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Properties

Label 16.0.21550427904...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{46}\cdot 5^{4}\cdot 7^{2}$
Root discriminant $13.99$
Ramified primes $2, 5, 7$
Class number $1$
Class group Trivial
Galois group $C_2^3.C_2^4.C_2$ (as 16T608)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -16, 56, -88, 32, 72, -44, -64, 36, 32, 8, -32, -6, 16, 0, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 16*x^13 - 6*x^12 - 32*x^11 + 8*x^10 + 32*x^9 + 36*x^8 - 64*x^7 - 44*x^6 + 72*x^5 + 32*x^4 - 88*x^3 + 56*x^2 - 16*x + 2)
 
gp: K = bnfinit(x^16 - 4*x^15 + 16*x^13 - 6*x^12 - 32*x^11 + 8*x^10 + 32*x^9 + 36*x^8 - 64*x^7 - 44*x^6 + 72*x^5 + 32*x^4 - 88*x^3 + 56*x^2 - 16*x + 2, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 16 x^{13} - 6 x^{12} - 32 x^{11} + 8 x^{10} + 32 x^{9} + 36 x^{8} - 64 x^{7} - 44 x^{6} + 72 x^{5} + 32 x^{4} - 88 x^{3} + 56 x^{2} - 16 x + 2 \)

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:  \(2155042790440960000=2^{46}\cdot 5^{4}\cdot 7^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.99$
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:  $2, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13} a^{14} + \frac{1}{13} a^{13} - \frac{2}{13} a^{12} - \frac{1}{13} a^{11} + \frac{3}{13} a^{10} + \frac{3}{13} a^{9} + \frac{2}{13} a^{8} - \frac{5}{13} a^{7} - \frac{3}{13} a^{6} - \frac{5}{13} a^{5} + \frac{4}{13} a^{4} - \frac{3}{13} a^{3} + \frac{2}{13} a^{2} - \frac{5}{13} a + \frac{4}{13}$, $\frac{1}{1385267} a^{15} - \frac{1334}{60229} a^{14} + \frac{1163}{60229} a^{13} + \frac{218171}{1385267} a^{12} + \frac{453641}{1385267} a^{11} + \frac{2062}{12259} a^{10} + \frac{672132}{1385267} a^{9} - \frac{498964}{1385267} a^{8} + \frac{123837}{1385267} a^{7} - \frac{243200}{1385267} a^{6} + \frac{9129}{33787} a^{5} + \frac{466529}{1385267} a^{4} + \frac{615132}{1385267} a^{3} - \frac{379715}{1385267} a^{2} + \frac{293182}{1385267} a - \frac{657812}{1385267}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{1708769}{1385267} a^{15} + \frac{270331}{60229} a^{14} + \frac{96498}{60229} a^{13} - \frac{26573737}{1385267} a^{12} + \frac{76074}{106559} a^{11} + \frac{489519}{12259} a^{10} + \frac{5052366}{1385267} a^{9} - \frac{54172921}{1385267} a^{8} - \frac{79301666}{1385267} a^{7} + \frac{84054981}{1385267} a^{6} + \frac{2587744}{33787} a^{5} - \frac{89332853}{1385267} a^{4} - \frac{89992976}{1385267} a^{3} + \frac{121381406}{1385267} a^{2} - \frac{50503200}{1385267} a + \frac{7412363}{1385267} \) (order $8$)
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:  \( 1908.29635418 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T608):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), 4.0.512.1 x2, 4.2.1024.1 x2, \(\Q(\zeta_{8})\), 8.0.4194304.1

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$