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Properties

Label 16.0.18205911982...4144.5
Degree $16$
Signature $[0, 8]$
Discriminant $2^{28}\cdot 7^{14}$
Root discriminant $18.46$
Ramified primes $2, 7$
Class number $2$
Class group $[2]$
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![16, 0, 144, 0, 392, 0, 196, 0, 105, 0, 56, 0, 14, 0, 4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 4*x^14 + 14*x^12 + 56*x^10 + 105*x^8 + 196*x^6 + 392*x^4 + 144*x^2 + 16)
 
gp: K = bnfinit(x^16 + 4*x^14 + 14*x^12 + 56*x^10 + 105*x^8 + 196*x^6 + 392*x^4 + 144*x^2 + 16, 1)
 

Normalized defining polynomial

\( x^{16} + 4 x^{14} + 14 x^{12} + 56 x^{10} + 105 x^{8} + 196 x^{6} + 392 x^{4} + 144 x^{2} + 16 \)

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:  \(182059119829942534144=2^{28}\cdot 7^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.46$
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:  $2, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{5} + \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{16} a^{10} + \frac{1}{16} a^{8} - \frac{1}{8} a^{7} - \frac{1}{16} a^{6} - \frac{1}{4} a^{5} + \frac{3}{16} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} - \frac{1}{16} a^{7} - \frac{1}{4} a^{6} - \frac{3}{16} a^{5} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{16} a^{12} - \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{3}{16} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{32} a^{13} - \frac{1}{32} a^{12} - \frac{1}{32} a^{11} - \frac{1}{32} a^{10} - \frac{1}{32} a^{9} + \frac{3}{32} a^{8} - \frac{3}{32} a^{7} - \frac{3}{32} a^{6} - \frac{1}{4} a^{5} - \frac{1}{16} a^{4} + \frac{3}{8} a^{3} - \frac{1}{8} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{2368} a^{14} + \frac{27}{1184} a^{12} + \frac{25}{1184} a^{10} + \frac{47}{592} a^{8} - \frac{1}{8} a^{7} - \frac{263}{2368} a^{6} - \frac{1}{4} a^{5} + \frac{35}{1184} a^{4} - \frac{3}{8} a^{3} - \frac{63}{592} a^{2} - \frac{1}{4} a + \frac{71}{296}$, $\frac{1}{2368} a^{15} - \frac{5}{592} a^{13} - \frac{1}{32} a^{12} - \frac{3}{296} a^{11} - \frac{1}{32} a^{10} + \frac{57}{1184} a^{9} + \frac{3}{32} a^{8} + \frac{107}{2368} a^{7} - \frac{3}{32} a^{6} + \frac{109}{1184} a^{5} - \frac{1}{16} a^{4} - \frac{285}{592} a^{3} - \frac{1}{8} a^{2} - \frac{3}{296} a - \frac{1}{4}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( -\frac{27}{2368} a^{15} - \frac{63}{1184} a^{13} - \frac{231}{1184} a^{11} - \frac{455}{592} a^{9} - \frac{3851}{2368} a^{7} - \frac{3535}{1184} a^{5} - \frac{3479}{592} a^{3} - \frac{955}{296} a \) (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:  \( 57122.4567908 \)
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

\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-1}) \), 4.0.2744.1, 4.0.10976.1, \(\Q(i, \sqrt{7})\), 8.0.1686616064.1, 8.0.1686616064.2, 8.0.481890304.4

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
7Data not computed