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Properties

Label 16.0.21767823360...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{12}\cdot 5^{12}$
Root discriminant $21.56$
Ramified primes $2, 3, 5$
Class number $4$
Class group $[2, 2]$
Galois group $C_4 \times D_4$ (as 16T19)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 4, -10, 23, 10, -4, -28, 13, -28, -4, 10, 23, -10, 4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 4*x^14 - 10*x^13 + 23*x^12 + 10*x^11 - 4*x^10 - 28*x^9 + 13*x^8 - 28*x^7 - 4*x^6 + 10*x^5 + 23*x^4 - 10*x^3 + 4*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^16 - 2*x^15 + 4*x^14 - 10*x^13 + 23*x^12 + 10*x^11 - 4*x^10 - 28*x^9 + 13*x^8 - 28*x^7 - 4*x^6 + 10*x^5 + 23*x^4 - 10*x^3 + 4*x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 4 x^{14} - 10 x^{13} + 23 x^{12} + 10 x^{11} - 4 x^{10} - 28 x^{9} + 13 x^{8} - 28 x^{7} - 4 x^{6} + 10 x^{5} + 23 x^{4} - 10 x^{3} + 4 x^{2} - 2 x + 1 \)

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:  \(2176782336000000000000=2^{24}\cdot 3^{12}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.56$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{19} a^{10} - \frac{7}{19} a^{5} + \frac{1}{19}$, $\frac{1}{19} a^{11} - \frac{7}{19} a^{6} + \frac{1}{19} a$, $\frac{1}{19} a^{12} - \frac{7}{19} a^{7} + \frac{1}{19} a^{2}$, $\frac{1}{209} a^{13} + \frac{5}{209} a^{12} + \frac{2}{209} a^{11} + \frac{5}{209} a^{10} - \frac{4}{11} a^{9} + \frac{50}{209} a^{8} - \frac{92}{209} a^{7} + \frac{62}{209} a^{6} - \frac{16}{209} a^{5} - \frac{4}{11} a^{4} - \frac{94}{209} a^{3} + \frac{24}{209} a^{2} - \frac{17}{209} a + \frac{100}{209}$, $\frac{1}{209} a^{14} - \frac{1}{209} a^{12} - \frac{5}{209} a^{11} - \frac{2}{209} a^{10} + \frac{12}{209} a^{9} + \frac{4}{11} a^{8} - \frac{50}{209} a^{7} + \frac{92}{209} a^{6} - \frac{62}{209} a^{5} + \frac{7}{19} a^{4} + \frac{4}{11} a^{3} + \frac{94}{209} a^{2} - \frac{24}{209} a + \frac{17}{209}$, $\frac{1}{209} a^{15} - \frac{5}{209} a^{10} + \frac{6}{209} a^{5} + \frac{78}{209}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

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:  \( \frac{80}{209} a^{15} - \frac{120}{209} a^{14} + \frac{244}{209} a^{13} - \frac{640}{209} a^{12} + \frac{1440}{209} a^{11} + \frac{1720}{209} a^{10} + \frac{80}{209} a^{9} - \frac{195}{19} a^{8} - \frac{80}{209} a^{7} - \frac{1720}{209} a^{6} - \frac{1440}{209} a^{5} + \frac{640}{209} a^{4} + \frac{2220}{209} a^{3} + \frac{120}{209} a^{2} - \frac{80}{209} a \) (order $10$)
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:  \( 18370.4449216 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times D_4$ (as 16T19):

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 $C_4 \times D_4$
Character table for $C_4 \times D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{15}) \), 4.0.18000.1, \(\Q(\zeta_{5})\), \(\Q(\sqrt{3}, \sqrt{5})\), 4.2.43200.3, 4.2.1728.1, 8.4.1866240000.3, 8.4.46656000000.1, 8.0.324000000.3

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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