 
gp: K = bnfinit(x^16 + 374*x^14 + 57596*x^12 + 4706416*x^10 + 219029360*x^8 + 5782375104*x^6 + 80953251456*x^4 + 508849009152*x^2 + 932889850112, 1)
 

Normalized defining polynomial

\( x^{16} + 374 x^{14} + 57596 x^{12} + 4706416 x^{10} + 219029360 x^{8} + 5782375104 x^{6} + 80953251456 x^{4} + 508849009152 x^{2} + 932889850112 \)

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:  \(10294261539221265797567188438089728=2^{24}\cdot 11^{8}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $133.59$
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:  $2, 11, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1496=2^{3}\cdot 11\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{1496}(1,·)$, $\chi_{1496}(131,·)$, $\chi_{1496}(1409,·)$, $\chi_{1496}(1099,·)$, $\chi_{1496}(1233,·)$, $\chi_{1496}(483,·)$, $\chi_{1496}(89,·)$, $\chi_{1496}(923,·)$, $\chi_{1496}(353,·)$, $\chi_{1496}(1187,·)$, $\chi_{1496}(529,·)$, $\chi_{1496}(1451,·)$, $\chi_{1496}(1363,·)$, $\chi_{1496}(441,·)$, $\chi_{1496}(705,·)$, $\chi_{1496}(571,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{22} a^{2}$, $\frac{1}{22} a^{3}$, $\frac{1}{484} a^{4}$, $\frac{1}{484} a^{5}$, $\frac{1}{10648} a^{6}$, $\frac{1}{10648} a^{7}$, $\frac{1}{234256} a^{8}$, $\frac{1}{234256} a^{9}$, $\frac{1}{5153632} a^{10}$, $\frac{1}{5153632} a^{11}$, $\frac{1}{113379904} a^{12}$, $\frac{1}{113379904} a^{13}$, $\frac{1}{2494357888} a^{14}$, $\frac{1}{2494357888} a^{15}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{736258}$, which has order $2945032$ (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:  \( 3640.012213375973 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{16}$ (as 16T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 16
The 16 conjugacy class representatives for $C_{16}$
Character table for $C_{16}$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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 $16$ $16$ $16$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$2.8.12.6$x^{8} + 2 x^{6} + 8 x^{4} + 80$$2$$4$$12$$C_8$$[3]^{4}$
2.8.12.6$x^{8} + 2 x^{6} + 8 x^{4} + 80$$2$$4$$12$$C_8$$[3]^{4}$
11Data not computed
17Data not computed