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Properties

Label 16.0.10995116277...000.10
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 5^{8}$
Root discriminant $17.89$
Ramified primes $2, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{8}$ (as 16T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, 0, 60, 0, 36, 0, -64, 0, 35, 0, 0, 0, 20, 0, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 + 20*x^12 + 35*x^8 - 64*x^6 + 36*x^4 + 60*x^2 + 25)
 
gp: K = bnfinit(x^16 - 4*x^14 + 20*x^12 + 35*x^8 - 64*x^6 + 36*x^4 + 60*x^2 + 25, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{14} + 20 x^{12} + 35 x^{8} - 64 x^{6} + 36 x^{4} + 60 x^{2} + 25 \)

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:  \(109951162777600000000=2^{48}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.89$
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:  $2, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5} a^{6} + \frac{2}{5} a^{4} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{7} + \frac{2}{5} a^{5} + \frac{1}{5} a^{3}$, $\frac{1}{5} a^{8} + \frac{2}{5} a^{4} - \frac{2}{5} a^{2}$, $\frac{1}{5} a^{9} + \frac{2}{5} a^{5} - \frac{2}{5} a^{3}$, $\frac{1}{5} a^{10} - \frac{1}{5} a^{4} - \frac{2}{5} a^{2}$, $\frac{1}{25} a^{11} - \frac{1}{25} a^{9} + \frac{1}{25} a^{7} - \frac{11}{25} a^{5} - \frac{9}{25} a^{3} + \frac{1}{5} a$, $\frac{1}{25} a^{12} - \frac{1}{25} a^{10} + \frac{1}{25} a^{8} - \frac{1}{25} a^{6} + \frac{11}{25} a^{4} - \frac{2}{5} a^{2}$, $\frac{1}{25} a^{13} + \frac{6}{25} a^{3} + \frac{1}{5} a$, $\frac{1}{129625} a^{14} - \frac{38}{5185} a^{12} + \frac{276}{5185} a^{10} - \frac{1971}{25925} a^{8} + \frac{1084}{25925} a^{6} + \frac{42061}{129625} a^{4} - \frac{1033}{25925} a^{2} - \frac{1584}{5185}$, $\frac{1}{129625} a^{15} - \frac{38}{5185} a^{13} + \frac{343}{25925} a^{11} - \frac{934}{25925} a^{9} + \frac{47}{25925} a^{7} - \frac{30529}{129625} a^{5} + \frac{332}{1037} a^{3} + \frac{2564}{5185} a$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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:  \( \frac{172}{7625} a^{14} - \frac{167}{1525} a^{12} + \frac{802}{1525} a^{10} - \frac{584}{1525} a^{8} + \frac{226}{305} a^{6} - \frac{15968}{7625} a^{4} + \frac{2274}{1525} a^{2} + \frac{222}{305} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 17780.6827043 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_8$ (as 16T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $D_{8}$
Character table for $D_{8}$

Intermediate fields

\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{5}) \), \(\Q(i, \sqrt{5})\), 4.0.1280.1 x2, 4.2.1600.1 x2, 8.0.40960000.2, 8.0.2097152000.6 x4, 8.2.2621440000.1 x4

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$