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Properties

Label 16.16.2329808512...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{12}\cdot 5^{12}\cdot 13^{12}$
Root discriminant $38.50$
Ramified primes $2, 5, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4^2:C_2$ (as 16T17)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![191, -1669, 3503, 3454, -13448, -621, 18083, -3484, -10981, 3615, 3039, -1402, -282, 210, -10, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 - 10*x^14 + 210*x^13 - 282*x^12 - 1402*x^11 + 3039*x^10 + 3615*x^9 - 10981*x^8 - 3484*x^7 + 18083*x^6 - 621*x^5 - 13448*x^4 + 3454*x^3 + 3503*x^2 - 1669*x + 191)
 
gp: K = bnfinit(x^16 - 8*x^15 - 10*x^14 + 210*x^13 - 282*x^12 - 1402*x^11 + 3039*x^10 + 3615*x^9 - 10981*x^8 - 3484*x^7 + 18083*x^6 - 621*x^5 - 13448*x^4 + 3454*x^3 + 3503*x^2 - 1669*x + 191, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} - 10 x^{14} + 210 x^{13} - 282 x^{12} - 1402 x^{11} + 3039 x^{10} + 3615 x^{9} - 10981 x^{8} - 3484 x^{7} + 18083 x^{6} - 621 x^{5} - 13448 x^{4} + 3454 x^{3} + 3503 x^{2} - 1669 x + 191 \)

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:  \(23298085122481000000000000=2^{12}\cdot 5^{12}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.50$
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:  $2, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{54} a^{12} - \frac{1}{9} a^{11} + \frac{1}{54} a^{10} - \frac{2}{27} a^{9} + \frac{1}{27} a^{8} + \frac{2}{27} a^{7} + \frac{10}{27} a^{6} + \frac{13}{54} a^{5} - \frac{1}{2} a^{4} + \frac{7}{18} a^{3} - \frac{1}{3} a^{2} - \frac{7}{54} a + \frac{11}{54}$, $\frac{1}{54} a^{13} + \frac{1}{54} a^{11} + \frac{1}{27} a^{10} - \frac{2}{27} a^{9} - \frac{1}{27} a^{8} + \frac{4}{27} a^{7} + \frac{7}{54} a^{6} + \frac{5}{18} a^{5} + \frac{1}{18} a^{4} - \frac{7}{54} a^{2} + \frac{5}{54} a + \frac{2}{9}$, $\frac{1}{10314} a^{14} - \frac{7}{10314} a^{13} + \frac{1}{382} a^{12} - \frac{71}{10314} a^{11} - \frac{824}{5157} a^{10} - \frac{265}{1719} a^{9} - \frac{674}{5157} a^{8} + \frac{4123}{10314} a^{7} + \frac{59}{573} a^{6} + \frac{2017}{5157} a^{5} - \frac{481}{3438} a^{4} - \frac{1243}{10314} a^{3} - \frac{46}{573} a^{2} - \frac{1069}{10314} a + \frac{1}{27}$, $\frac{1}{13418514} a^{15} + \frac{643}{13418514} a^{14} + \frac{61628}{6709257} a^{13} + \frac{9439}{2236419} a^{12} + \frac{1560613}{13418514} a^{11} + \frac{52213}{1490946} a^{10} + \frac{79031}{745473} a^{9} + \frac{44449}{4472838} a^{8} + \frac{2158559}{6709257} a^{7} - \frac{720107}{1490946} a^{6} - \frac{1988519}{4472838} a^{5} + \frac{4974167}{13418514} a^{4} - \frac{4146503}{13418514} a^{3} + \frac{525976}{6709257} a^{2} + \frac{1117988}{2236419} a - \frac{3325}{7806}$

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:  \( 43291466.6921 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_4):C_4$ (as 16T17):

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 $C_4^2:C_2$
Character table for $C_4^2:C_2$

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{65}) \), 4.4.274625.1, \(\Q(\sqrt{5}, \sqrt{13})\), 4.4.274625.2, 8.8.75418890625.1, 8.8.4826809000000.1, 8.8.1142440000.1

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 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
5Data not computed
13Data not computed