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Properties

Label 16.0.33152641631...9344.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{8}\cdot 7^{6}$
Root discriminant $14.37$
Ramified primes $2, 3, 7$
Class number $1$
Class group Trivial
Galois group $C_8:C_2^2$ (as 16T45)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13, -40, 14, 64, 52, -552, 1208, -1604, 1591, -1312, 944, -588, 304, -124, 38, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 38*x^14 - 124*x^13 + 304*x^12 - 588*x^11 + 944*x^10 - 1312*x^9 + 1591*x^8 - 1604*x^7 + 1208*x^6 - 552*x^5 + 52*x^4 + 64*x^3 + 14*x^2 - 40*x + 13)
 
gp: K = bnfinit(x^16 - 8*x^15 + 38*x^14 - 124*x^13 + 304*x^12 - 588*x^11 + 944*x^10 - 1312*x^9 + 1591*x^8 - 1604*x^7 + 1208*x^6 - 552*x^5 + 52*x^4 + 64*x^3 + 14*x^2 - 40*x + 13, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 38 x^{14} - 124 x^{13} + 304 x^{12} - 588 x^{11} + 944 x^{10} - 1312 x^{9} + 1591 x^{8} - 1604 x^{7} + 1208 x^{6} - 552 x^{5} + 52 x^{4} + 64 x^{3} + 14 x^{2} - 40 x + 13 \)

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:  \(3315264163198009344=2^{32}\cdot 3^{8}\cdot 7^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{11} - \frac{1}{9} a^{7} - \frac{1}{9} a^{6} + \frac{4}{9} a^{5} - \frac{1}{3} a^{3} + \frac{2}{9} a - \frac{2}{9}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{11} - \frac{1}{9} a^{8} - \frac{4}{9} a^{6} - \frac{4}{9} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{2}{9} a^{2} - \frac{4}{9} a + \frac{2}{9}$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{11} - \frac{1}{9} a^{9} + \frac{4}{9} a^{7} + \frac{4}{9} a^{6} + \frac{1}{9} a^{5} + \frac{1}{3} a^{4} - \frac{1}{9} a^{3} - \frac{4}{9} a^{2} + \frac{4}{9} a - \frac{2}{9}$, $\frac{1}{621387} a^{15} + \frac{1216}{207129} a^{14} + \frac{11827}{621387} a^{13} + \frac{18470}{621387} a^{12} + \frac{7957}{69043} a^{11} - \frac{63559}{621387} a^{10} + \frac{9326}{207129} a^{9} + \frac{11191}{207129} a^{8} + \frac{2241}{5311} a^{7} - \frac{140470}{621387} a^{6} - \frac{9369}{69043} a^{5} - \frac{139219}{621387} a^{4} + \frac{138470}{621387} a^{3} - \frac{38326}{207129} a^{2} - \frac{301942}{621387} a + \frac{7813}{15933}$

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:  \( \frac{37858}{69043} a^{15} - \frac{2374735}{621387} a^{14} + \frac{10449892}{621387} a^{13} - \frac{31203460}{621387} a^{12} + \frac{7815332}{69043} a^{11} - \frac{41699059}{207129} a^{10} + \frac{187335808}{621387} a^{9} - \frac{245570485}{621387} a^{8} + \frac{7113958}{15933} a^{7} - \frac{246473863}{621387} a^{6} + \frac{47920946}{207129} a^{5} - \frac{3400518}{69043} a^{4} - \frac{17190944}{621387} a^{3} + \frac{1524422}{207129} a^{2} + \frac{8629874}{621387} a - \frac{93697}{15933} \) (order $12$)
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:  \( 3545.42621346 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8:C_2^2$ (as 16T45):

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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), 4.0.1008.2, 4.0.1008.1, \(\Q(\zeta_{12})\), 8.0.16257024.2

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 ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.4.2.1$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$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$7.4.2.2$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}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$