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Properties

Label 16.8.67184640000...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 3^{8}\cdot 5^{14}$
Root discriminant $20.03$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_8:C_2):C_2$ (as 16T16)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 14, 66, 216, 378, 44, -356, -232, -12, 34, -34, 48, 23, -2, -6, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 6*x^14 - 2*x^13 + 23*x^12 + 48*x^11 - 34*x^10 + 34*x^9 - 12*x^8 - 232*x^7 - 356*x^6 + 44*x^5 + 378*x^4 + 216*x^3 + 66*x^2 + 14*x + 1)
 
gp: K = bnfinit(x^16 - 2*x^15 - 6*x^14 - 2*x^13 + 23*x^12 + 48*x^11 - 34*x^10 + 34*x^9 - 12*x^8 - 232*x^7 - 356*x^6 + 44*x^5 + 378*x^4 + 216*x^3 + 66*x^2 + 14*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 6 x^{14} - 2 x^{13} + 23 x^{12} + 48 x^{11} - 34 x^{10} + 34 x^{9} - 12 x^{8} - 232 x^{7} - 356 x^{6} + 44 x^{5} + 378 x^{4} + 216 x^{3} + 66 x^{2} + 14 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(671846400000000000000=2^{24}\cdot 3^{8}\cdot 5^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.03$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{1121423241232136831} a^{15} - \frac{505253844738543781}{1121423241232136831} a^{14} - \frac{201725014494372613}{1121423241232136831} a^{13} - \frac{369477782428659638}{1121423241232136831} a^{12} - \frac{181322380678150592}{1121423241232136831} a^{11} + \frac{197500871653365609}{1121423241232136831} a^{10} + \frac{436493190230302693}{1121423241232136831} a^{9} + \frac{132286856085050633}{1121423241232136831} a^{8} - \frac{102280603838667205}{1121423241232136831} a^{7} - \frac{77093216610502829}{1121423241232136831} a^{6} - \frac{478505934970715210}{1121423241232136831} a^{5} - \frac{334686705914947786}{1121423241232136831} a^{4} + \frac{263128639480659734}{1121423241232136831} a^{3} + \frac{442542478022620742}{1121423241232136831} a^{2} - \frac{306125330657754618}{1121423241232136831} a - \frac{519220889392287319}{1121423241232136831}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 24091.2329353 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}:C_2$ (as 16T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $(C_8:C_2):C_2$
Character table for $(C_8:C_2):C_2$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{15}) \), \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{20})^+\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\zeta_{60})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 16 siblings: data not computed

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
2Data not computed
3Data not computed
5Data not computed