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Properties

Label 16.0.164025000000000000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{12}\cdot 3^{8}\cdot 5^{14}$
Root discriminant $11.91$
Ramified primes $2, 3, 5$
Class number $1$
Class group Trivial
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, 5, 21, 35, 37, 40, 13, -25, 150, -225, 237, -180, 107, -50, 19, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 19*x^14 - 50*x^13 + 107*x^12 - 180*x^11 + 237*x^10 - 225*x^9 + 150*x^8 - 25*x^7 + 13*x^6 + 40*x^5 + 37*x^4 + 35*x^3 + 21*x^2 + 5*x + 1)
 
gp: K = bnfinit(x^16 - 5*x^15 + 19*x^14 - 50*x^13 + 107*x^12 - 180*x^11 + 237*x^10 - 225*x^9 + 150*x^8 - 25*x^7 + 13*x^6 + 40*x^5 + 37*x^4 + 35*x^3 + 21*x^2 + 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 19 x^{14} - 50 x^{13} + 107 x^{12} - 180 x^{11} + 237 x^{10} - 225 x^{9} + 150 x^{8} - 25 x^{7} + 13 x^{6} + 40 x^{5} + 37 x^{4} + 35 x^{3} + 21 x^{2} + 5 x + 1 \)

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:  \(164025000000000000=2^{12}\cdot 3^{8}\cdot 5^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11.91$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{11} a^{13} - \frac{3}{11} a^{12} - \frac{3}{11} a^{11} + \frac{1}{11} a^{10} - \frac{2}{11} a^{9} + \frac{4}{11} a^{8} + \frac{1}{11} a^{6} + \frac{1}{11} a^{5} + \frac{5}{11} a^{4} + \frac{5}{11} a^{3} + \frac{1}{11} a^{2} + \frac{4}{11} a - \frac{5}{11}$, $\frac{1}{341} a^{14} - \frac{8}{341} a^{13} + \frac{111}{341} a^{12} - \frac{28}{341} a^{11} - \frac{73}{341} a^{10} - \frac{129}{341} a^{9} - \frac{1}{11} a^{8} - \frac{131}{341} a^{7} - \frac{15}{341} a^{6} - \frac{2}{31} a^{5} - \frac{42}{341} a^{4} - \frac{90}{341} a^{3} - \frac{100}{341} a^{2} + \frac{74}{341} a + \frac{36}{341}$, $\frac{1}{25984541} a^{15} - \frac{2117}{25984541} a^{14} + \frac{584872}{25984541} a^{13} + \frac{578104}{25984541} a^{12} + \frac{8318991}{25984541} a^{11} - \frac{1644823}{25984541} a^{10} - \frac{746785}{25984541} a^{9} + \frac{6859487}{25984541} a^{8} - \frac{11484826}{25984541} a^{7} + \frac{1783795}{25984541} a^{6} - \frac{9827516}{25984541} a^{5} + \frac{11058396}{25984541} a^{4} + \frac{7360878}{25984541} a^{3} - \frac{179688}{25984541} a^{2} - \frac{1961129}{25984541} a - \frac{5563575}{25984541}$

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:  \( -\frac{2474092}{25984541} a^{15} + \frac{15445832}{25984541} a^{14} - \frac{62197032}{25984541} a^{13} + \frac{178845044}{25984541} a^{12} - \frac{404705109}{25984541} a^{11} + \frac{729207417}{25984541} a^{10} - \frac{1030457498}{25984541} a^{9} + \frac{1072699911}{25984541} a^{8} - \frac{734662668}{25984541} a^{7} + \frac{134369031}{25984541} a^{6} + \frac{216013184}{25984541} a^{5} - \frac{262376574}{25984541} a^{4} + \frac{54625528}{25984541} a^{3} - \frac{220412}{25984541} a^{2} + \frac{30034579}{25984541} a + \frac{27821863}{25984541} \) (order $30$)
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:  \( 1005.23975626 \)
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

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

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.12.6$x^{8} + 2 x^{6} + 8 x^{4} + 80$$2$$4$$12$$C_8$$[3]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
3Data not computed
5Data not computed