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Properties

Label 16.0.41995136315...4144.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{12}\cdot 3^{14}\cdot 11^{8}$
Root discriminant $14.59$
Ramified primes $2, 3, 11$
Class number $1$
Class group Trivial
Galois group $D_8:C_2$ (as 16T44)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -12, 64, -189, 349, -468, 496, -366, 196, -24, -50, 60, -29, 6, 4, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 4*x^14 + 6*x^13 - 29*x^12 + 60*x^11 - 50*x^10 - 24*x^9 + 196*x^8 - 366*x^7 + 496*x^6 - 468*x^5 + 349*x^4 - 189*x^3 + 64*x^2 - 12*x + 1)
 
gp: K = bnfinit(x^16 - 3*x^15 + 4*x^14 + 6*x^13 - 29*x^12 + 60*x^11 - 50*x^10 - 24*x^9 + 196*x^8 - 366*x^7 + 496*x^6 - 468*x^5 + 349*x^4 - 189*x^3 + 64*x^2 - 12*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 4 x^{14} + 6 x^{13} - 29 x^{12} + 60 x^{11} - 50 x^{10} - 24 x^{9} + 196 x^{8} - 366 x^{7} + 496 x^{6} - 468 x^{5} + 349 x^{4} - 189 x^{3} + 64 x^{2} - 12 x + 1 \)

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:  \(4199513631529734144=2^{12}\cdot 3^{14}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{12} + \frac{1}{6} a^{11} - \frac{1}{6} a^{10} - \frac{1}{2} a^{9} + \frac{1}{6} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{6} a^{2} - \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{6} a^{13} + \frac{1}{6} a^{11} + \frac{1}{6} a^{10} - \frac{1}{3} a^{9} + \frac{1}{6} a^{8} - \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{6} a^{5} - \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{6} a^{2} - \frac{1}{6}$, $\frac{1}{6} a^{14} - \frac{1}{6} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{6} a^{7} + \frac{1}{6} a^{6} + \frac{1}{6} a^{4} + \frac{1}{3} a^{3} + \frac{1}{6} a^{2} + \frac{1}{6} a - \frac{1}{6}$, $\frac{1}{1310322} a^{15} - \frac{31899}{436774} a^{14} + \frac{2217}{436774} a^{13} - \frac{40535}{655161} a^{12} + \frac{75575}{655161} a^{11} + \frac{29872}{655161} a^{10} - \frac{103750}{655161} a^{9} + \frac{161081}{655161} a^{8} + \frac{67397}{1310322} a^{7} - \frac{260387}{655161} a^{6} - \frac{46100}{655161} a^{5} - \frac{142621}{655161} a^{4} + \frac{630715}{1310322} a^{3} - \frac{26209}{1310322} a^{2} + \frac{3377}{50397} a + \frac{266806}{655161}$

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:  \( -\frac{35597}{16799} a^{15} + \frac{569309}{100794} a^{14} - \frac{665603}{100794} a^{13} - \frac{1501249}{100794} a^{12} + \frac{1894215}{33598} a^{11} - \frac{1818457}{16799} a^{10} + \frac{2351843}{33598} a^{9} + \frac{3676060}{50397} a^{8} - \frac{39217403}{100794} a^{7} + \frac{32468854}{50397} a^{6} - \frac{28193579}{33598} a^{5} + \frac{12085581}{16799} a^{4} - \frac{17220449}{33598} a^{3} + \frac{24282413}{100794} a^{2} - \frac{3209347}{50397} a + \frac{797441}{100794} \) (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:  \( 2673.79224339 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_8:C_2$ (as 16T44):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $D_8:C_2$
Character table for $D_8:C_2$

Intermediate fields

\(\Q(\sqrt{33}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-3}) \), 4.4.13068.1, 4.0.13068.1, \(\Q(\sqrt{-3}, \sqrt{-11})\), 8.0.170772624.1

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 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 ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$3$3.8.7.1$x^{8} + 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
3.8.7.1$x^{8} + 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
$11$11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$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}$