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Properties

Label 16.4.34828517376...0000.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{24}\cdot 3^{12}\cdot 5^{8}$
Root discriminant $14.42$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2:D_4$ (as 16T34)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, -12, 32, 60, -18, -114, -38, 90, 18, -54, 6, 23, -2, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^14 - 2*x^13 + 23*x^12 + 6*x^11 - 54*x^10 + 18*x^9 + 90*x^8 - 38*x^7 - 114*x^6 - 18*x^5 + 60*x^4 + 32*x^3 - 12*x^2 + 2*x + 1)
 
gp: K = bnfinit(x^16 - 6*x^14 - 2*x^13 + 23*x^12 + 6*x^11 - 54*x^10 + 18*x^9 + 90*x^8 - 38*x^7 - 114*x^6 - 18*x^5 + 60*x^4 + 32*x^3 - 12*x^2 + 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{14} - 2 x^{13} + 23 x^{12} + 6 x^{11} - 54 x^{10} + 18 x^{9} + 90 x^{8} - 38 x^{7} - 114 x^{6} - 18 x^{5} + 60 x^{4} + 32 x^{3} - 12 x^{2} + 2 x + 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3482851737600000000=2^{24}\cdot 3^{12}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.42$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{19} a^{14} - \frac{7}{19} a^{12} - \frac{2}{19} a^{11} - \frac{8}{19} a^{10} + \frac{8}{19} a^{9} - \frac{8}{19} a^{8} - \frac{9}{19} a^{7} + \frac{3}{19} a^{6} + \frac{9}{19} a^{5} - \frac{3}{19} a^{4} - \frac{8}{19} a^{3} + \frac{6}{19} a^{2} + \frac{2}{19} a + \frac{1}{19}$, $\frac{1}{5437162419641} a^{15} - \frac{120928096956}{5437162419641} a^{14} + \frac{795413926447}{5437162419641} a^{13} + \frac{814247674880}{5437162419641} a^{12} - \frac{1539005288643}{5437162419641} a^{11} - \frac{1120258441522}{5437162419641} a^{10} + \frac{831501595785}{5437162419641} a^{9} - \frac{6727618445}{28466818951} a^{8} - \frac{2246672012273}{5437162419641} a^{7} + \frac{1360162996124}{5437162419641} a^{6} - \frac{741950770074}{5437162419641} a^{5} + \frac{1965262397537}{5437162419641} a^{4} - \frac{2315452979862}{5437162419641} a^{3} - \frac{547379686254}{5437162419641} a^{2} + \frac{2379855332870}{5437162419641} a + \frac{2216556910268}{5437162419641}$

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:  $9$
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:  \( 822.319546088 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:D_4$ (as 16T34):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_2^2:D_4$
Character table for $C_2^2:D_4$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{15}) \), 4.2.400.1, 4.2.3600.1, 4.2.1728.1, \(\Q(\sqrt{3}, \sqrt{5})\), 4.2.43200.3, 8.4.1866240000.3, 8.4.207360000.1, 8.4.116640000.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
3Data not computed
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$