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Properties

Label 16.0.29863208046...5024.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{18}\cdot 3^{12}\cdot 11^{8}$
Root discriminant $16.49$
Ramified primes $2, 3, 11$
Class number $1$
Class group Trivial
Galois group $C_8:C_2^2$ (as 16T38)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![64, -192, 216, -72, -23, -39, 102, -93, 72, -27, 12, -15, 16, -9, 6, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 6*x^14 - 9*x^13 + 16*x^12 - 15*x^11 + 12*x^10 - 27*x^9 + 72*x^8 - 93*x^7 + 102*x^6 - 39*x^5 - 23*x^4 - 72*x^3 + 216*x^2 - 192*x + 64)
 
gp: K = bnfinit(x^16 - 3*x^15 + 6*x^14 - 9*x^13 + 16*x^12 - 15*x^11 + 12*x^10 - 27*x^9 + 72*x^8 - 93*x^7 + 102*x^6 - 39*x^5 - 23*x^4 - 72*x^3 + 216*x^2 - 192*x + 64, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 6 x^{14} - 9 x^{13} + 16 x^{12} - 15 x^{11} + 12 x^{10} - 27 x^{9} + 72 x^{8} - 93 x^{7} + 102 x^{6} - 39 x^{5} - 23 x^{4} - 72 x^{3} + 216 x^{2} - 192 x + 64 \)

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:  \(29863208046433665024=2^{18}\cdot 3^{12}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.49$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{4} + \frac{1}{8} a^{3} + \frac{1}{8} a^{2}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{10} + \frac{1}{16} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{3}{16} a^{5} + \frac{3}{16} a^{4} + \frac{1}{16} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{9} - \frac{1}{16} a^{6} - \frac{1}{4} a^{5} - \frac{7}{16} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{10} - \frac{1}{16} a^{7} - \frac{1}{4} a^{6} + \frac{1}{16} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{14} - \frac{1}{32} a^{13} + \frac{1}{64} a^{12} - \frac{1}{32} a^{11} + \frac{3}{64} a^{10} - \frac{3}{32} a^{9} + \frac{5}{64} a^{8} - \frac{1}{16} a^{7} - \frac{3}{64} a^{6} - \frac{1}{16} a^{5} + \frac{3}{64} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{8376128} a^{15} + \frac{54051}{8376128} a^{14} - \frac{156265}{8376128} a^{13} - \frac{208493}{8376128} a^{12} + \frac{68741}{8376128} a^{11} - \frac{395539}{8376128} a^{10} - \frac{791005}{8376128} a^{9} + \frac{793357}{8376128} a^{8} - \frac{739871}{8376128} a^{7} - \frac{1687499}{8376128} a^{6} + \frac{355707}{8376128} a^{5} + \frac{2078347}{8376128} a^{4} + \frac{1010345}{2094032} a^{3} + \frac{208513}{523508} a^{2} - \frac{50043}{261754} a - \frac{29246}{130877}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{62277}{2094032} a^{15} - \frac{76595}{1047016} a^{14} + \frac{298315}{2094032} a^{13} - \frac{202011}{1047016} a^{12} + \frac{781849}{2094032} a^{11} - \frac{61407}{261754} a^{10} + \frac{422661}{2094032} a^{9} - \frac{591891}{1047016} a^{8} + \frac{3536663}{2094032} a^{7} - \frac{877545}{523508} a^{6} + \frac{4050129}{2094032} a^{5} + \frac{221907}{1047016} a^{4} - \frac{87212}{130877} a^{3} - \frac{1854359}{1047016} a^{2} + \frac{615688}{130877} a - \frac{275240}{130877} \) (order $6$)
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:  \( 10720.0696853 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8:C_2^2$ (as 16T38):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_8:C_2^2$
Character table for $C_8:C_2^2$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-11}) \), 4.2.8712.1, 4.2.8712.2, \(\Q(\sqrt{-3}, \sqrt{-11})\), 8.2.5464723968.2, 8.2.5464723968.1, 8.0.75898944.1

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 8 siblings: 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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
$3$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$11$11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$