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Properties

Label 16.0.12577916805...0641.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 61^{8}$
Root discriminant $13.53$
Ramified primes $3, 61$
Class number $1$
Class group Trivial
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, 5, 61, 58, 101, 99, 142, 96, 120, 43, 45, -21, -6, -14, 2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 2*x^14 - 14*x^13 - 6*x^12 - 21*x^11 + 45*x^10 + 43*x^9 + 120*x^8 + 96*x^7 + 142*x^6 + 99*x^5 + 101*x^4 + 58*x^3 + 61*x^2 + 5*x + 25)
 
gp: K = bnfinit(x^16 + 2*x^14 - 14*x^13 - 6*x^12 - 21*x^11 + 45*x^10 + 43*x^9 + 120*x^8 + 96*x^7 + 142*x^6 + 99*x^5 + 101*x^4 + 58*x^3 + 61*x^2 + 5*x + 25, 1)
 

Normalized defining polynomial

\( x^{16} + 2 x^{14} - 14 x^{13} - 6 x^{12} - 21 x^{11} + 45 x^{10} + 43 x^{9} + 120 x^{8} + 96 x^{7} + 142 x^{6} + 99 x^{5} + 101 x^{4} + 58 x^{3} + 61 x^{2} + 5 x + 25 \)

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:  \(1257791680575160641=3^{8}\cdot 61^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.53$
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:  $3, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{15} a^{12} + \frac{2}{15} a^{11} + \frac{1}{3} a^{9} - \frac{4}{15} a^{8} - \frac{4}{15} a^{7} + \frac{1}{15} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{4}{15} a + \frac{1}{3}$, $\frac{1}{15} a^{13} + \frac{1}{15} a^{11} + \frac{2}{5} a^{9} + \frac{4}{15} a^{8} + \frac{4}{15} a^{7} + \frac{1}{5} a^{6} + \frac{7}{15} a^{4} + \frac{7}{15} a^{3} - \frac{1}{5} a^{2} + \frac{2}{15} a - \frac{1}{3}$, $\frac{1}{75} a^{14} - \frac{2}{75} a^{13} + \frac{2}{75} a^{12} + \frac{1}{75} a^{10} - \frac{8}{75} a^{9} + \frac{7}{75} a^{8} + \frac{7}{25} a^{7} + \frac{2}{5} a^{6} + \frac{32}{75} a^{5} + \frac{28}{75} a^{4} - \frac{2}{15} a^{3} - \frac{7}{25} a^{2} - \frac{7}{15} a + \frac{1}{3}$, $\frac{1}{9872393925} a^{15} - \frac{19826417}{9872393925} a^{14} + \frac{203152627}{9872393925} a^{13} + \frac{40241498}{1974478785} a^{12} - \frac{468608088}{3290797975} a^{11} + \frac{138242302}{9872393925} a^{10} - \frac{3917269453}{9872393925} a^{9} - \frac{2680251434}{9872393925} a^{8} + \frac{153332831}{658159595} a^{7} + \frac{2051900287}{9872393925} a^{6} - \frac{3533256202}{9872393925} a^{5} - \frac{49302958}{1974478785} a^{4} - \frac{3003669316}{9872393925} a^{3} + \frac{294097111}{658159595} a^{2} - \frac{97023181}{1974478785} a + \frac{20342232}{131631919}$

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:  \( -\frac{2760391}{1974478785} a^{15} - \frac{71545502}{1974478785} a^{14} - \frac{2499763}{394895757} a^{13} - \frac{71438177}{1974478785} a^{12} + \frac{68160385}{131631919} a^{11} + \frac{627927379}{1974478785} a^{10} + \frac{976367809}{1974478785} a^{9} - \frac{1231248489}{658159595} a^{8} - \frac{4229184002}{1974478785} a^{7} - \frac{7617247754}{1974478785} a^{6} - \frac{1740737122}{658159595} a^{5} - \frac{7292411429}{1974478785} a^{4} - \frac{5255462869}{1974478785} a^{3} - \frac{5761925512}{1974478785} a^{2} - \frac{647306611}{658159595} a - \frac{111879376}{131631919} \) (order $6$)
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:  \( 809.594531745 \)
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

\(\Q(\sqrt{-183}) \), \(\Q(\sqrt{61}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{61})\), 4.2.11163.1 x2, 4.0.549.1 x2, 8.0.1121513121.2, 8.2.373837707.1 x4, 8.0.18385461.1 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$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}$
$61$61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$