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Properties

Label 16.0.15369536160...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{8}\cdot 5^{12}\cdot 11^{4}$
Root discriminant $21.09$
Ramified primes $2, 3, 5, 11$
Class number $4$
Class group $[4]$
Galois group $C_2^3:(C_2\times C_4)$ (as 16T68)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 18, -26, 27, 10, -8, 12, 25, 12, -8, 10, 27, -26, 18, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 18*x^14 - 26*x^13 + 27*x^12 + 10*x^11 - 8*x^10 + 12*x^9 + 25*x^8 + 12*x^7 - 8*x^6 + 10*x^5 + 27*x^4 - 26*x^3 + 18*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^16 - 6*x^15 + 18*x^14 - 26*x^13 + 27*x^12 + 10*x^11 - 8*x^10 + 12*x^9 + 25*x^8 + 12*x^7 - 8*x^6 + 10*x^5 + 27*x^4 - 26*x^3 + 18*x^2 - 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 18 x^{14} - 26 x^{13} + 27 x^{12} + 10 x^{11} - 8 x^{10} + 12 x^{9} + 25 x^{8} + 12 x^{7} - 8 x^{6} + 10 x^{5} + 27 x^{4} - 26 x^{3} + 18 x^{2} - 6 x + 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1536953616000000000000=2^{16}\cdot 3^{8}\cdot 5^{12}\cdot 11^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.09$
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:  $2, 3, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{6} a^{10} - \frac{1}{3} a^{5} + \frac{1}{6}$, $\frac{1}{6} a^{11} - \frac{1}{3} a^{6} + \frac{1}{6} a$, $\frac{1}{18} a^{12} + \frac{1}{18} a^{11} + \frac{1}{18} a^{10} - \frac{1}{6} a^{9} + \frac{1}{18} a^{7} - \frac{4}{9} a^{6} + \frac{7}{18} a^{5} - \frac{1}{6} a^{3} - \frac{5}{18} a^{2} - \frac{4}{9} a + \frac{7}{18}$, $\frac{1}{54} a^{13} - \frac{1}{18} a^{11} - \frac{1}{54} a^{10} + \frac{2}{9} a^{9} + \frac{5}{27} a^{8} - \frac{1}{3} a^{7} - \frac{1}{9} a^{6} + \frac{7}{27} a^{5} + \frac{4}{9} a^{4} + \frac{25}{54} a^{3} + \frac{4}{9} a^{2} + \frac{1}{18} a + \frac{23}{54}$, $\frac{1}{54} a^{14} + \frac{1}{27} a^{11} - \frac{1}{18} a^{10} + \frac{1}{54} a^{9} + \frac{1}{6} a^{8} - \frac{1}{18} a^{7} + \frac{17}{54} a^{6} - \frac{1}{2} a^{5} - \frac{1}{27} a^{4} + \frac{5}{18} a^{3} + \frac{5}{18} a^{2} - \frac{1}{54} a - \frac{4}{9}$, $\frac{1}{1782} a^{15} + \frac{2}{891} a^{14} - \frac{4}{891} a^{13} - \frac{20}{891} a^{12} + \frac{23}{1782} a^{11} + \frac{1}{198} a^{10} + \frac{8}{891} a^{9} - \frac{145}{891} a^{8} - \frac{68}{891} a^{7} - \frac{80}{891} a^{6} + \frac{179}{594} a^{5} - \frac{214}{891} a^{4} + \frac{68}{891} a^{3} + \frac{403}{891} a^{2} + \frac{455}{1782} a - \frac{439}{1782}$

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

Class group and class number

$C_{4}$, which has order $4$

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:  \( -\frac{280}{297} a^{15} + \frac{1586}{297} a^{14} - \frac{997}{66} a^{13} + \frac{5689}{297} a^{12} - \frac{10933}{594} a^{11} - \frac{4720}{297} a^{10} + \frac{866}{297} a^{9} - \frac{800}{99} a^{8} - \frac{7130}{297} a^{7} - \frac{4535}{297} a^{6} + \frac{1814}{297} a^{5} - \frac{1270}{297} a^{4} - \frac{4769}{198} a^{3} + \frac{5650}{297} a^{2} - \frac{5287}{594} a + \frac{74}{33} \) (order $10$)
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:  \( 16507.2216551 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3:(C_2\times C_4)$ (as 16T68):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 34 conjugacy class representatives for $C_2^3:(C_2\times C_4)$
Character table for $C_2^3:(C_2\times C_4)$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{15}) \), 4.0.18000.1, \(\Q(\zeta_{5})\), \(\Q(\sqrt{3}, \sqrt{5})\), 8.4.39204000000.1, 8.4.1568160000.5, 8.0.324000000.3

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$