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Properties

Label 16.0.68158561146...0416.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{12}\cdot 13^{4}\cdot 17^{12}$
Root discriminant $26.74$
Ramified primes $2, 13, 17$
Class number $2$
Class group $[2]$
Galois group $C_2^5.C_2$ (as 16T79)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4096, 0, 1472, 0, -2316, 0, 79, 0, 1023, 0, 131, 0, 1, 0, 5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 5*x^14 + x^12 + 131*x^10 + 1023*x^8 + 79*x^6 - 2316*x^4 + 1472*x^2 + 4096)
 
gp: K = bnfinit(x^16 + 5*x^14 + x^12 + 131*x^10 + 1023*x^8 + 79*x^6 - 2316*x^4 + 1472*x^2 + 4096, 1)
 

Normalized defining polynomial

\( x^{16} + 5 x^{14} + x^{12} + 131 x^{10} + 1023 x^{8} + 79 x^{6} - 2316 x^{4} + 1472 x^{2} + 4096 \)

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:  \(68158561146958659260416=2^{12}\cdot 13^{4}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.74$
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:  $2, 13, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{11} + \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{168} a^{12} - \frac{1}{56} a^{10} + \frac{5}{168} a^{8} - \frac{1}{4} a^{7} - \frac{17}{168} a^{6} - \frac{1}{2} a^{5} + \frac{17}{56} a^{4} - \frac{1}{2} a^{3} + \frac{11}{168} a^{2} - \frac{1}{4} a - \frac{8}{21}$, $\frac{1}{336} a^{13} - \frac{1}{112} a^{11} + \frac{5}{336} a^{9} + \frac{67}{336} a^{7} + \frac{45}{112} a^{5} - \frac{1}{2} a^{4} + \frac{11}{336} a^{3} - \frac{1}{2} a^{2} + \frac{5}{84} a$, $\frac{1}{137225778816} a^{14} - \frac{1}{672} a^{13} + \frac{75839095}{45741926272} a^{12} + \frac{1}{224} a^{11} + \frac{8122601057}{137225778816} a^{10} + \frac{79}{672} a^{9} - \frac{10658227229}{137225778816} a^{8} + \frac{101}{672} a^{7} - \frac{4462554763}{45741926272} a^{6} - \frac{45}{224} a^{5} - \frac{6470542951}{19603682688} a^{4} - \frac{95}{672} a^{3} - \frac{9430148683}{34306444704} a^{2} - \frac{47}{168} a + \frac{55616146}{357358799}$, $\frac{1}{1097806230528} a^{15} + \frac{75839095}{365935410176} a^{13} - \frac{129103177759}{1097806230528} a^{11} - \frac{10658227229}{1097806230528} a^{9} - \frac{27333517899}{365935410176} a^{7} - \frac{16272384295}{156829461504} a^{5} - \frac{26583371035}{274451557632} a^{3} - \frac{2390279301}{5717740784} a - \frac{1}{2}$

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

Class group and class number

$C_{2}$, which has order $2$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 132688.682342 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2$ (as 16T79):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $C_2^5.C_2$
Character table for $C_2^5.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.30056.2, 4.4.4913.1, 4.0.3757.1, 4.0.39304.1, 4.4.510952.2, 4.0.2312.1, 4.0.63869.1, 8.0.261071946304.2, 8.8.261071946304.3, 8.0.261071946304.5, 8.0.903363136.2, 8.0.1544804416.1, 8.0.4079249161.1, 8.0.261071946304.13

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
$13$13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
$17$17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$