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Properties

Label 16.16.5441955840...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{24}\cdot 3^{12}\cdot 5^{14}$
Root discriminant $26.36$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_8: C_2$ (as 16T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 36, 60, -350, -655, 1068, 1908, -1770, -2280, 1840, 1038, -1002, -30, 170, -30, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 30*x^14 + 170*x^13 - 30*x^12 - 1002*x^11 + 1038*x^10 + 1840*x^9 - 2280*x^8 - 1770*x^7 + 1908*x^6 + 1068*x^5 - 655*x^4 - 350*x^3 + 60*x^2 + 36*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 - 30*x^14 + 170*x^13 - 30*x^12 - 1002*x^11 + 1038*x^10 + 1840*x^9 - 2280*x^8 - 1770*x^7 + 1908*x^6 + 1068*x^5 - 655*x^4 - 350*x^3 + 60*x^2 + 36*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 30 x^{14} + 170 x^{13} - 30 x^{12} - 1002 x^{11} + 1038 x^{10} + 1840 x^{9} - 2280 x^{8} - 1770 x^{7} + 1908 x^{6} + 1068 x^{5} - 655 x^{4} - 350 x^{3} + 60 x^{2} + 36 x + 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(54419558400000000000000=2^{24}\cdot 3^{12}\cdot 5^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.36$
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 Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{9} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{10} + \frac{2}{5} a^{5} + \frac{1}{5}$, $\frac{1}{5} a^{11} + \frac{2}{5} a^{6} + \frac{1}{5} a$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{7} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{13} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{14} + \frac{2}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{1102072505} a^{15} + \frac{91828939}{1102072505} a^{14} - \frac{18940053}{220414501} a^{13} + \frac{109933237}{1102072505} a^{12} + \frac{39154149}{1102072505} a^{11} - \frac{1685269}{18679195} a^{10} - \frac{94939992}{1102072505} a^{9} - \frac{17842156}{220414501} a^{8} - \frac{420297886}{1102072505} a^{7} - \frac{19735042}{1102072505} a^{6} - \frac{92105140}{220414501} a^{5} - \frac{52360621}{1102072505} a^{4} + \frac{37851238}{220414501} a^{3} + \frac{36933032}{1102072505} a^{2} - \frac{383588281}{1102072505} a - \frac{184743653}{1102072505}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $15$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1123278.07398 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}$ (as 16T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_8: C_2$
Character table for $C_8: C_2$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{20})^+\), \(\Q(\zeta_{60})^+\), 8.8.14580000000.1 x2

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

Sibling fields

Degree 8 sibling: 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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{16}$

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
2Data not computed
3Data not computed
5Data not computed