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Properties

Label 16.0.63880676485...7601.1
Degree $16$
Signature $[0, 8]$
Discriminant $13^{8}\cdot 23^{8}$
Root discriminant $17.29$
Ramified primes $13, 23$
Class number $3$
Class group $[3]$
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![16, 0, -172, 72, 573, -702, 296, -100, 266, -280, 74, 38, 0, -36, 26, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 26*x^14 - 36*x^13 + 38*x^11 + 74*x^10 - 280*x^9 + 266*x^8 - 100*x^7 + 296*x^6 - 702*x^5 + 573*x^4 + 72*x^3 - 172*x^2 + 16)
 
gp: K = bnfinit(x^16 - 8*x^15 + 26*x^14 - 36*x^13 + 38*x^11 + 74*x^10 - 280*x^9 + 266*x^8 - 100*x^7 + 296*x^6 - 702*x^5 + 573*x^4 + 72*x^3 - 172*x^2 + 16, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 26 x^{14} - 36 x^{13} + 38 x^{11} + 74 x^{10} - 280 x^{9} + 266 x^{8} - 100 x^{7} + 296 x^{6} - 702 x^{5} + 573 x^{4} + 72 x^{3} - 172 x^{2} + 16 \)

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:  \(63880676485490517601=13^{8}\cdot 23^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.29$
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:  $13, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{8} a^{5} - \frac{1}{2} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{16} a^{12} + \frac{1}{16} a^{9} - \frac{1}{8} a^{8} + \frac{3}{16} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{5}{16} a^{3} + \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{13} + \frac{1}{16} a^{10} - \frac{1}{8} a^{9} + \frac{3}{16} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{3}{16} a^{4} + \frac{3}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{48} a^{14} - \frac{1}{48} a^{13} - \frac{1}{48} a^{12} + \frac{1}{48} a^{11} - \frac{1}{16} a^{10} - \frac{1}{16} a^{9} + \frac{5}{48} a^{8} + \frac{1}{48} a^{7} + \frac{5}{48} a^{6} + \frac{11}{48} a^{5} - \frac{3}{16} a^{4} - \frac{1}{48} a^{3} + \frac{7}{24} a^{2} - \frac{5}{12} a - \frac{1}{3}$, $\frac{1}{89355988704} a^{15} + \frac{3458645}{347688672} a^{14} - \frac{134354537}{29785329568} a^{13} + \frac{2144124701}{89355988704} a^{12} - \frac{1184010217}{89355988704} a^{11} - \frac{3362136367}{29785329568} a^{10} - \frac{4588754755}{89355988704} a^{9} - \frac{3943486393}{89355988704} a^{8} - \frac{1184478617}{89355988704} a^{7} - \frac{6716588011}{29785329568} a^{6} - \frac{9735985139}{89355988704} a^{5} - \frac{2202002983}{89355988704} a^{4} - \frac{1262539087}{14892664784} a^{3} + \frac{1282059191}{5584749294} a^{2} - \frac{1387688197}{11169498588} a - \frac{1198932467}{5584749294}$

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

Class group and class number

$C_{3}$, which has order $3$

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:  \( 2823.55404905 \)
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

\(\Q(\sqrt{-299}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{13}, \sqrt{-23})\), 4.2.3887.1 x2, 4.0.6877.1 x2, 8.0.7992538801.1, 8.2.347501687.1 x4, 8.0.614810677.1 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$23$23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$