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Properties

Label 16.0.72823647931...6576.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{30}\cdot 7^{14}$
Root discriminant $20.13$
Ramified primes $2, 7$
Class number $2$
Class group $[2]$
Galois group $C_2\times D_8$ (as 16T29)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 0, 26, 0, 49, 0, 0, 0, 0, 0, 14, 0, 0, 0, -2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^14 + 14*x^10 + 49*x^4 + 26*x^2 + 4)
 
gp: K = bnfinit(x^16 - 2*x^14 + 14*x^10 + 49*x^4 + 26*x^2 + 4, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{14} + 14 x^{10} + 49 x^{4} + 26 x^{2} + 4 \)

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:  \(728236479319770136576=2^{30}\cdot 7^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{10} a^{10} + \frac{1}{5} a^{8} + \frac{1}{10} a^{4} + \frac{1}{5}$, $\frac{1}{10} a^{11} + \frac{1}{5} a^{9} + \frac{1}{10} a^{5} + \frac{1}{5} a$, $\frac{1}{20} a^{12} - \frac{1}{20} a^{11} - \frac{1}{20} a^{10} - \frac{1}{10} a^{9} + \frac{1}{5} a^{8} - \frac{9}{20} a^{6} + \frac{9}{20} a^{5} + \frac{7}{20} a^{4} + \frac{1}{10} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{20} a^{13} - \frac{1}{20} a^{10} - \frac{1}{5} a^{9} - \frac{1}{10} a^{8} - \frac{9}{20} a^{7} - \frac{1}{10} a^{5} + \frac{9}{20} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{260} a^{14} + \frac{1}{130} a^{12} - \frac{1}{20} a^{11} + \frac{2}{65} a^{10} - \frac{1}{10} a^{9} + \frac{7}{260} a^{8} + \frac{4}{13} a^{6} + \frac{9}{20} a^{5} + \frac{3}{13} a^{4} + \frac{6}{13} a^{2} + \frac{2}{5} a + \frac{29}{65}$, $\frac{1}{520} a^{15} - \frac{1}{520} a^{14} - \frac{11}{520} a^{13} + \frac{11}{520} a^{12} - \frac{1}{104} a^{11} + \frac{1}{104} a^{10} - \frac{97}{520} a^{9} + \frac{97}{520} a^{8} - \frac{63}{520} a^{7} + \frac{63}{520} a^{6} - \frac{57}{520} a^{5} + \frac{57}{520} a^{4} + \frac{47}{260} a^{3} - \frac{47}{260} a^{2} + \frac{3}{130} a - \frac{3}{130}$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 11142.4336746 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_8$ (as 16T29):

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 $C_2\times D_8$
Character table for $C_2\times D_8$

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{2}) \), 4.0.2744.1, 4.0.10976.1, \(\Q(\sqrt{2}, \sqrt{-7})\), 8.0.6746464256.1, 8.0.421654016.1, 8.0.1927561216.8

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.8.3$x^{4} + 6 x^{2} + 4 x + 14$$4$$1$$8$$C_2^2$$[2, 3]$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
$7$7.8.7.1$x^{8} + 14$$8$$1$$7$$D_{8}$$[\ ]_{8}^{2}$
7.8.7.1$x^{8} + 14$$8$$1$$7$$D_{8}$$[\ ]_{8}^{2}$