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Properties

Label 16.0.10265933934...0608.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{79}\cdot 19^{8}$
Root discriminant $133.57$
Ramified primes $2, 19$
Class number $2319554$ (GRH)
Class group $[2319554]$ (GRH)
Galois group $C_{16}$ (as 16T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![33967126082, 0, 57207791296, 0, 15807416016, 0, 1663938528, 0, 86011860, 0, 2414368, 0, 37544, 0, 304, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 304*x^14 + 37544*x^12 + 2414368*x^10 + 86011860*x^8 + 1663938528*x^6 + 15807416016*x^4 + 57207791296*x^2 + 33967126082)
 
gp: K = bnfinit(x^16 + 304*x^14 + 37544*x^12 + 2414368*x^10 + 86011860*x^8 + 1663938528*x^6 + 15807416016*x^4 + 57207791296*x^2 + 33967126082, 1)
 

Normalized defining polynomial

\( x^{16} + 304 x^{14} + 37544 x^{12} + 2414368 x^{10} + 86011860 x^{8} + 1663938528 x^{6} + 15807416016 x^{4} + 57207791296 x^{2} + 33967126082 \)

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:  \(10265933934658824457230071374020608=2^{79}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $133.57$
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:  $2, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1216=2^{6}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{1216}(1,·)$, $\chi_{1216}(645,·)$, $\chi_{1216}(457,·)$, $\chi_{1216}(1101,·)$, $\chi_{1216}(913,·)$, $\chi_{1216}(341,·)$, $\chi_{1216}(153,·)$, $\chi_{1216}(797,·)$, $\chi_{1216}(609,·)$, $\chi_{1216}(37,·)$, $\chi_{1216}(1065,·)$, $\chi_{1216}(493,·)$, $\chi_{1216}(305,·)$, $\chi_{1216}(949,·)$, $\chi_{1216}(761,·)$, $\chi_{1216}(189,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{19} a^{2}$, $\frac{1}{19} a^{3}$, $\frac{1}{361} a^{4}$, $\frac{1}{361} a^{5}$, $\frac{1}{6859} a^{6}$, $\frac{1}{6859} a^{7}$, $\frac{1}{130321} a^{8}$, $\frac{1}{130321} a^{9}$, $\frac{1}{2476099} a^{10}$, $\frac{1}{2476099} a^{11}$, $\frac{1}{47045881} a^{12}$, $\frac{1}{47045881} a^{13}$, $\frac{1}{893871739} a^{14}$, $\frac{1}{893871739} a^{15}$

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

Class group and class number

$C_{2319554}$, which has order $2319554$ (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

$C_{16}$ (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

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), \(\Q(\zeta_{32})^+\)

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 $16$ $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ $16$ $16$

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
19Data not computed