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Properties

Label 16.8.21767823360...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 3^{12}\cdot 5^{12}$
Root discriminant $21.56$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
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, -6, -138, 46, 278, -324, -162, 306, -36, -80, 90, -18, -33, 20, 0, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 20*x^13 - 33*x^12 - 18*x^11 + 90*x^10 - 80*x^9 - 36*x^8 + 306*x^7 - 162*x^6 - 324*x^5 + 278*x^4 + 46*x^3 - 138*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 + 20*x^13 - 33*x^12 - 18*x^11 + 90*x^10 - 80*x^9 - 36*x^8 + 306*x^7 - 162*x^6 - 324*x^5 + 278*x^4 + 46*x^3 - 138*x^2 - 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 20 x^{13} - 33 x^{12} - 18 x^{11} + 90 x^{10} - 80 x^{9} - 36 x^{8} + 306 x^{7} - 162 x^{6} - 324 x^{5} + 278 x^{4} + 46 x^{3} - 138 x^{2} - 6 x + 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{12749} a^{14} - \frac{3219}{12749} a^{13} + \frac{5370}{12749} a^{12} + \frac{5889}{12749} a^{11} - \frac{34}{209} a^{10} - \frac{1924}{12749} a^{9} - \frac{2395}{12749} a^{8} + \frac{3368}{12749} a^{7} - \frac{62}{1159} a^{6} + \frac{4910}{12749} a^{5} - \frac{5839}{12749} a^{4} - \frac{4773}{12749} a^{3} + \frac{699}{12749} a^{2} - \frac{5216}{12749} a - \frac{1291}{12749}$, $\frac{1}{116754060330281} a^{15} + \frac{2334211079}{116754060330281} a^{14} + \frac{22862515458810}{116754060330281} a^{13} - \frac{2236110258942}{10614005484571} a^{12} - \frac{21747538318124}{116754060330281} a^{11} + \frac{2906375022520}{116754060330281} a^{10} + \frac{57895123411698}{116754060330281} a^{9} - \frac{48362649921832}{116754060330281} a^{8} + \frac{36183862443725}{116754060330281} a^{7} + \frac{26619849407210}{116754060330281} a^{6} + \frac{4387972842720}{116754060330281} a^{5} - \frac{27199646984777}{116754060330281} a^{4} + \frac{352030689700}{1914000989021} a^{3} - \frac{29386428917772}{116754060330281} a^{2} + \frac{173097922260}{558631867609} a + \frac{7006901986807}{116754060330281}$

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:  $11$
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:  \( 42826.6103026 \) (assuming GRH)
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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{15}) \), \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{20})^+\), 4.2.43200.3, 4.2.1728.1, \(\Q(\sqrt{3}, \sqrt{5})\), 8.4.1866240000.3, \(\Q(\zeta_{60})^+\), 8.4.46656000000.1

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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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