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Properties

Label 16.0.18049771555...1984.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{6}\cdot 7^{8}$
Root discriminant $15.98$
Ramified primes $2, 3, 7$
Class number $1$
Class group Trivial
Galois group $C_8:C_2^2$ (as 16T38)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -12, 44, -116, 252, -472, 766, -1044, 1171, -1076, 818, -516, 268, -112, 36, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 36*x^14 - 112*x^13 + 268*x^12 - 516*x^11 + 818*x^10 - 1076*x^9 + 1171*x^8 - 1044*x^7 + 766*x^6 - 472*x^5 + 252*x^4 - 116*x^3 + 44*x^2 - 12*x + 2)
 
gp: K = bnfinit(x^16 - 8*x^15 + 36*x^14 - 112*x^13 + 268*x^12 - 516*x^11 + 818*x^10 - 1076*x^9 + 1171*x^8 - 1044*x^7 + 766*x^6 - 472*x^5 + 252*x^4 - 116*x^3 + 44*x^2 - 12*x + 2, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 36 x^{14} - 112 x^{13} + 268 x^{12} - 516 x^{11} + 818 x^{10} - 1076 x^{9} + 1171 x^{8} - 1044 x^{7} + 766 x^{6} - 472 x^{5} + 252 x^{4} - 116 x^{3} + 44 x^{2} - 12 x + 2 \)

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:  \(18049771555189161984=2^{32}\cdot 3^{6}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.98$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{39} a^{14} + \frac{2}{13} a^{13} - \frac{5}{39} a^{12} - \frac{3}{13} a^{11} + \frac{1}{3} a^{10} - \frac{5}{13} a^{9} + \frac{10}{39} a^{8} + \frac{1}{13} a^{7} + \frac{2}{39} a^{6} - \frac{4}{39} a^{4} + \frac{6}{13} a^{3} - \frac{10}{39} a^{2} + \frac{1}{13} a - \frac{16}{39}$, $\frac{1}{2067} a^{15} + \frac{19}{2067} a^{14} + \frac{125}{2067} a^{13} + \frac{719}{2067} a^{12} - \frac{24}{53} a^{11} + \frac{255}{689} a^{10} - \frac{575}{2067} a^{9} + \frac{94}{2067} a^{8} - \frac{177}{689} a^{7} - \frac{3}{53} a^{6} + \frac{893}{2067} a^{5} + \frac{902}{2067} a^{4} - \frac{569}{2067} a^{3} - \frac{374}{2067} a^{2} + \frac{129}{689} a + \frac{23}{53}$

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:  \( -\frac{5672}{2067} a^{15} + \frac{14180}{689} a^{14} - \frac{180590}{2067} a^{13} + \frac{13555}{53} a^{12} - \frac{1189232}{2067} a^{11} + \frac{715033}{689} a^{10} - \frac{3160292}{2067} a^{9} + \frac{1271368}{689} a^{8} - \frac{3727648}{2067} a^{7} + \frac{963791}{689} a^{6} - \frac{1815772}{2067} a^{5} + \frac{327257}{689} a^{4} - \frac{473410}{2067} a^{3} + \frac{61852}{689} a^{2} - \frac{50578}{2067} a + \frac{2351}{689} \) (order $4$)
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:  \( 7453.60013153 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8:C_2^2$ (as 16T38):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_8:C_2^2$
Character table for $C_8:C_2^2$

Intermediate fields

\(\Q(\sqrt{7}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-7}) \), 4.2.9408.2, 4.2.9408.1, \(\Q(i, \sqrt{7})\), 8.2.4248502272.1, 8.2.4248502272.2, 8.0.88510464.1

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 8 siblings: 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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.16.65$x^{8} + 4 x^{6} + 28 x^{4} + 20$$8$$1$$16$$QD_{16}$$[2, 2, 5/2]^{2}$
2.8.16.65$x^{8} + 4 x^{6} + 28 x^{4} + 20$$8$$1$$16$$QD_{16}$$[2, 2, 5/2]^{2}$
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$